Connect 9 points without tearing. Models of logical solutions

Non-standard for its reasoning the task about how to connect 9 points with 4 lines, makes itching stereotypes and turn on creativity.

How to position the points and drawing?

On a sheet of paper, better if it is in a cell, you need to draw 9 points. They must be located three in a row. It will look like a square, in the center of which is worth the point, and in the middle of each of the sides it also has it. It is better if this drawing is located away from the edges of the sheet. Such a location of the square will be required to correctly solve the task of how to connect 9 points with 4 lines.

The task

Requirements that need to be taken into account:

Observing these rules, you need to connect 9 points with 4 lines. Very often, after a couple of minutes, a person starts thinking over this pattern, a person begins to argue that this task has no answer.

The solution of the problem

The main thing is to forget everything that was taught at school. There they give stereotypical views, which here only hurt.

The main reason is that the task of how to connect 9 points 4 lines, do not solve In the following case: they end in drawn points.

This is fundamentally wrong. Points are the ends of segments, and the task explicitly refers to the lines. This is necessary to take advantage.

You can start with any vertex square. The main thing is exactly the angle, which specifically, not fundamentally. Let the points be denoted to the left, moving to the right, and from above, moving down. That is, in the first row there are 1, 2 and 3, the second consists of 4, 5 and 6, and the third is formed 7, 8 and 9.

Let the beginning be in the first point. Then to connect 9 points with 4 lines, you will need to do the following.

1. Mark a ray diagonally to points 5 and 9.
2. On the latter you need to stop - this is the end of the first line.
3. There are two ways, they are both equal and lead to the same result. The first will go to the number 8, that is, to the left. The second is to the six or up. Let the last option be.
4. The second line begins at point 9 and goes through 6 and 3. But on the last digit it does not end. It needs to be continued on such a segment as if another point was drawn there. Here will be the end of the second line.
5. Now again, the diagonal, which will pass through the numbers 2 and 4. It is not difficult to guess that the second number is not the end of the third line. It needs to be continued as it was with the second. Thus ended the third line.
6. It remains to spend the fourth through points 7 and 8, which should end in figure 9.

This task completed and all the conditions are observed. Someone this figure resembles an umbrella, and someone claims that she is an arrow.

If you record a shorter plan of how to connect 9 points 4 lines, then the following will be: Start in 1, continue in 5, turn to 9, spend in 6 and 3, extend to (0), turn on 2 and 4, continue to ( 0), roll to 7, 8 and 9. Here (0) the ends of the segments that do not have numbers are indicated.

As a conclusion

Now you can still break your head over a more complex task. It has 16 points located similarly to the task. And they already need to connect 6 lines.

If this task turned out to be on the teeth, you can try to solve others, with the same requirements, but differ in dots and direct, from the following list:

• 25 points in order of square, as well as all subsequent, and 8 straight lines;
• 36 points per 10 lines that are not interrupted because the handle cannot be broken from the sheet;
• 49 points connected by 12 lines.

If you hit this page, you probably have already tried to solve the "test 9 points", namely to connect nine points to four straight lines without taking the handles from the paper sheet. If you didn't manage to solve this puzzle, do not despair. On this page you can find several solutions to this famous difficult task of nine points that strained the minds of many thousands, if not millions of people.

The task

Condition:

Condition: It is necessary to connect nine points to four straight lines without taking the handles from the sheet of paper.

This task is not as easy as it may seem. To solve it, you need to think no standard and apply your creative thinking, otherwise nothing will work. If you try to act in the forehead to start connecting all the points with standard lines, you can spend a lot of time and never solve the task of nine points. Our standard thinking to which we are taught in school, sends us to look for a solution, based on only six typical lines: 4 sides of the square and 2 of it are diagonal. Most people seem to be the decision of the puzzle of 9 points should be based on this. But it is not there. It does not even find if you connect another 2 lines between the centers of the Square Parties:

In general, only 20 straight lines can be held between all ninepoints: 4 sides of the square; 2 diagonals; 6 lines connecting the centers of the parties of a large square; 8 lines connecting the centers of the parties of a large square with its corners. How to draw all segments connecting our 9 points, shown in the figure below:

But even using this scheme, it is impossible to find 4 lines that could be connected to all nine points without taking hands.

The right decision of the "dough 9 points"

The solution to this puzzle lies somewhat wider than our standard perception of the task. In order to independently find the right approach remember that:

1. Through any 2 points you can spend only one straight line.
2. The straight line is not a segment and, therefore, we are not necessarily limited when drawing the lines by our nine blue circles.

Thus, let's try to continue the lines beyond the limiting of us until recently the square. It shows that the area of \u200b\u200bour search has increased significantly. Bottling a little can come to one of the right solutions.

Sequence of ninth points compounds with four lines:

1. To begin with, swipe the line connecting point number 1 and point number 7, through point number 4. Do not stop the movement and draw further about as much as from point number 4 to point number 7.
2. Next, move diagonally to up, connecting points number 8 and No. 6. Do not stop at point number 6 and continue to the line to a mental straight line passing through the top side of our square.
3. Draw a line to the right left sequentially through points No. 3, No. 2 and No. 1. Stay at point number 1.
4. Now spend the final segment through points number 1, №5 and №9. All 9 points, and the truth are connected by four lines, as was required in the condition of the problem.

Other options. This method is not the only one, you can start from any angle and move one of two directions. On the site 4Brain such options for solving the problem of "9 points 4 lines" is represented by a minimum of 12:

Just think about the task that many can not solve, has 12 ways to solve. Also, see the simplified version of this task: how to connect 4 points with three lines so that the lines are closed into a whole figure.

Creative approach in this puzzle

Most people who solved this task have not been able to get out of standard thinking, which in this test is expressed by a square formed by ninepoints. It is comfortable to look at any vital task straight, most simple. On the other hand, a person can spend a lot of time and effort in order to, using a standard approach, to find a right decision when this decision is better to look for, originally approaching the process creatively.

In our life, we often encounter such tasks about "nine points and four lines", and in order to decide to develop their creative thinking, including with the help of our training. After all, the task of 9 points has other solutions (read on).

Other solutions

By changing our frame or applying the lateral gap, you can find other solutions to this task. For example, the hyperbolization method when creating a lateral gap may result in thought that no one clarifies that the task should apply the standard geometry conditions (on the infinite smallness of points and endless subtleties of lines). Let our line be so broad that can immediately cross several points on their width. Then we are not the fact that 4 lines can connect all 9 points, but even one.

In addition, even in our image of 4 points, which is given in our condition a puzzle about 9 points, the points-mug themselves are large enough so that they can be connected to 3 lines like this:

And may not be limited to a two-dimensional space or use the concept of curvature space. We can also focus on the phrase "without opening the handles from the paper sheet", and just putting the handle on the side to move it and thus draw just 3 parallel lines.

This is a rather complicated puzzle, because it is not so easy to guess, as with the help of only 4-lines to connect as many as 9 points, without taking your hands from paper. There was a time that I after a variety of fruitless attempts decided that this is simply impossible, but in fact the solution is very simple.

We start drawing from the lowest right point.

We solved this logical task for a long time at school, as far as I know, there is one single solution there, without a variety of options. You need to take one point and not pulling the handles to draw something on the like shooter, as shown in the picture. Arrower This can look in any direction as you like)



Carolina already led visual examplebut I will use to explain your drawing



You need to start from the point in which the basis of the red line will be. We drive a red diagonal. Without taking hands, drive the line the corresponding blue in the picture. Blue goes into a lilac. And complete the line is similar green. Everything is clearly and simple. Four lines, the hand does not leave, all points are connected with each other. At the same time, the order of lines is very important, otherwise it will not work. The only order of blue and green lines can be changed in places. Red necessarily first.

Connect the data of the points of one line, without taking the hands, does not turn out to be difficult, as it seems at first glance. To do this, draw a triangle whose angles go beyond points, you can start with direct cornerAnd when they returned to a straight corner, we divide it in half connecting the remaining points.

