Rice. 4. Connect nine dots with four lines

Everything ingenious is simple! Why doesn't everyone find a solution!? The problem is in the implicit (hidden, disguised) premise, which consists in the fact that the lines should rest on the vertices of the figure outlined by nine points. As soon as such restrictions are removed, explicitly declaring this to the subject, then the latter seems to have an epiphany, and the solution is found instantly ...

A similar implicit premise underpins the desire of many managers to cut costs. They proceed from the fact that the amount of income (sales volume) is much more difficult to manage than the amount of expenses, and they strive to minimize the latter. Not taking into account that some expenses are very important, so to speak, generating income, and the reduction of such expenses will inevitably lead to a drop in sales. On the other hand, an increase in profit-generating spending is likely to lead to outpacing revenue growth.

Eliyahu Goldratt describes this situation very well in his book "Goldratt Rules".

The approach to conflict resolution should be to try to eliminate the interfering initial premise, which neutralizes the conflict situation itself. Removing the conflict opens the way to the desired changes. We can focus on increasing the size of the pie, instead of fighting for a larger share in the process of sharing a small piece. This will be a solution in which everyone wins.

It must be initially borne in mind that in any relationship changes are possible, thanks to which each of the parties comes to meet their needs. It doesn't matter if there is such a possibility at the moment. It is important for any tension in a relationship to be sure that such a possibility exists. Seek her, not the other side's fault. If we allow ourselves to judge others, our emotions blind us. What are the chances of concentrating energy and time on the search for changes that will restore harmony? Insignificant.

Finding a solution that benefits both parties involves finding a premise to be eliminated. But finding it is not always easy. A win-win solution increases the size of the overall pie. The larger the pie, the larger the piece we can get. …when conflicts arise, the focus should be on finding a solution where both parties win. And given that subconsciously we always strive for our own victory, shouldn't we consciously look for a solution that will ensure the victory of the other side? Wouldn't such an approach increase the chances of our own success as well?

It is amazing how everything is interconnected - the statement that harmony exists in any relationship; an approach that benefits both parties; advice to start by looking for a large (or greater) interest of the second party; the ability to identify the biggest gains in solving hidden problems. All this complements each other, forming a single picture.

Let's summarize briefly:

The situation where the gain of one side turns into a loss of the other is not immutable.

If we move from a one-dimensional view to a two-dimensional one (or, moreover, to a multidimensional one), one can find options when both sides win

Since we function within the framework of various systems, and these systems have emergent properties, we should strive for a large number of dimensions for the manifestation of these properties.

There is some implicit premise underlying the one-dimensional win-lose view; it is necessary to open it and translate the situation into a (two-dimensional) win-win plane.

Related information:

1. IV. Learning new material. Despite the fact that the definition of a circle is not given to students, it is necessary to acquaint them with the property of the points of a circle

Non-standard in its reasoning, the problem of how to connect 9 dots with 4 lines makes you break stereotypes and turn on creativity.

How to correctly position the dots and pattern?

On a sheet of paper, it is better if it is in a box, you need to draw 9 dots. They should be arranged three in a row. The diagram will look like a square, in the center of which there is a dot, and in the middle of each of the sides there is also one. It is better if this pattern is placed away from the edges of the sheet. This placement of the square will be required in order to correctly solve the problem of how to connect 9 points with 4 lines.

The task

Requirements that must be taken into account:

Following these rules, you need to connect 9 points with 4 lines. Very often, after a couple of minutes of thinking about this drawing, a person begins to assert that this task does not have an answer.

The solution of the problem

The main thing is to forget everything that was taught in school. They give stereotypical ideas, which will only get in the way here.

The main reason that the task is about how to connect 9 dots with 4 lines, not unraveled in the following case: they end at the drawn points.

This is fundamentally wrong. The points are the ends of the segments, and the problem explicitly talks about lines. This is what you should definitely use.

You can start at any corner of the square. The main thing is the angle, which one specifically, it doesn’t matter. Let the points be marked on the left, moving to the right, and from above, moving down. That is, the first row contains 1, 2 and 3, the second consists of 4, 5 and 6, and the third is formed by 7, 8 and 9.

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

1. Lead the beam diagonally to points 5 and 9.
2. You need to stop at the last one - this is the end of the first line.
3. Then there are two ways, they are both equivalent and will lead to the same result. The first will go to the number 8, that is, to the left. The second - to the six or up. Let there be the last option.
4. The second line starts at point 9 and goes through 6 and 3. But it does not end at the last digit. It needs to be continued up for another segment, as if another point was drawn there. This will be the end of the second line.
5. Now again the diagonal, which will pass through the numbers 2 and 4. It is easy to guess that the second number is not the end of the third line. It must be continued, as it was with the second. Thus ended the third line.
6. It remains to draw the fourth through points 7 and 8, which should end in the number 9.

