Knot Theory?

Ring Puzzle brings a puzzle that is very interesting. A Japanese man who is above 70 yo could not solve this puzzle and he had been trying to do that for more than ten years. I think I am sure people from Knot Theory would have serious mathematical explanations to give here. It is worth watching!

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Possible Opening Screen for Playing with Maths

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                                                                                                         Source of the Problems

Kids and Wigs: Hint

Once more, just express it all by means of a mathematical system. A bit of Algebra and done!

Solution

Back to the problem

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Kids and Wigs: Solution

The problem is easily solved with a system formed by three equations with three variables.

We will call the number of wigs X.

Y will be the number of her kids.

Z will be the exact number of her grandchildren.

She said that the sum of these three numbers was 19. Thus:

X + Y + Z = 19  (A)

We know that the number X, the number of wigs, was equal to twice as much as the number of grandchildren less 1. 


Thus:

X = 2(Z-1)  (B)

The number of grandchildren is twice as much as the number of kids:

Z = 2Y  (C)

Solving this system, formed by A, B and C, we find out that

X = 10            Y = 3              Z = 6

Back to the problem

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Kids and Wigs

You have just entered a train. You hear a girl wearing a black costume asking a lady wearing a blue dress about her wigs.

-         But, in short, how many wigs do you have?

The lady, in a joking tone, shaking her earrings (which matched the dress), answered:

-         The number of my wigs, my dear, is nothing out of the ordinary. It is very reasonable. It is equal to twice as much as the number of my grandchildren less one.

-         Less one? How? – inquired the girl who was showing a lack of understanding in terms of calculations.

-         Less one, for sure – confirmed the lady.  From the total number of my grandchildren, take out one and multiply the result by two. You will then get exactly the number of my wigs. Do you understand?

-         And, lady, have you got many grandchildren? – asked the indiscreet girl in black.

-         Well – replied the lady, very sweetly.  My kids are already all married (and well married!) and each one of them has already given me two grandchildren. I noticed, the other day, by chance, that if I add the number of my wigs to the number of my kids and the number of my grandchildren, I would get 19. Do you understand?

We ask:

Can you, my friend, having read this curious and indiscreet conversation, calculate the number of wigs of the lovely lady in the blue dress?

Hint 

Solution

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                                                                                                       Source of the Problem

Solution: Anatole, Carla, and Border

Take X to be the price of one barrel. For the 59 that crossed the border, Anatole paid 5 barrels plus 40 dollars. Thus, he paid 5X + 40.

This was the tax due for the 59 barrels; to find out the value of the tax paid for one barrel we need to divide it by 59. So the tax paid by Anatole per barrel was

(5X+40)/59

For the 18 barrels that she crossed the border with, Carla paid 2 barrels and received 40 dollars in return. Thus, Carla paid 2X-40 for the 18 barrels.

This way, the tax paid by Carla per barrel was

(2X-40)/18

Since both fractions represent the value paid for one barrel, they must be equal. Thus, we have:

(5X+40)/59 = (2X-40)/18

∴    90X + 720 = 118X – 2,360 =>28X = 3,080 => X = 110

Thus, 110 dollars is the price of one barrel.

The total amount paid in tax for the 77 barrels was 7 barrels. Thus, the total amount of tax charged was 770 dollars. We know that 77 barrels crossed the border. Thus, each barrel had

770 / 77

of tax on them, that is, each barrel was taxed 10 dollars.

Back to the problem

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Hint: Anatole, Carla, and the Border

Try to express all by means of a mathematical system.

  Solution

  Back to the problem

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