Connect 9 points evenly separated from each other and having inside the right squares can be using a triangle in which 3 sides and straight line from its vertex leaves.

Do not break off these lines can be drawn as follows: I carry out a direct line on the outside, connecting 4 points, then diagonally by the opposite point, another 3 points, then we return to the starting point - the top, capturing more points and descend at right angles Down and crossing the hypotenuse finish the figure at 9 points.

The condition was completed, no point was crossed twice, the hand did not break off.

You can solve the task in the second way, on the contrary.

An option, as far as I know, there is only one (or rather, I am known to me only one, it is possible that our thinkers have come up with some way :-)). It is best visible in the picture, where all 9 points are connected by four straight lines.



Connect 4 straight lines 9-point very simple. To do this, display the line beyond these points to create the desired angle. To clearly understand how nine points are connected by four lines, look this video.

here immediately, here is a question with a trick. Rather, the atmosphere itself, like most similar tasks, is not based on logic, but rather at the creative principle.M. we are sure: desire and work, self-education and the experience gained will help you achieve a new, non-trivial level of thinking.

Riddling without unnecessary words: how to connect nine points with four lines?

This mystery intrigued hundreds of thousands of people. The following conditions must be observed: cross all nine points forming a square using direct lines (no more than four).

Hand, but rather, a pencil, in this case it is impossible to tear off the sheet. The subsequent line should begin where the previous one has ended.At first glance, it is not so difficult, but in fact, each subsequent attempt often removes an inquisitive mind from a positive result.

The thing is that from the very childhood we were taught to think, pushing out of certain templates and rules.First of all evolved logical thinking, on the principles of which our world is built. So, not so.

Here you need to go beyond logic and stop thinking within the borders of the four sides of the square and its diagonals.

We analyze the task based on knowledge of the object, and you should simply remember that the straight line is completely not necessarily limited to the form frames, i.e. it is possible to go beyond the boundaries.

Conditionally correct each point from 1 to 9:

1. We carry out the first line, starting from point 1 after 4, 7 and go beyond the borders of the figure.
2. Without leaving the hand from the sheet, make an angle and strive for a point at number 8 and 6 and just go beyond the scope.
3. Next, turn and pass through 3, 2, 1.
4. We turn through the corner of the square by passing the pathway through the points at number 1.5 and 9. It turns out a peculiar arrow-cursor that can be sent to any of the four angles according to your desire.

There is also a "hardcore" method for those who own spatial thinking.On a square sheet (sticker), draw nine circles (as in the task). Under 7 and 8 point, apply glue.

Take the basis of the cylindrical form. The tube from decorative cosmetics (lipstick or tone cream) is perfect. Connect space under 7 and 8 point with place under 2 and 3.

Conduct one solid line, starting from point number 1 and dropping down the spiral.When you return the left view, you will see that three lines cover all the points that fit into the puzzle conditions are fired.

"Advanced" individuals can solve it even without the help of glue, the main thing is to imagine the final result.

To solve this and similar puzzles, it is worth developing and opening unusual approaches to the problem.Try the fun exercises listed below.

Tip: Prix points directly on paper, it will be easier to deal with the solution.

Games for home leisure

At one time, Steve Jobs, a man-synonym for the words "creativity," stressed that people who own the skill of creative thinking do not invent, but rather notice the relationship between several things.

This is what makes it possible to synthesize something new.Therefore, first of all it is worth "pumping" such observation, over the surrounding phenomena and things more often.

Game number 1.

We offer the following exercise: look back and name as many things that are located in the same room with you and begin on one letter, without excluding mental concepts.

For example, "M":

1. Furniture, Lightning (on clothes), chalk (feeding for an animal)
2. Opinion, peacefulness, manners
3. Milk, Materials (Upholstery), T-shirt
4. Ointment, makeup, gauze, etc.

Simple version of the game: Letters "B", "C", "P", "K". If you are confident in your abilities, choose - "T", "A", "D".Do not limit yourself and congenital imagination.

If you wish in one room, you can find about 40+ words. Experts find approximately 100 words in each room.

Game number 2.

The next game was very popular in the XVII century. If you are offered to have fun "nonsense" - do not rush to refuse, the other name is "Burim".

For immersion in the process, you will need several sheets of paper, handle and a good company, which is not against practicing the collective essay of poems. Z.argea stipulates theme and limitations.

C. in addition, obvious combinations of single, pronouns, verbous forms and beaten baffles are excluded (Hi-lunch, love-carrot). Sometimes a certain topic is negotiated.

It happens like this: someone writes a line, and the other complements the verse next, until it turns out a full-fledged work.

Game number 3.

This different age, even the smallest.It develops spatial, which is probably useful in an adult future.

Put the kid at the table and give him a black pencil and a sheet of paper. Turn on pleasant music and ask him to close your eyes.Let the child draws, twists the random lines among themselves, without thinking about accuracy.

Sometimes it is better to create some drawings that are superimposed by one to another.Later sit down with him and with the help of color pencils allocate the outlines similar to animals, objects, all kinds of images. Let the child himself be a source of ideas.

Select entertainment that will help kill time with memory benefit

Tip: excellent warm-up for the mind will be a puzzle with matches (chopsticks). Such small tasks will be interesting for both children and adults. They are available to everyone!

Exercises for the development of creative thinking

Stand up. Take any book from the shelf. On two different pages, blindly choose a few words.Now try to find everything in common that they can have. For example, the word "carpet" and "tree": they are both lying on the ground, their images are found in fairy tales (carpet-plane, the tree for which the cat is walking) and so on.

If you play with the child, choose words easier: cat-dog, tomato pear, table chair.Write a dozen any nouns on a sheet: "Strawberry", "Fish", "Water", etc. And now imagine that this sheet is the requirements of the customer, and you are the architect's builder itself.

Build a house using them as basic requirements.For example, wallpapers will be a red "strawberry" color, and the walls of the house - glitter in the sun, like fish scales. The house itself stands on the top of the mountain, where the sky is simply boundless blue, like water, etc.

Sitting in the room, find within your visibility the subject that you know and is interesting. For example, "Apple."

You will come to the rescue

Come up with five adjectives, which are perfectly combined with the subject:

1. Green
2. Sour
3. Delicious
4. Soft
5. Juicy

And now more complicate the task and come up with another five adjectives, but those that are absolutely not suitable in meaning: barbed, coarse, plush, tin, slim.With some words, it is not so easy to work, but in more interesting the task: well, train, wind, wall.

Take in the hands of a pencil and in a notebook in the cell. Draw a column of crosses.Width and height arbitrary, just make sure that they are at a sufficient distance from each other.

Then these crosses turn into small pictures, drawing the necessary details (fish, crossed axes, sword, dragonfly, etc.).Similarly, drew the letter "O", "T", "B" and invent new, interesting images. On an advanced level, you can turn sketches in small plots with action.

Come up with a whole story! It is not so difficult as it may seem at first glance.

Use a computer with mind

Tip: Read the words backwards: Tale-aqua, Akulatub Bottle, Akzhol Spoon. This is definitely a useful occupation, will help pass the waiting time in the queue or in public transport.

Online Games Improving Creative Thinking

IQ-Ball

You are a small, round, alive ball with a paw-sucking out of the body.The goal is to get a candy at each level, overcoming all sorts of obstacles. You will be interfered with fixed and moving elements, time limit, inertia.

It is not possible to repel or cling to the "paw" from all surfaces. Thinking will be able to quickly, the reach of the goal depends on it.

"Black cat"

Before you, the field created from the circles. In the midst of him sits a black cat. With the click of the mouse click you can fill the mini-area, through which the cat can no longer pass.

One move is making you following - a tricky animal.Your task is not to let him run over the edge of the playing field, because it means losing. Here you have to connect all your intelligence and creative thinkingAnd most importantly - choose the right battle strategy.

