## Odd One

Which of the following objects below is the odd one?

**SOLUTION**

The second square is the odd one – each of the others has some unique feature (shape, boundary, color, size), except for it.

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Which of the following objects below is the odd one?

The second square is the odd one – each of the others has some unique feature (shape, boundary, color, size), except for it.

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A challenge is given to 100 people. A hat will be placed on each of their heads, and each hat will have an integer between 1 and 100 written on it (numbers can repeat). Every person will be able to see the hats of the other 99, but not his own. After that, everyone will have to guess what is the number on their hat (without others hearing). If at least one person guesses correctly, they will be awarded 1 million dollars. What strategy should the people come up with in order to optimize their chance of winning?

Label the people with numbers 1, 2, 3, … , 100. A strategy which ensures 100% success is the following:

Person X should sum the numbers on the hats of the other 99 people, then subtract the result from X, and take the residue modulo 100 of the answer (say “100” if the residue is 0).

This way if the sum of the numbers on all hats has residue R when divided by 100, then person R will guess correctly the number on his hat.

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You are sitting in a motionless car, which is tightly sealed, i.e. no open windows, holes in the car, etc. A helium balloon on a string is tied to the floor. If you start accelerating the car, is the balloon going to move back, forward, or stay in place?

The reason the balloon floats up in the air is that the helium has a lower density, so when gravity pulls the denser air around down, the balloon gets pushed up. Similarly, when the car accelerates, the air around gets drawn to the back of the car, which makes the helium balloon go forward.

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A princess is living in a palace which has 17 bedrooms, arranged in a line. There is a door between

You knock on doors:

2, 3,…, 15, 16, 16, 15,…, 3, 2.

This adds up to a total of 30 days exactly. If during the first 15 days you don’t find the princess, this means that every time you were knocking on an even door, she was in an odd room, and vice versa. Now it is easy to see that in the next 15 days you can’t miss her.

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What is 3/7 chicken, 2/3 cat, and a half goat?

The answer is CHICAGO. CHI is 3/7 of CHICKEN, CA is 2/3 of CAT, GO is a half of GOAT.

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Use exactly two threes (3) and two eights (8) to get the number 24. You can use multiplication (×), division (÷), addition (+), subtraction (-) signs, and brackets. You can not use any advanced arithmetic operations, such as exponential, factorial, etc.

The answer is shown below.

8 ÷ (3 – 8 ÷ 3) = 24

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Two friend mathematicians meet each after

– I have 3 daughters, the product of their ages is 36.

– I can’t figure out how old they are, can you tell me more?

– Sure, the sum of their ages is equal to the number of my house.

– I know your house number, but still can’t figure out the ages of your daughters.

– Also, my eldest daughter is called Monica.

– OK, now I know how old your daughters are.

What ages are the three daughters of the mathematician?

Using the first clue, we find that there are 8 possibilities:

(1, 1, 36) -> sum 38

(1, 2, 18) -> sum 21

(1, 3, 12) -> sum 16

(1, 4, 9) -> sum 14

(1, 6, 6) -> sum 13

(2, 2, 9) -> sum 13

(2, 3, 6) -> sum 11

(3, 3, 4) -> sum 10

Since the second mathematician couldn’t guess the ages even after the second clue, the sum has to be 13. Therefore the only possible options are (1, 6, 6) and (2, 2, 9). However, the third clue suggests that there is an “eldest” daughter and then the correct answer is 2, 2 and 9.

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100 prisoners are given the following challenge: They will be taken to a room and will be arranged in a column, such that each of them faces the backs of the prisoners in front. After that, black and red hats will be placed on their heads, and the prisoners will be asked one at a time what is the color of their hat, starting from the one at the back of the column. If a prisoner guesses his color correctly, he is spared; if not – he is executed. If every prisoner can see only the hats of the prisoners in front of him in the line, what strategy should they come up with, so that their losses are minimized?

There is a strategy which ensures that 99 prisoners are spared and there is 50% chance that one of them is executed. Clearly, one can not do better.

The strategy is as follows: The first prisoner (at the back of the line) counts the number of black hats worn by the prisoners in front. If the number is odd, he says “BLACK”. If the number is even, he says “RED”. Then, the second prisoner counts the black hats in front of him, figures out the color of his own hat, and answers the question.The third prisoner sees the number of black hats in front of him and uses this information, along with what the second prisoner’s hat is, to determine the color of his own hat. The prisoners continue in the same manner until all 99 prisoners in the front guess their hat colors correctly. The chance for survival of the first prisoner is 50%.

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Manifold is a brilliant puzzle invented by Jerome Morin-Drouin. Based on the origami principle, the goal of Manifold is to fold the printed paper several times, so that eventually you will end up with a 4 × 4 square which is white on one side and black on the other. The Manifolds here are provided by The Incredible Company and are part of their Manifold game which contains a total of 100 puzzles. Click the images, download them, print them, and solve the puzzles.

Source:

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A chess king starts on one cell of a chessboard and takes a non-intersecting tour, passing through each square once, and ending up on the initial square. Show that the king has made no more than 36 diagonal moves.

The king must visit the 28 perimeter squares in order; otherwise, he will create a portion of the board which is inaccessible for him. However, he can not travel from one square to a neighboring one using only diagonal moves. Therefore, he must make at least 28 horizontal/vertical moves and at most 64 – 28 = 36 diagonal moves.

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