This task is completed and all conditions are met. To some, this figure resembles an umbrella, while others claim that it is an arrow.

If you write down a short plan of how to connect 9 points with 4 lines, you get the following: start at 1, continue at 5, turn at 9, draw at 6 and 3, extend to (0), turn to 2 and 4, continue to ( 0), turn to 7, 8 and 9. Here (0) are the ends of the segments that do not have numbers.

As a conclusion

Now you can still break your head over a more complex problem. There are already 16 points in it, located similarly to the considered task. And you need to connect them already with 6 lines.

If this task turned out to be too tough, then you can try to solve others, with the same requirements, but differing in the set of points and lines, from the following list:

• 25 points in the order of a square, like all subsequent ones, and 8 straight lines;
• 36 points on 10 lines that are not interrupted because the pen cannot be torn off the sheet;
• 49 dots connected by 12 lines.

MYSTERIES ABOUT NUMBERS.

Riddles about numbers and numbers. It would seem that the mysterious can be in numbers. Numbers have been used for thousands of years and have been studied far and wide. But the peculiarity of the human brain is that it needs miracles, secrets and riddles. And it is precisely with this that such an abundance of riddles about numbers is connected. Let's look at just a few of them.

Very interesting and educational

Live in a smart book
Cunning brothers.
Ten of them, but these brothers
They count everything in the world.
(Numbers)

The numbers stood up like a squad,
In a friendly numerical series.
First in order role
We will play the figure ... (0)

He looks like a bun
He is round and round.
Looks like a cat
If it folds into a ball.
(0)

The ball bounces through the pages.
He is looking for his sister
What looks like a ring -
Without beginning and end.
(0)

How old is a chicken in an egg
How many wings does a kitten have
How many numbers are in the alphabet
How many mountains the tiger will swallow
How many tons does a mouse weigh?
How many crows in a flock of fish
How many hares did the moth eat
Only the number knows...
(Zero)

This figure is similar
To the letter of the alphabet O.
But without other numbers
She doesn't mean anything.
(0)

Like a leafless branch
I am straight, dry, thin.
You met me often
In a student's diary.
(Unit)

She looks like a stalk
It stands respectfully, like a nobleman.
Straight, always straight
She goes after zero.
(1)

Don't score a goal so easily
There is a stake on the gate.
And you can't fight him in battle
This is a number ... (Unit)

How many suns behind the clouds
How many refills are in a fountain pen
How many noses does an elephant have
How many watches are on your hand?
How many legs does a fly agaric have
And attempts at the sapper,
Knows and is proud of himself
Digit-column...
(Unit)

She stands among the sheet.
One when the notebook is empty.
Sticking your nose up to the ceiling
She scolds the student.
And like a heron in the swamps
He is pecked at for his laziness.
At least she has one leg
She is slim, proud, strict.
Neither the crane nor the titmouse.
And just...
(Unit)

Sister with a sly nose
The account will open...
(Unit)

Flexibly bends the neck
And beautiful and slim
Deftly raises the tail,
What's the number? Digit... (2)

If you get this figure at school,
Your parents will scold you at home.
It is easy to compare with a swan,
Do all the kids know about this number?
(2)

The swan swims in a notebook,
So something is wrong.
If you are completely ignorant,
Get this number.

The sun is shining, the pond is blooming,
The swan floats on it
He barely swam closer -
It turned out to be a number ... (2)

How many ears on top
How many legs does a half frog have
How many mustaches does a catfish have
At the planet of the poles,
How many halves in total
In a pair of brand new shoes,
And the front paws of a lion
Only the number knows...
(2)

With a light stroke of the pen
A number has appeared ... (2)

What slides on a bright surface
Student's notebook
beautiful white swan
Turned red from shame
For a loafer, a rascal
Naughty boy?
What he is criticized for
And they deprive them of sweets at lunchtime.
With a light stroke of the pen
A number has appeared... (2)

I will circle the letter Z,
I'll take you to the numbers.
You take a closer look -
The result is a figure ... (3)

That's such a miracle! Come on, come on
You better look -
It looks like a letter
But also a number... (3)

This number is unusual.
Chubby, you know.
And it looks like the letter Z
What's the number? Guess!
(3)