In this case, it can be advised not to hurry, but thinking out your turn to the challenge, mark the circles through one.In this case, you will always have time to close the way a fluffy animal.

Present various images that you can easily find on the Internet.This is not just sketching, but pictures with plowing potential.

The same "doodle" can carry at the same time several values:

1. Facade
2. Wedge
3. Cloak
4. Diamond, etc.

The positive result of the game is to accelerate mental processes, the versatile development of imagination and creative start. Such a simple fun can capture you for a long time.

"Memory Matrix"

Many and adults and children are familiar with this game. Before you, the field that is filled with squares for a few seconds.Next they disappear. Your task is to search these most features "by memory".

With each subsequent level, the field grows and the task is complicated. The game is well developing memory, creative skills and the ability to quickly concentrate.

Tip: Try to play Lines 98. It in parallel develops logical thinking.

Clear, developing tasks

Draw a rectangular island on a sheet of paper, in the middle of which indiscreet treasures are hidden. It is surrounded by the same in the form of the mo.

You are a hunter for jewels, which is outside this land. Arsenal is only two planks, the length of each is a little less than the width of the RVA.

It is impossible to jump over or fly it, there are no ropes to fasten the boards together, like nails, and each individually falls into the abyss.

The goal is to get to the treasure. The answer to this riddle is based on the principles of geometry: the first board "Put" at the corner of the RVA so that it does not fail.

By this you most reduce the width of the RVA, and the second board will be free to the island with a treasure.Put a fat point in the middle of the sheet. The goal is to drawlit around it the right circle, but so that the beginning of the line starts from the point itself.

Ripstand: Bend the corner of the paper, in front of the angle itself, put the point without taking the hand, swipe from the point to the rest of the sheet, align the corner and continue the move until you draw a circle.

And finally, a simple question: why all over the world make exclusively round pizza, but deliver in square boxes?

Contradiction here only at first glance. And the answer is as follows: the pizza is the round, then that the angles do not burn, which inevitably happens when the dishes are baked rectangular forms.

In the case of the box, several factors are important:

1. So eat easier to get out of it
2. Produce square boxes much cheaper and easier than round
3. Pizza in it seems impressive

Tip: Training your brain with small tasks at least several times a week and already quite soon you will feel that you have become much easier to find non-standard solutions at work and in life, think outside the strict framework of logical thinking.

The patterns listed by us in the preceding section are most rigidly related to the receipt of an intuitive effect. They clearly manifest themselves in situations, the volume complex complexity of which is minimal, and the solution found coincides (or almost coincides) with the decision itself, i.e. it does not arise the need for a special implementation of this method associated with the transformation of it in principle. Such tasks, remaining creative, are not problematic. In problem situations, the solution obtained by one simplest cognitive task must be re-used as the principle of operation in another, more complex situation. However, the way

actions developed as a result of solving the initial task, first is still very limited and directly leads to success only in a very close situation. Actions at this stage are not yet abstracts enough. To transform a private method in principle, it is necessary to deepen the level of abstraction, "filtering" action, objectively The principle containing the principle, the sensory elements of the situation, often random, i.e., in a certain sense, formalize the intuitive effect.

Concrete material experimental research He served as a specially developed cycle of tasks, which was based on the basis of which the principle of solving one of the well-known tasks - puzzles was laid. We met with some tasks of this cycle already in the preceding sections. Here we give their full systematic description.

The simplest and however, the initial task of this cycle was called the task of "three points" (I). The conditions of the task of "three points" are as follows: connect three points with two straight lines, without crossing the T-shaped barrier (Fig. 21).

The second in order of the task was the "4 points" (II) known to us.

The third was the only described in the previous section of the task "9 points" (III) 4.

The fourth task is also familiar to us - "16 points" (IV).

The fifth task is "25 points" (V): 25 points are given; It is required to spend through these points without taking a pencil from paper, eight straight lines.

The sixth task is "36 points" (VI): 36 points are given; It is required to spend through these points without taking a pencil from paper, 10 straight lines.

The seventh task is "49 points" (VII): 49 points are given; It is required to spend through these points, without taking a pencil from paper, 12 straight lines.

It is easy to see that a series of such tasks can be continued to be impossible. In this case, it is necessary to be guided by the following pattern: the number of points should correspond to the squares of the natural range of numbers; The number of

4 The requirement to "return to the starting point" is only necessary for the "4 point" task. For all other tasks it is unnecessary.

with which you need to connect the points must increase by two, respectively, each square. In all cases, this number of lines will constitute a limit; A smaller number that does not disrupt the requirements of the task, it is impossible to connect points.

The desired number of lines according to the elected number of points is easy to determine, Taking advantage of the equation

where w.- number of lines, and h.- The number of points increasing as the squares of the natural range of numbers (9, 16, 25, 36, 49, 64,81, 100, 122, 144, etc.).

Accordingly, this pattern we could use tasks: "64 points" (VIII); "81 point" (IX); "100 points" (x); "122 points" (xi); "144 points" (XII), etc.

In the table, the cycle of tasks could be considered as a complex cognitive task - the problem. However, this problem was given to the subject not immediately (for example, "144 points"), but according to individual tasks - links. The decision of the first link ("3 points") revealed the initial principle ("go beyond the plane limited points"), permeating the entire subsequent path of "climbing".

Adult subjects one after another were presented to the tasks of this cycle (I, II, III, IV, V, VI, VII, etc.) until the subject revealed the principle that satisfies the solution of any link, i.e. not yet The whole complex cognitive task was solved.

In other series of experiments, along with this technique, various kinds of formulations were used with the subsequent accounting of their effectiveness both along the line direct and along the side of the side product.

First of all, the general course of solving problems of this cycle was traced, i.e., the consistent solution of a complex cognitive task.

The solution of the problem«-points. "The easiest in cognitive attitude among all other tasks is the "3 points" task. In this task, finding the decision completely coincides with the decision itself, since the need for any specification of the found principle, its refinement for applying to this specific working conditions is completely absent. This task would be the most successful object to study intuitive solutions. However, in this respect, it is inherent in the lack of: the principle to go beyond the area of \u200b\u200bthe plane, limited by points, is overlapped with a simpler reception - the ability to connect three points just two straight, without leaving the specified limits. Therefore, for the formation of psychological difficulty, this task needs to be complicated by the conditions expressing in the introduction

T-shaped barrier, excluding this overlapping given principle the possibility.

As a rule, the task "3 points" (with the T-shaped barrier) is solved without the help of a special forming task. The fact is that, acting according to additional benchmarks (T-shaped barrier), the test itself builds in this situation the forming task, which coincides with the solution of the revealing problem, and by-product in such conditions in

all cases coincides with a direct product, since acting by landmarks, the subject does not have a specific plan of the decision plan, and the guidelines seem to lead it to it.

Most often, the solution of the task "3 points" subjects is built according to the scheme shown in Fig. 22. First, not two line data are used, and three (one straight line turns into a broken). The ends of this line are connected with the end of the barrier (Fig. 22, a), then the drawing takes the view shown in Fig. 22, b, B,and only further, after many other attempts, there is a solution (Fig. 22, d).

If you use this task in the forming function and prevent it "4 points", then the latter is easily solved, even if the 3-point forming task is given without stimulating, i.e., with direct order of presentation. It follows that this task ("3 points") arises relative to the "4 point" task to another ratio than all previously encountered forming tasks. The fact is that, as we have already noted, the end route of the test hand, which is the key to the decision "4 points", is no longer as a side, but as a direct product of the action: The "3 point" task itself performs and stimulating and forming functions. .

As a result of solving the problem of "3 points", the subject produces the initial principle of solving the entire task cycle with the growing number of points.

The feature of the task "3 points", as we have already noted, is that the addition is introduced in its condition, the end of which is considered subjects as an additional point with which it connects the first line conducted by it (according to the principle of elementary compound). Next, analyzing the task using the elementary reception (the connection of the points for the shortest distance), the subject comes to what lines the broken line.

