Expert – Page 6 – Puzzle Prime

Policeman and Thief

Section: Brain Teasers Category: Mathematics

Difficulty: Expert

A town consists of 3 horizontal and 3 vertical roads, separated by 4 square blocks. A policeman and a thief are running along the roads with speeds of 21km/h and 10km/h respectively. Show that the policeman has a strategy ensuring he will eventually see the thief.

Remark: The policeman can see the thief if they are on the same road at some moment. He has no idea about his position at any time.

SOLUTION

A working strategy for the policeman would be to go to the center and to start encompassing the four blocks clock-wise one by one, in a clockwise manner.

Since the policeman is twice as fast as the thief if the thief is in the center of the town at some point, then there exists a moment in which the policeman is in the center, and the thief is not on the boundary, i.e. he gets shot.

Now, assuming the thief never visits the center, his angle with respect to the coordinate system defined by the two middle roads changes continuously. The angle of the policeman with respect to the same coordinate system can be defined to change continuously as well. Since the policeman needs less time to increase his angle with 360 degrees than the thief, there will be a moment when the two have the same angle. However, this implies that the policeman will be able to see the thief and shoot him.

Unknown Author August 16, 2017

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Knocked Off Piece

Section: Brain Teasers Category: Chess

Difficulty: Expert

The following position occurs in a real game, right after one of the pieces gets knocked off the board. What was the piece?

SOLUTION

It was a black knight. First, notice that the black pawns have moved 14 times diagonally and thus they have taken 14 pieces. Therefore the knocked off piece is black. Since it is impossible for both kings to be checked at the same time, the missing piece was positioned on a2. It couldn’t be a queen or a rook, because the white king would be checked both by it and the pawn on b3, which is impossible. Therefore the missing piece is either the black white-squared bishop or the black knight. However, the pawns on b7 and d7 haven’t been moved the entire game and then the black white-squared bishop hasn’t either. Thus we conclude that the knocked off piece is a black knight.

Unknown Author July 19, 2017

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15 Puzzle

Section: Brain Teasers Category: Mathematics

Difficulty: Expert

On the picture, you can see the famous “15 Puzzle”. The rules are simple – you can slide any of the 15 squares to the empty spot if it neighbors with it. The question is: if the squares with numbers 14 and 15 are exchanged, can you solve the puzzle, i.e. can you bring it to the state shown on the picture?

SOLUTION

No, you can’t. In order to see this, at each moment count the number of pairs of little squares, which are wrongly ordered. For example, if the numbers on the first row are 7, 2, 12 and 5 in this order, then 7 and 2, 7 and 5, and 12 and 5 are wrongly ordered. Notice that after every move you make, the number of wrongly ordered pairs changes with an odd number – ± 3 or ± 1. If you want to go from the state in which squares 14 and 15 are exchanged to the solved state on the picture, you must make an even number of moves and therefore you would change the number wrongly ordered pairs by an even number. However, the number of wrongly ordered squares in the starting state is 1, whereas in the ending state is 0, which yields a contradiction.

Unknown Author June 10, 2017

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