Leave No Squares

How many matchsticks do you need to remove so that no squares of any size remain?

Nine matchsticks are enough, as seen from the solution below.

To see that eight matchsticks are not enough, first we consider the center 2 by 2 square. It must have at least one matchstick removed and we can assume without loss of generality that it is the top left one.

Then, we consider the top left and the bottom right 1 by 1 squares, as well as the top right and the bottom left 2 by 2 squares. Each of the former two must have at least one matchstick removed and each of the latter two must have at least three matchsticks removed.

Therefore, the total number of matchsticks that need to be removed is at least 1+1+1+3+3=9.

Chessboard Madness

You have unlimited number of knights, bishops, rooks and kings. What is the biggest number of pieces (any combination) you can place on a chessboard, so that no piece is attacked by another one?

If we put 32 knights on all black squares, then no two pieces will attack each other. Now let’s see that if we have more than 32 pieces, then there will be two which attack each other. Split the chessboard in 8 rectangular sectors of size 2×4. It is not hard to see that if we have more than 4 pieces in the same 2×4 sector, then 2 of them will attack each other. Therefore we can place at most 4 × 8 = 32 pieces on the chessboard.

Shark Attack

A man stands in the center of a circular field which is encompassed by a narrow ring of water. In the water there is a shark which is swimming four times as fast as the man is running. Can the man escape the field and get past the water to safety?

Yes, he can. Let the radius of the field is R and its center I. First the man should start running along a circle with center I and radius R/4. His angular speed will be bigger than the angular speed of the shark, so he can keep running until gets opposite to it with respect to I. Then he should dash away (in a straight line) towards the water. Since he will need to cover approximately 3R/4 distance and the shark will have to cover approximately 3.14R distance, the man will have enough time to escape.

The Monty Hall Show

You are in Monty Hall’s TV show where in the final round the host gives you the option to open one of three boxes and to receive the reward inside. Two of the boxes contain just a penny, while the third box contains $1.000.000. In order to make the game more exciting, after you pick your choice, the rules require the host to open one of the two remaining boxes, such that it contains a penny inside. After that he asks you whether you want to keep your chosen box or to switch it with the third remaining one. What should you do?

This is the so called “Monty Hall” problem. The answer is that in order to maximize your chances of winning $1.000.000, you should switch your box. The reason is that if initially you picked a box with a penny, then after switching you will get a box with $1.000.000. If initially you picked a box with $1.000.000, then after switching you will get a box with a penny. Since in the beginning the chance to get a penny is 2/3, then after switching your chance to get $1.000.000 is also 2/3. If you stay with your current box, then your chance to get $1.000.000 will be just 1/3.

Pinned Men

The following game is played under very specific rules – no pinned piece checks the opposite king. How can White mate Black in 2 moves?

First, White plays f3 and threatens mate with Qxe2. Indeed, blocking with the black rook on d4 will not help, because it will become pinned, which means that the rook on d6 will become unpinned, which will make the bishop on b6 pinned, and that will unpin the knight on c7, resulting in a mate. Below are listed all variations of the game.

1. … Rd5 2. Qxe2#
2. … Bxa5 2. Kc8#
3. … Bxc7 2. Nxc7#
4. … Bxe8 2. Kxe8#
5. … Qxe7+ 2. Kxe7#
6. … Rd2 2. Bxd2#
7. … Rxd6+ 2. Qxd6#