Today's puzzles are inspired by chess. If you have not yet watched the recent documentaries on Judit Polgár and Hans Niemann, they are highly recommended.
1. Oddities
A chess tournament is taking place with several participants. Not every player played against every other player, and some players may have played many more games than others. Some of the players played an odd number of games. Prove that the number of such players must be even.
2. L of a Trip
A knight in chess moves in an "L" pattern — two squares in one direction and one square in a perpendicular direction. Starting in the bottom right corner of a regular 8x8 chessboard, is it possible for a knight to visit every square on the chessboard exactly once and end up in the top left corner?
3. Pawn Return
Take a chessboard with the standard initial setup of pieces. What is the fewest number of moves needed for a pawn to leave its initial place, get promoted to a queen, and then return to its original position? Assume the two players are collaborating to achieve this, not that one is scuppering the other.
4. Four Knights
Show how to swap the two pairs of knights on the following strangely-shaped grid. The knights make one move at a time. You are trying to get the black knights to where the white knights are, and the white knights to where the black knights are. If you try to solve this problem using knights on a physical grid, you will get very confused. Try to think abstractly. With one simple insight, the problem is quickly solvable.
I will be back at 5pm UK with the solution. No spoilers, please! Instead, discuss chess.
Today's puzzles come from We Solve Problems, a charity that runs free maths circles for secondary school pupils (years 7 to 11) between September and May in more than a dozen cities across the UK. A maths circle is a social club for maths-loving children, led by postgraduates and PhD students. You can register now for the next school year.
I have been setting a puzzle here on alternate Mondays since 2015. I am always on the lookout for great puzzles. If you would like to suggest one, please email me.
