Precious Necklace

Section: Brain Teasers Category: Insight
Difficulty:

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.

unknown-author
Unknown Author December 25, 2018
2 Comments

Free Rice

Section: Article Category: Discovery

What is better than solving problems and learning new things? Doing exactly that, while fighting the hunger in Africa! “Free Rice” is a website, on which you choose a topic (vocabulary, science, math, etc.), and every time you answer correctly to a question, you contribute 10 grains of rice to people who need them. By clicking the banner below, you will be redirected to freerice.com. Just make sure to disable your ad-blockers, since hiding the (non-intrusive) ads may prevent you from accumulating donations.

puzzle-prime
Puzzle Prime December 4, 2018
0 Comments

Brick in the Wall

Section: Brain Teasers Category: Mathematics
Difficulty:

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: 

  1. wall 5×6 with 15 bricks;
  2. wall 6×6 with 18 bricks?

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.

unknown-author
Unknown Author December 1, 2018
0 Comments

Integer Dimensions

Section: Brain Teasers Category: Mathematics
Difficulty:

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.

unknown-author
Unknown Author November 28, 2018
0 Comments

Spades and Bridge

Section: Brain Teasers Category: Insight
Difficulty:

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.

unknown-author
Unknown Author November 25, 2018
1 Comment

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