After that, the search is organized by the internal limits of the figure formed by points, which makes it possible to transfer the cash method of the "elementary association" into several other conditions. Finally, the subject, highlighting the first angle as another point, connects it from the third and as a result reaches a solution.

Experience shows that if the subject does not know the principle of the decision, then the task of type "4 points" can be solved only in

the case if there are benchmarks lying outside the figure formed by a direct connection of the points in which the subject must act in the zone. In this case, that is, when the subject decides the problem of the "3 points", the presence of a barrier, the requirement to circumvent the barrier leads to the need to break out of the figure formed by points, and the successful attempt is fixed. Thus, a method of action is produced, which can then be transferred to the solution of the "4 point" task.

The role of the features of the interaction of a subject with an object that cause the possibility of developing a new method of action is clearly acting in the event that you compare the task "3 points" on the other, externally, a completely similar: it is required to connect four points, located, as shown in Fig. 23, two connected straight. As a result of this exercise, it will be impossible to achieve the direct work out of the method, with the help of which the subject could solve the "4 point" task.

So, acting by landmarks by "elementary association" in a situation determining the particular content of the interaction of the subject with the object, the subject produces a method of action, as if absorbing the content of the situation in which it is produced.

In the future experiments of this series of the test, who solved the "3 points" task, turned to the next task to "4 points". Features of solving this task, we have repeatedly described. We will add only one thing: referring to the "4 points" task, after the decision "3 points" the subject almost immediately found the right decision, since the implementation of the principle was not particularly difficulty in this case.

After solving "4 points", the subject appealed to the next cycle task - to "9 points".

Solving the problem of "9 points".We present the protocols to solve this problem with two subjects (Fig. 24, a, b).

As can be seen from the protocol, the first test (V.) has found the solution of the task of the 22nd attempt, and the test of N. - on the 16th.

The subjects, solved the problem of "9 points", was set by the problem of "16 points" (in the future we will lead the protocols of decisions of subsequent tasks with the same subjects) (Fig. 25, a, b).

In the task of "16 points", the first test (V.) achieved a decision on the 18th attempt: the second (N.) - on the 12th.

Fig. 25.

The task of "16 points" followed the problem of "25 points" (Fig. 26, a, b).

In this problem, the test of V. has achieved a decision on the 6th attempt, and the testable N. - on the 12th.

We present the protocols of solutions of the following task (Fig. 27, a, b).

In the event of the "36 points" task, the test of V. has achieved a decision on the 10th attempt, the test of N. - on the 7th.

The task of "49 points" Tested V. decided on the 2nd attempt, Tested by N. - on the 4th (Fig. 28, a, b).

The task of "64 points" both subjects decided on the first attempt (Fig. 29, a, b).

Following finding the solution of the problem of the "64 points" task (from the first attempt), both subjects covered the "144 points" reference task (Fig. 30, a, b).

The solution of the control task as well as the previous one has been achieved from the first attempt.

Thus, being a link of a wide cognitive task, each task of the link itself represents an independent mental task. The process of solving this problem, the final product of which becomes a new functional stage of the development of the principle, itself proceeds in the internal structural levels, which is differentiated by a number of peculiar processes of interaction, the products of which become the conditions of internal development and determine the flow

new processes. In domestic development, a number of stages are detected (the number of which in different cases is not the same). The most characteristic of them are as follows: a) rational use of the result of the decision of the preceding problem; b) refusal from the chosen path and the transition to the "spontaneous" manipulation by means of elementary, unconscious empirically generalized techniques; c) return to the initial principle ("go beyond") - the adjustment of the rationally used principle by means of unconscious empirically generalized elementary processes; d) solving the problem.

Span. AND Fig. 28.

AND en. IN Fig. 29.

Fig. thirty

but -the result of solutions of the problem "3 points"; b.- the result of solving the problem of "4 points"; oh, G.- ~ The first and second attempt to solve the problem of "9 points" characteristic of one group of subjects (direct angle in zone A); d, E.- The first and second attempts by the ressenter of the task of "9 points" characteristic of another group of subjects (direct angle in the zone C).

Consider each of these stages.

Rational use of the result of the decision of the previous task.In the overwhelming majority of subjects, the orientation in the situation of each next task in the first stage is determined by the direct product of the action in the situation of the previous task. In other words, at the first stage, the tests, as a rule, carry out the direct transfer of this product to the conditions of the new task; Previously, the result resulting is now as a solution; The product goes into the process.

In the task of "4 points", this first stage usually coincides with the decision and therefore does not act here with all distinction. The most characteristic of this stage is detected when analyzing the solution of the problems of "9 points", "16 points", "25 points", "36 points", and sometimes "49 points", i.e., where the problem obtained in solving Points »The principle needs special concretization.

For example, in the "9 point" task, the first searches for the test solutions of this problem are strikingly similar.

In the overwhelming majority of cases, the drawings of the first first attempts turn out to be completely similar (Fig. 31).

Each of these drawings is a clearly pronounced transfer of the result of the decision of the previous task.

It should be noted that the graphical expression of this transfer has some originality compared to attempts to solve the "4 points" task. This originality is as follows.

As can be seen from fig. 31, when identifying the principle of solutions in the situation "3 points", all subjects, obeying the peculiarities

but- the preceding solution of the problem of "9 points"; b.- The first, second and third attempts to solve the problem of "16 points" (second group of subjects). The figure shows only a small part of the options.

a - solving the problem of "19 points"; b.- The first attempts to solve the problem of "25 points"

th - solving the problem "25.points "; b.- The first attempts to solve the problem of "36 points"

a - solving the problem of "36 points"; b.- The first attempts to solve the problem of "49 points"

conditions, orient the acute angle, formed by two specified direct, in the part of the space that is highlighted by us as the zone "C". The exact same orientation of the acute angle we discover and in the drawing of solving the problem of "4 points". Accordingly, the direct angle in the drawing of the solution of this task turns out to be oriented in the zone "A". When the principle of solving the problem of "4 points" in the situation "9 points", some variability of the drawing construction is observed: one part of the subjects focuses the straight corner in exactly the same way as it was done in the situation "4 points", i.e. in the zone "A "However, the other part of the subjects changes the spatial orientation of this angle, placing it into the zone" C ".

A similar picture is observed and when analyzing the solution of the following tasks (Fig. 32-35).

As the task-links have progressed on the system, the transpitability of the transfer marked by us is somewhat modified, the nature of the tolerable drawing is stabilized. Each of the subjects produces any one of the two possible principles of solving the problem (see Fig. 33-35) and strictly adheres to it later. According to experiments, switching the subject with one principle of solutions to another in these conditions is almost impossible.

The found facts suggest that, as a result of solving the problem of the "3 points" problem, the principle of solving the entire chain of tasks, the tests have not yet realized with the full relevance of the significance of this principle and do not exist from the entire set of conditions of the situation. Insufficient awareness of the importance of the principle and manifests itself in the fact that the drawing of the solution of the "4 point" task accurately copies the spatial layout of the lines on the drawing of the "3 points" solutions. In some subjects, this phenomenon applies to the decision of the subsequent task - "9 points". However, other subjects, moving to solving the "4 points" task and reaching this decision, aware of the importance of the principle with which they have to deal with. As a result of such a realization, the subjects under some extent abstract this principle from specific characteristics of the situation and fix it in the expression "need to break out." In the future, this expression becomes guidance to action. The tests of the tests in the course of solving the problem are disclosed than the reorientation of the spatial arrangement of the drawing of the solution is motivated - the tests first seek to implement the condition "it is necessary to break out", therefore the construction of the drawing (when solving the problem of "9 points") and begins in some cases not from the point in The zone "A", as it was in the situation of the previous task ("4 points"), and immediately goes beyond the site limited by points.