How many months in winter
In summer, in autumn, in spring,
How many eyes does a traffic light have
Bases on the baseball field
The edges of a sports sword
And the stripes on our flag
Whatever anyone tells us
The figure knows the truth ... (3)

Guess this number!
She's a big know-ka.
You add one with two,
And you'll get a number... (3)

This number is simply amazing.
She has family everywhere.
Even in the alphabet
She has a twin sister.
(3)

Someone at night an old chair
Turned back down.
And now in our apartment
He became a number ... (4)

How many legs does a mongoose have
Petals in a cabbage flower
Fingers on a chicken leg
And on the back paw of a cat,
Tanya's hand with Petya
And all sides in the world
Yes, and the oceans in the world,
Knows the number ... (4)

Either a number, or a fork,
Or a fork in the road.
In a student's notebook
I know for sure - everyone is happy with her.
(4)

This figure is similar
On a beautiful sail!
Expanding wider
Number ... (4) At school, you should not be lazy:
draw, write, study,
Answer in class
And put in a diary ... (5)

If TWO flip
And take a close look
So and so look again,
Then we get a number ... (5)

How many fingers are on the hand
And a penny in a patch,
At the starfish rays,
The beaks of five rooks,
Blades near maple leaves
And the corners of the bastion
Tell about it all
The number will help us ... (5)

Mom looks with impatience
On the pages of the diary.
Waiting for the cherished assessment
With a naughty son.
But again, only fours.
No beauty...
(Fives)

The number looks like a hook
What's on the truck.
And looks like a double
If upside down.
(5)

And my friend Tonya
I had to sit on a pony,
And Tony with a pony
It became legs immediately... (6)

If the padlock
Lift up the proboscis,
Then we'll see here
Not a lock, but a number... (6)

How many letters does the dragon have
And zeros for a million
Various chess pieces
Wings of three white hens,
May beetle legs
And the sides of the chest.
If we can't count ourselves
The number will tell us ... (6)

Tsiferka, like a hippo,
She has a round belly
The neck deftly bends,
Follows the five.
(6)

I made a parade
I'm building numbers like a soldier!
And there is a clear order -
After five steps... (6)

Spit curls in the wind
And a stripe down the back.
(7)

We have a proverb
And she is known to all -
Cut only once
And you need to measure ... (7)

She looks like a braid
But he can't mow the grass -
Not sharpened at all
And the figure does not mow ... (7)

She looks like an ax
But he does not split firewood into the yard.
Like a scythe, but not quite
It's just a number... (7)

How many colors are in the rainbow
Days in the week for whales.
Gnomes at Snow White,
Twin brothers at the pawn
Notes that even children know
And all the wonders in the world
deal with it all
The number will help us ... (7)

What a heron, everyone knows
He steps behind the six.
Yes, everyone knows her
This heron is a number... (7)

I am a number less than ten.
I'm easy to find.
But if you order the letter "I" to stand next to you,
I am everything: father, and you, and grandfather, and mother ... (7)

Two circles stood side by side -
What are these amazing glasses?
Turn it upside down
They are the same!
(8)

Two rings on top of each other
Deleted two good fellows.
In their numerical series we will ask
And get a number... (8)

How many winds on the sea
And the hooves of two donkeys,
Octopus tentacles,
And the fangs of a pair of Great Danes?
How many legs does a spider have
Spider-cross?
If we ask about it
The number will answer us ... (8)

How many legs do you have? -
We will quietly ask the octopus.
He can't answer us.
But even so we know - ... (8)

The figure looks like a toy -
Roly-poly rattle.
Don't hit her on the ground.
Everyone understands - this is... (8)

This number is a secret.
Both winter and hot summer
You can barely distinguish
Where are her legs, head.
(8)

Oh what a comma
There is a big one on the sheet!
You can even measure it
It's just a number... (9)

That's the number, take a look
She has a focus
You turn it over
And get the number six!
(9)

How many pirates in a dozen
If three have gone somewhere,
Months of the year without summer
Nonet performers,
Lives of a stray cat
And in a dozen flies without midges?
Don't look for the answer anywhere, because
The number has an answer ... (9)

The number six is ​​upside down
Turned into a new number!
(9)

This number is so tricky
You turn it over
And miracles begin
You will see the number six!
(9)

Zero, stand behind the unit,
For my own sister.
Only when you are together
They will call you... (10)

How many stars are in the sky at night
How many bread crumbs are in bread
How many drops are in the rain
How many fish live in the water
How many legs does a polypod have?
Very very very…
(many)