What is the secret in the pattern of this stained glass?
The image is a superposition of a blue shape and a yellow shape. The places where they coincide are colored in green (blue + yellow = green). The blue shape is consisting of horizontal stripes with lengths 3, 1, 4, 1, 5, 9, 2, 6, 5, representing the number pi, and the yellow shape is consisting of vertical stripes with lengths 4, 6, 6, 9, 2, 0, 1, 6, 1, representing the Feigenbaum constant.
A man must mail a precious necklace to his wife, but anything sent through the mail will be stolen unless it is sent in a padlocked box. A box can bear any number of padlocks, but neither of the spouses has the key to a lock owned by the other. How can the husband mail the necklace safely to his wife?
The man can put a lock on the box and send it to his wife. Then she can put her own lock and send it back. Once the man receives the box, he can remove his lock and send the box once again to his wife. When she gets it, she can finally unlock the box using her own key.
On the picture, you can see an example of a wall made of 2×1 bricks. On the wall, there are 2 cracks, which are straight lines passing through the whole wall from top to bottom and from left to right, without intersecting any bricks.
Can you make the following walls without any cracks:
The solution for a 5×6 wall is shown below. However, if the wall has dimensions 6×6, it is impossible to build it without any cracks. Indeed, assume the wall does not have any cracks. Therefore every line passing through it must intersect 2, 4, or 6 bricks. Since there are in total 10 lines passing through the wall and each brick is intersected by exactly one of them, the total number of bricks must be at least 10 x 2 = 20 > 18. This yields a contradiction.
A large rectangle is partitioned into smaller rectangles, each of which has integer length or integer width. Prove that the large rectangle also has integer length or integer width.
This problem can be solved using graph theory, but the most elegant solution is based on some basic calculus.
Place the big rectangle in the plane so that its sides are parallel to the X and Y axes. Now integrate the function f(x)=sin(πx)sin(πy) over the boundary of any small rectangle. Since at least one of its sides has integer length, the result will be 0. If you sum all integrals taken over the boundaries of the small rectangles and cancel the opposite terms, you will get that the integral of f(x) over the boundary of the large rectangle is also equal to 0. Therefore at least one of its sides has integer length.
Four years ago, Meg put a nail on a tree in order to mark her height. If the tree grows 10 inches per year, and currently the nail is 5 inches lower than Meg, how much has Meg grown over these four years?
Five inches. Trees grow at their tops.
You are playing a game of Bridge. Which probability is greater – that you and your partner do not have any spades, or that you and your partner have all the spades in the game?
You and your partner do not have any spades if and only if your opponents have all the spades. Therefore the probabilities are equal.
A person enters a pet shop and sees a beautiful parrot. The seller guarantees him that the bird repeats everything it hears, so the person buys it. However, when he goes back home and tests the bird, it turns out that it doesn’t say a word. The buyer goes to the shop to complain to the seller, but the seller argues he has not lied. How is this possible?
The parrot is deaf.
A Sheikh dies, leaving behind three sons, 17 camels, and the following order:
1. The oldest son shall inherit one in two camels.
2. The middle son shall inherit one in three camels.
3. The youngest son shall inherit one in nine camels.
Now the three sons do not know what to do. They ask an old friend of the family for advice, and he finds a solution. What does the friend propose?
The friend lent them one camel to make the camels 18. The first son took 9 camels, the second son took 6 camels, the third son took 2 camels. Then the friend took back his camel.
There are five pirates, one monkey, and lots of coconuts on an island. The pirates are supposed to share the coconuts on the next day, but while everybody is sleeping, the first pirate gives 1 coconut to the monkey, splits the remaining coconuts into 5 equal piles, and secretly keeps one of the piles for himself. Later, the second pirate does the same, then the third one, the fourth one, and the fifth one. On the morning, the pirates wake up and split all the remaining coconuts in five, leaving one last coconut for the monkey.
What is a possible number for the number of coconuts on the island?
Notice that if we find a certain number of coconuts which works, then we can add 56 and get a new one. Now imagine the pirates start with -4 coconuts, i.e. they have a total loan of 4 coconuts. Every time a pirate wakes up, he gives 1 coconut to the monkey, which makes the total loan 5 coconuts. Then the pirate keeps a loan of 1 coconut for himself and leaves -4 coconuts. Now we just add 56 coconuts to -4 to make the number positive and get 56 – 4 coconuts as a possible answer.
Researchers have discovered
Older students have bigger feet and perform better at spelling bee contests.
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