Refusal from the chosen path and the transition to the "spontaneous" manipulation by means of elementary, unconscious, empirically generalized techniques.The first stage of the decision is completed by refusing from the elected path and the transition to this natural manipulation in the area of \u200b\u200bthe area, limited points, which is extremely characteristic of the actions of the subjects, unfamiliar with the principle of solving the preceding problem (this stage is characteristic of the "9 points" tasks and "16 points ").

In fig. 36 Samples of such manipulation are given.

Transition from the first stage to the second.Used at the first stage of solving a mental problem of action, being an adequate condition of the task, it requires, however, additional concretization and development, therefore this method of action does not directly satisfy the characteristics of the situation.

but- attempts to solve the problem of "9 points", b.- Attempts to solve the problem of "16 points"

The new product arising in the end attempts to solve the problem (we mean the task "9 points"), only in the first case (at the first attempt), it cuts out one of the possible options and opens some (seemingly) perspective (crossing the hypothenuclear two points at once), What is carried out in the next attempt. The first attempt based on the product appears on the product leads to an unpromising product. The question of the ways through which the transition from the first stage to the second is not yet found out (it is possible that there are several peculiar paths).

It should be thought that a leading role in this change cannot be attributed to any subject, none of the object - the cause of the subject with the object itself is the cause of the subject. . Sube deforms the initial situation. However, the effect of this deformation is determined not only by the method of action of the subject, but also the characteristics of the object to which the action is directed, i.e. the interaction of the subject and the object.

Another characteristic feature of this transition is the fact that, by varying the drawing, the testes, as a rule, do not give themselves a clear report on the true causes of their actions, they evaluate only their effect.

The fact that the second stage in all cases is presented attempts to achieve solutions by elementary merging points in the shortest distance is not surprising. The situation of this problem actualizes only one specific reception at the subjectable. And if this reception disappears, it naturally replaces the "universal method", which in this case "does not have competitors."

Return to the original principle("Go beyond") - adjustment of the rationally used principle by means of unconscious empirically generalized techniques. The second stage is usually completed after 3-10 attempts. The mechanism of this stage largely coincides with the mechanism of the preceding one. Differences are consisting only in the way to operate the subject. But, as in the previous stage, the method of the second stage does not lead to the desired result. The subjects of the subject detect the search loan. The dynamics of the situation goes out. The critical moment appears again, that some uncertainty in choosing a path of further attempts, some "declaring" situation, which is characteristic of the climax of the application of a particular action method, i.e., the conditions conducive to the change of action.

As the experimental data show, in the third stage, the subject again uses the method of action that he has already operated on the first stage. (As it should be expected, since in the experience of most subjects there are no other methods that could be updated with this situation.) However, now in operations is also detected and something new. First, there is no exactly accurate, literal transfer of the drawing of the decision of the previous task (although in the first attempts of this stage in some subjects, such a literal transfer still occurred). Apparently, the first and second stage did not disappear, they contributed to the deepening of the abstraction of the principle of solving obtained in the previous task. In the third stage, the subjects are guided by only one requirement - "break beyond". This clearly acts on the drawings attempts to solve (Fig. 37) - the third stage is characterized by the conciseness of samples, which often consist of only two lines.

We give the drawings attempted at the third stage in the conditions of the problem of "9 points" (Fig. 37). As can be seen from the drawings, the subject seeks to rationally use the identified principle of the decision and is looking for its adequate use. However, without having a special method (method) of the organization of such a search, he reiterates unconsciously resorts to the "universal" reception of manipulation by landmarks, i.e., this principle admits the situation to the situation of the task through unconscious empirically generalized techniques. Thus, both previously used methods are combined, and this gives a qualitatively different nature of the action, since it turns out to be an adequate this complex of conditions of the situation.

The third stage prepares the decision, and sometimes ends with them (in the case when the decision is achieved completely suddenly, thanks to a successful coincidence). A more prepared solution consists in the fourth stage.

Decision.The allocation of the fourth stage as relatively independent is justified by the fact that the method of action at this stage acquires in some subjects a different quality. At a certain point, the subject, departing from the processes of manipulation, begins not only

Fig. 37.

it is organized by the revealed principle, but organizes a conscious targeted analysis of the situation (a feature of such an analysis is, however, the fact that only the assessment of the result obtained is recognized in it, and the process of production itself, as in previous cases, remains unconscious).

During this kind of manipulation, the visual component of the problem is differentiated on a certain kind of group of points; By combining these groups with elementary reception (the connection of the points for the shortest distance) is achieved.

To illustrate this provision, we analyze the protocols of experiments.

In the drawings (Fig. 38), it was clearly captured by the analysis of the test "16 points" tasks. "Alternating" to these points drawing a solution to the problem of "9 points", the subject broke the entire complex "16 points" into two subgroups and combined them then by "elementary compound".

The opposite in shape, but the identity of its meaning clearly performed and in the case when one of the subjects could not solve this task on its own.

We give a protocol experience.

The solution of the task "9 points" is known to the subject.

The task of "4 points" (Fig. 39, but).

Fig. 38. Solution of the task "9 points" is known to the subject

Fig.40

The task "9 points" (Fig. 39.6).

The test is proposed for the "16 points" task (Fig. 40).

The subject recognized the task of unresolved.

A differentiating table is proposed (Fig. 41).

With this table, the subject found a solution at the first attempt.

With the solution of the task of "9 points", the subject familiarized himself about a year before these experiments and immediately remember him could not.

Fig.41 ®®®

However, the "4 points" were resolved by the subjects for 1.5 minutes, after which the test was spent on the solution of the "9 points" task for less than one minute (i.e. the decision was almost "from the place"). Then the subject was proposed for the task of "16 points". In the first first attempts, the test fully changed the drawing of the problem of the problem of "9 points", however, making sure that it did not lead to success, he refused such transfer and "closed" in the area of \u200b\u200bthe square, limited points. Next, the second stage of solving the subject has not advanced. After

14 unsuccessful attempts (not overlooking their content outside the second stage), while spending a decision of 20 minutes, the subject refused the task, recognizing it unreserved.

Then it was proposed so-called differentiating table, containing the same 16 points, but with such a change: 9 points (3x3) on this table were applied with a red carcel, and the rest are black (see the drawing of the differentiating table - Fig. 41, protocol of experiments with this subject). With the help of a differentiating table, the test less than 1 minute found a solution ("from the place"). Experience has shown what "lacked" to solve that it was necessary to see in the drawing and that the subject could not get alone, as was done in the previous case.

Describing all the stages all selected by us as a whole, it is necessary to note the following. The duration of each stage is determined by the features of the speakers of the situation. A certain type of manipulation is preserved until the situation remains dynamic, i.e., some variability of attempts remains. As soon as the repetition and novelty appear, which makes an action in a situation, disappears, during the decision it comes a fracture, leading either to the abandonment of the decision, or to the transition to a new stage, i.e., to a fundamental change in the action method.

The solution of each of the intermediate tasks-links is built in the same principle, with the only difference, which, as the task chain progresses, the number of manipulations is gradually reduced. To illustrate this pattern, we give an example of an average number of attempts made by 30 subjects when solving the chain of tasks-links.

Thus, in most cases, the "81 point" task is solved from the first attempt. Here, the subjects, as a rule, on their own initiative, were verbally formulated the principle of solutions: "Before you need to strike out all the extra points, and then solve the problem of" 9 points ". If after that the test "144 points" was given to the subject, it was solved from the first attempt. The subject has developed the ability to solve "from the place" any such task, regardless of the elected number of points and without support for a visual component (in a verbal plan), i.e., the principle of solving this problem was finally developed. In the experiments carried out, very large variability of indicators from various subjects. However, everyone clearly performed a tendency to reduce the number of attempts during the transition to each subsequent task (despite the permanent increase in the objective complexity of the task). The fact that, to solve the control problem ("144 points"), each subject passed at least 6-7 previous tasks was passed.

Since the place of each link in a number of this cycle of tasks (starting with the second) is determined by purely objective quantitative dependencies, it was decided to investigate how much it was necessary to move on this chain in the development of the principle. To do this, it was necessary to find out what the exclusion of some individual units would lead to.

In a series of experiences dedicated to this series, the following technique was used.

The following "incomplete" tasks cycles were proposed to various groups of subjects (five people each).

The first cycle is the tasks I, II, IV, V and G. d. (Omitted the problem of "9 points").

The second cycle is the tasks I, II, III, V, VI, etc. (lowered the problem "16 points");

The third cycle is the faces I, II, III, IV, VI, VII, etc. (omitted the task of "25 points");

The fourth cycle is the faces I, II, III, IV, V, VII, VIII, etc. (omitted the task of "36 points").

As indicators, the difficulty of solving one or another cycle was used: first, the number of subjects that solved this cycle (from the total number of a group of five people), secondly, the average number of attempts necessary for the subject to solve those challenges that followed For the missed link. This number of attempts was compared with those average data, which were obtained at the "normal" cycle on 30 subjects in the previous series of experiments.

The results obtained in the second series of experiments are presented in Table. one.

As can be seen from the table, compared with the full cycle, the difficulty of abbreviated (incomplete) cycle increases significantly. Moreover, the first cycle in which the "9 points" task was omitted, turned out to be the most difficult. In the conditions of these experiments (at which the time for the solution of each link was limited to 30 minutes), none of the subjects found a solution. In the remaining cycles, as the lowered task is removed from the beginning of a number, the difficulty gradually decreased.

Thus, it was found that the full cycle of tasks represents the optimal conditions for the development of the decision principle. This provision was of particular interest, since objectively the principle of solving any task in the finished form was already in solving the problem of "16 points".

Table 1

		Abbreviated cycles
								fourth
"16 points" "25" "" 36 "" "49" »				Passed		Passed		Missed 4 14

Note.1 - the average number of attempts to solve the 30 subjects (data of the first series of experiments); a - the number of subjects solved this cycle (out of 5 people); B - the average number of attempts for all subjects that solved this cycle.

However, the method of action developed as a result of solving this task was still very limited and directly led to success only in a very close situation (the task "25 points"). The actions of the subjects at this stage were still still a sensual side, they were not quite abstracted. To transform a private method, it was necessary to deepen the level of abstraction, "filtering" the action, an objectively expressing principle, from the guides of its sensual elements of the situation, often random. Such "filtering" was carried out in solving subsequent tasks.

These experiments pushing the idea of \u200b\u200bthe dependence of the principle of solving on the inclusion of the test under the condition broader, or, as we say, a promising task, in which the result of the previous decision is already an operation as a method of action.

It was found that in order to successfully identify the general principle of solving the problems of the cycle used, it is necessary that this cycle is complete (especially in its first 4-5 units). This fact cannot be explained only by the rupture itself between the tasks.

Skipping of any link leads, of course, to the complication of transfer conditions due to the increase in the number of possible options for attempts. This circumstance, of course, plays a certain role, but this reason cannot be the only one, since the skipping of more retractable links (from the task of "25 points" and below) does not cause more special difficulties in solving the next task of the abbreviated cycle, although objectively complexity Each subsequent task increases in geometric progression. Apparently, the characteristic of the method itself is important here, which uses the subject.

It was natural to assume that the skipping of those or other links at the beginning of the chain (as long as the principle of operation was not yet completely detected) provided such a negative effect because these links are needed to identify the principle and when they are dedicated, the principle concluded in solving the previous task was not enough Revealed. This is sometimes irresistible difficulty in solving a follow-up task. Consequently, to identify the principle, it is necessary to include the prosecution in the conditions of a wider (promising) problem, but this task should not contain too much difficulties associated with the specificization of the principle.

Thus, as a result of the experiments, we managed to allocate one of the conditions that contribute to the abstraction of the method of action, and thus the development of the principle of decision. Such a condition was the inclusion of the test in the situation promising, i.e., a broader task in which the result of the preceding task was to be used as a solution.

In further experiments, we investigated other conditions, also contributing to the abstraction of a method of action from private elements of a particular situation in which this action was first implemented.

Previously, we have already emphasized that for the awareness of the method used in solving the practical task of the action method, and therefore, and to identify the principle of solution to the subject, the theoretical task should be delivered. Naturally, the identification and awareness of the method of action to some extent implies its abstraction. Hence it followed that the formulation of the theoretical problem should be one of the conditions for abstracting the method of action.

To identify this dependence, the following methodological technique was used. The subject dealt with the usual ("full") cycle of tasks.

However, the first task-link ("3 points") was not given to the subject for an independent decision, but was explained by the experimenter.

Explanations were made approximately. "We are given a task to connect three points with two straight, without crossing the barriers. Look: the shortest way cannot be implemented. We use another opportunity - carry a line down and wage the barrier. "

Immediately after such an explanation of the solution of the problem of "3 points", the test "4 points" was given to the subject. An ordinary instruction was modified for this task. The experimenter spoke to the subject: "Now add three points another one - the fourth - and delete the barrier. Try to connect all these points without leaning a pencil from paper, so that Randas returns to the starting point. What is quite possible, it is only necessary to supplement the drawing (connecting three points with a barrier) at the right upper part. "

Following this, the subject without any difficulty found the correct solution of the "4 points" task.

Thus, the subject to some extent acquainted with the initial principle of solving the task-link cycle. However, since in a situation of these tasks, its own activity was reduced almost to a minimum, it was possible to assume that the principle identified by the subject was very little abstracted from a particular shell of the situation.

After such preparation, we introduced the experience of the "9 points" task, offering it to the subject for an independent decision.

In total, in this series, we conducted 7 experiments (with 7 subjects). Of these 7 experiments, it was possible to select 4 cases (4 experience with 2 subjects), which satisfied the plan of these experiments.

In the specified 4 cases, the tests, making up 8-12 unsuccessful attempts to solve the "9 points" task, refused to continue the decision, recognizing the task of the unresolved. Comparing these indicators with those that we were obtained in experiments where the activity of the tests in the decision of the previous tasks ("3 points" and "4 points") was not limited to anything, it was possible to conclude that the reason for the failure of the subjects in this kind of experiments was precisely Restriction of activity.

Since, from our point of view, the deprivation of the subjects necessary for the success of activity had a negative impact primarily on the abstraction of the principle of decision in the situation of previous tasks, we concluded that one of the conditions for the success of such abstraction is the activity, the independence of the subject's actions in the problem situation 5.

The task of the described experiments was not reduced only to identifying the activity factor. Continuing the experiments, we calculated to detect a fruitful influence on the abstraction of the principle from the side of the theoretical task assigned to the subject.

It seemed to us that, even acting in the situation of the task of "9 points", under certain conditions the subject will be able

Since such a conclusion seemed to us theoretically obvious and even banal, we did not find the necessary further actual experimental analysis of his parcels (giving themselves, of course, the report is that for this conclusion itself, the actual material itself does not give a sufficient basis).

to abstract to some extent that the principle that he was given in solving the previous tasks, and if such an abstraction occurs, it will have to lead the test to solve the problem of "9 points" (if our assumption that connects failure subjects with the insufficient abstraction of the principle in solving preceding tasks, it was right).

To make the theoretical task of maximum naturalness, it was decided to use the test with the experimenter for this. Chatting with the subjects, I refused to continue the search for the decision "9 points", the experimenter requested them to give explanations just done by unsuccessful attempts to solve. At the same time, the subject asked the question: "Why did you exactly solve the task?"

At the first moment, this question in all four subjects caused obvious bewilderment - none of them could quickly find even any satisfactory motivation.

Then the experimenter requested the subjects to explain why each individual line was carried out. The subjects (all four behaved exactly the same way), having mastered the question somewhat, began to invent motivation, at the beginning very remote, as it seemed to us, from the true state of things. However, so they included in the situation of the theoretical task.

As the experiments have shown, such inclusion pretty quickly led to a positive effect. All four have found the solution of the problem of "9 points" when analyzing only the 3-4th drawing attempts to solve.

At the same time, the subjects stated that, thinking about why they needed to spend a different line, they unexpectedly noticed how to solve the task. At the same time, such "insight", according to the subjects, was so fleeting that to answer the question of how nevertheless managed to solve the task, there was no possibility, despite the fact that the task itself and its decision became for the tests perfectly clear.

The subsequent actions of these tests in the situation of further tasks of the cycle showed that these actions did not differ from the actions of the tests that solved the cycle in the usual way, that is, without any restriction of activity. The number of attempts to solve those and other categories of subjects were approximately equal. Hence it followed that the formulation of the theoretical problem led to the same effect of abstraction of the principle, to which the active activity of the prior tasks under the situation also led.

Thus, we had reason to consider the formulation of the theoretical problem as one of the conditions for the success of the abstraction of the principle of solutions and thereby its development.

To identify further conditions contributing to the abstraction of the principle of the decision, we used the transition from the third link of the cycle to the fourth (i.e., from solving the problem of "9 points" to the task of "16 points").

Based on the already said earlier, it was necessary to assume that the success of the decision "16 points" is within a certain dependence on the degree of abstraction of the principle of "9 points" solutions.

This provision was primarily confirmed experimentally. For this, the method of limiting the activity of the subjects was also used. However, if in previous experiments, the activity of the subjects was limited only when solving the two first cycle tasks ("3 points" and "4 points"), now we have spread this limitation to the third task, that is, on "9 points". This task, as well as previous, was not solved by the test active - the experimenter simply showed its solution in the finished form. After such a show, the subjects had to solve the problem of "16 points".

As the experiments showed, none of the subjects in such conditions could not find the solution of "16 points". It was obvious that when showing the test solutions of the problem of "9 points", any of them could not be sufficiently abstracted by the principle of its decision.

It would be very simple to achieve the necessary abstraction of this principle if we decided to take advantage of training. To do this, it would be sufficient to suggest any wording, for example: "connecting points, guided the following rule: first three down and then two sides; You can start with the diagonal. " However, we were interested in creative solutions, so we found ways to contribute to abstraction that the subject could use without direct learning. With such an idea, the following methodological technique was used.

Those testes that refused to continue the search for solving the problem of "16 points" were to return to the "9 points" task, but to solve it not in the usual way, as all other subjects did, but with some modification. The experimenter pointed out the subject and direction of the first line with which the test had to start building the drawing. Despite the fact that the solution of the problem of "9 points" was already given to the subject, the new task was very difficult to fulfill. This confirmed that, knowing the solution of the decision, the subjects had not yet owned them completely.

In order to create conditions for complete mastering in this way, we proposed the subject to execute 12 solutions of the "9 points" task using a special table (Fig. 42). On the table, 12 sets of points were applied (9 points each) and each complex had a line that it was necessary to use, starting building a drawing.

Fig. 42. Table of options for solving "9 points"

The fulfillment of the first 4-5 constructions of the subjects spent a relatively long time, the remaining construction was done much faster. After the subject performed all 12 constructions, he was again offered the task of "16 points". This time the decision "16 points" was very soon 6.

Such a reception that stimulates the abstraction of the principle has proven to be very effective. This was specially shown in experiments with another group of subjects, also consisting of 5 people. The new tests were performed by 12 preceding constructions of the options for solving "9 points" before it was proposed by the problem "16 points" (the first two tasks were given in the same way as in the previous case, that is, with a limitation of activity). All of these 5 subjects that fulfill the preliminary constructing options for solving "9 points" found a solution "16 points" after the fourth, sometimes fifth attempts. This result was undoubtedly much more successful than the usual results with which we encountered with the "natural" path of solving a cycle (15-20 attempts).

The effectiveness of the described reception was decided to compare with the effectiveness of other possible receptions. For such a comparison, the following methods were used.

It should be noted that some of the tests of the "9 points" task in the course of construction of various options themselves put theoretical task, analyzed the situation under its influence and verbally formulated the principle of construction. These formulations were different from each subject, but in general they all reminded the one we have already spoke about ("first three down, then two lobs; you can begin with a diagonal).

1. Acceptance of learning, in which 5 subjects after displaying "9 points" (two first cycle tasks were also given with a limitation of activity) reported the defining principle of the wording ("Two down, three lobs; you can start with a diagonal").

2. Acceptance of pre-automation of action, where 5 subjects (under the same preliminary conditions) earlier than to start solving the problem of "16 points", it was necessary to repeat the decision of the "9 points" problem 12 times, but not from different provisions, i.e. . without varying the drawing, and repeating the same it is shown first by the experimenter.

3. Combined reception in which the wording message (first reception) was combined with the automation of the constructions of the solution in one embodiment (second reception).

4. The second combination reception in which the wording message was combined with a single construction of two drawings of solving the "9 points" task for two different options.

The average number of attempts to solve the problem of "16 points", undertaken by the subjects of each group, served as an indicator of the efficiency of each reception.

We present the results of these experiments, indicating for the comparison and the number of attempts necessary to solve the problem of "16 points" with the "natural" cycle passage (without limiting the activity and introduction of any additional receptions), as well as under the condition of limited activity in the situation of preceding tasks But when taking a preliminary construction of 12 different options for solving "9 points".

1. "Natural" path of passing cycle 15-20

2. Implementation of the construction of 12 options 4-5

3. When wording without additional receptions, 30-35

4. Prn Automation of one of the options 6 *

5. Combined reception (wording + automation of one option) 10

6. Combined reception (wording + Building 2 options) 5

* In conditions when the first line is carried out by an experimenter.

It can be seen that the most effective was the reception associated with the implementation of the construction of 12 different solutions (4-5 attempts), as well as a combination reception, in which the verbal wording of the principle was accompanied by a single construction of two different solutions (5 attempts).

Acceptance of the construction of one of the solution options was also very effective (6 attempts), but in assessing its effectiveness it is necessary to take into account one important circumstance that occurred in these experiments,

based on which the effectiveness of this reception noted by us cannot be directly compared with the effectiveness of other techniques. The case is that when automating one of the solutions, high efficiency was achieved only in exceptional circumstances, which were additionally created by the experimenter. These circumstances were as follows. In the first experiments, it was found that out of five subjects alone found the decision "16 points", having done only six attempts for this. Three subjects could not solve "16 points" at all, and one, the last, did more than 30 preliminary attempts for this. It should be noted that although we automated each test only one of the solution options, at the same time, each test, these options were different. So, the first automated version number 1 7 (Fig. 43, but),second - No. 2 (Fig. 43, b)in the third - number 3 (Fig. 43, in),fourth - No. 4 (Fig. 43, d) and in the fifth - No. 5 (Fig. 43, e).

It turned out that the test, solved "16 points" only after six preliminary attempts, was dealing with option number 3 (Fig. 43, in).Moreover, in the first and second attempt to solve "16 points", this subject began building a drawing from an extreme upper left point marked in Fig. 44 arrow "/", and in further attempts (probably by random circumstances) he moved the beginning of constructing to the lower left left point (in fig. 44 marked with the "2" arrow). After which they found a solution expressed by the drawing shown in Fig. 45, a.

We noticed that the second part of its construction allocated in Fig. 45, and fat lines, accurately appropriate

Options are numbered by us according to the table of building 12 solution options.

Fig. 46. \u200b\u200bMethodology H Result of an additional series of experiments: [- Automated options; II - the first line conducted by the experimenter (directions indicated by arrows); III - Drawings of solving the problem found by the subject (not. 3 task did not decide)

shaft is the option of solving "9 points", which has been automated before. Other subjects have no such coincidences.

The notched fact made us carry out an additional series of experiments with five subjects, which revealed the cause of this case.

Experiments of the additional series were built so. Initially, the same conditions were created as in the previous experiments, that is, the subjects were familiar with the three first tasks of the cycle when restricting activity. Then, as in the previous case, they automated one of the variants of the 9-point solutions (one that was previously shown by the experimenter). Thus, the first subject automated version number 2 (P, Is. 46, 1a)in the second - number 3 (Fig. 46, 16), in the third - number 5 (Fig. 46, / c), in fourth - No. 6 (Fig. 46, / d) and in the fifth - No. 8 (Fig. 46, Id). After automation, the subjects refer to the "16 points" task. In contrast to previous cases, in these experiments, the experimenter imposed the beginning of the construction of the drawing (the experimenter conducted the first line himself and only then passed the pencil to the test) (Fig. 46, // - b, B, G, D).

The results of these onvitob were as follows. Of the five subjects, only one did not find the solution to the problem. The rest spent on the search for a very small number of attempts.

From here it follows that all solutions had a strictly definite character - the previously automated version was the second part of the final drawing. Consequently, the automation of action that the preceding problem was carried out led to a very tangible effect in solving the task of subsequent. However, this effect was possible only in much conditions, where the variability of the subjects of the subjects was reduced to a minimum.

To finally prove the provision that the automation of action was crucial, we repeated these experiments, somewhat modifying them. The modification was that, while maintaining all other conditions, we excluded the automation of the convention of the solution, limited only to its once demonstration of the subject.

Of the three people who participated in these control experiments, no one has found solving the problem. Thus, the role of automation of the method of solution in these circumstances was finally proved.

Describing the previous series of experiments, we have repeatedly noted a significant impact of the verbal wording of the method of solving the preceding task for the success of actions in a situation of a follow-up. In a new series of experiences, this issue was subjected to a special experimental consideration.

The following technique was used. In the first part, all the subjects (12 people were involved in these experiments, divided into 2 groups of B person in each) after a quick showing of solutions of the "3 points" task, "4 points" and "9 points" additionally performed the construction of the drawing of four different options Solutions "9 points" (options 2, 3, 9 and 12 - see Fig. 42).

Representatives of the first group did not give any additional instructions. When building these test options, the experimenter carefully monitored that it was not accompanied by attempts to verbally formulate the principle of solving the problem. Those testes that have noticed a tendency to such formulation were excluded from experiments. Thus, out of 13 people managed to select B, whose actions did not have any hints on trying to verbally formulate the principle of decision.

Representatives of the second group after constructing the first first variants of the decision, an additional instruction was given, requiring the verbal wording of the principle (using the experimenter).

Thus, in the initial part of the experiments, two groups of subjects were drawn up: in the first - the construction of four options for the decision "9 points" was not accompanied by the verbal wording of the principle; In the second, this is a construction, on the contrary, has been completed by such a wording.

The final part of the experiments was carried out after a weekly interruption and consisted in the following. 6 subjects (3 people

Table 2

		Task "16 points"	Task "9 points"
I.group
		There is no decision
			Decision after 7 attempts
		Decision after 8 attempts
			Solution at 1 attempt

the ka from each group) was given the problem of "16 points" (the time for solving was limited to ten minutes). The remaining 6 subjects (also 3 people from each group) was proposed a re-solution of the problem of "9 points".

The results of the experiments are given in Table. 2.

The table shows that the subjects of the second group (i.e., those that verbally formulated the principle of solving the problem of "9 points") in the final part of the experience were discovered incomparably greater success than the subjects of the first group (i.e., those who did not formulate verbally The principle of decision). So, for example, none of the subjects of the first group could not find the solution "16 points" for 10 minutes, while all the tests of the second group successfully completed this task; For the test first group, the reconciliation of the "9 points" task has become a problem, and each of them turned out to be necessary to make an average of 8 attempts, while the subjects of the second group reproduced this Reiization "from the place" (two people at the first attempt and One - with the second).

In this case, we consider it important to emphasize the following circumstance. Those testes that verbally formulated the principle of solutions and thus knew the rule of this action (for example, "two down, three lobs; it is possible to start with a diagonal"), have never been shot down when the "9 points" task is never shot down. If such a rule was not given or not formulated by him, the details of the 9-point solutions were very soon "forgotten", only the principle of "break" 8 remained in active memory. After some time (a few days, and maybe hours and even minutes), repeating the solution of the task, the subject can no longer use the previously found solution, it develops this decision again, guided by the general principle - "break out!", And again makes it possible to concretize it The principle is applied to the situation "9 points" (it is for this reason that the subjects under the first group and it turns out to be necessary in re-deciding the problem of "9 points" to do an average of 8 attempts). In the same case, if in the preceding decision of the problem, the method of action was formulated verbally, even after a week (and maybe through significantly long time) the solution of the problem does not cause any difficulty - it is not produced again, but is reproduced in the finished form.

Thus, the process of the development of the principle of solving the problem performed as a complex, controversial, discrete process that is constantly united by the interaction of the subject with the object and, at the same time, the guide of this interaction.

It should be noted that the rule is "three down, two lobs; Moznp to start and diagonally suggests and includes knowledge of the original principle "break!" And at the same time, the OIO contains a product of the specification of this principle in relation to the "9 points" task.

The creative element in solving those used in the experiments of mental problems is composed of elementary action - connecting two points in the shortest distance. The conditions for creative solutions occur when the relevant groups of points turned out to be allocated on the basis of knowledge purchased in solving previous tasks or by the same elementary techniques (gradually binding to certain structures). During the decision of the previous task, the signs necessary for solving, which further and united, giving a creative decision. However, the relationship between these signs, their unified structure has not yet been realized. This structure was realized when solving the subsequent stimulating task, which contributed to the transition of abstraction to a new, higher level.

The main quality characterizing such a stimulating task is its ability to convert a practical goal to theoretical.

Such a transformation involves the activity, the independence of the subject, it can be successfully implemented in the closest broader (promising) task, where the action of the preceding situation is acting as a link in resolving the subsequent. Such a circumstance with necessity leads to the fact that the result of the previous decision is now as an operation as a method of action. However, as a stimulating task, not only a promising situation can perform. The same task is to become stimulating, if necessary, finding various ways to solve it.

To some extent, the abstraction of the principle contributes to the automation of that method, which turns into principle. This is explained by the fact that the result of the decision of the previous task, speaking as a solution to the subsequent decision, must meet the requirements that are usually presented to objects playing the role of funds. Any means need to act as an instrument, not constantly engaged in the analysis of how this instrument is created. The use of funds should not be due to the need to pay attention to its structure; The subject should be used by the ready-made product of the past decision, and not to produce constantly and again this product during solving a more complex task. In other words, the success of action in this case contributes to the monolithization of the action, the concentration of all efforts around one goal, which excludes the need to spray the activities in connection with the occurrence within its utility tasks. These utility tasks must be resolved before.

At the same time, receiving automation of the action of the preceding task is not the best way. It detects the effect only in very narrow transfer boundaries. A much greater effect is achieved in the case when the necessary method of action is verbalized.

In all cases, the success of the development of the principle of solving the problem is associated with the transition of a subject to the highest level of interaction with the object. The highest level of interaction, implementing first through the preceding, reorganizes it then according to its own characteristics.

It should be assumed that the change in the content of the forming principle is due to the reduction in the elements of the reflection of a by-product and due to the translation of some of these elements into the reflection category of the direct product.

So, the success of the formalization of intuitive effects is favored by the following, experimentally identified conditions: the inclusion of activities in the context of a broader task, in which the result of the previous action should perform already as an operation; formulation of the theoretical problem, i.e., where the goal is not to achieve a practical result, but in clarifying the method that such a result has already been obtained; For the success of formalization, the method of solving the previous task is advisable, without moving a certain limit, to bring to a certain extent of automation sufficient to act this method as a means, i.e. to operate as a holistic education. In all these cases, the optimal choice of the surrounding complexity of the situation is important.

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