Earlier today four chess puzzles were presented. They are repeated here along with their solutions. 1. Oddities. A chess tournament occurs with multiple participants. Not every player faces every other, and some play more games than others. Some players complete an odd number of games. Prove that the count of such players must be even. Solution: The overall games played by all must be even, as each game involves two participants. Adding odd and even numbers to reach an even total requires an even count of odd numbers, since an odd count of odds yields an odd sum. 2. L of a trip. A knight moves in an L shape: two squares one way and one perpendicular. From the bottom right corner of an 8×8 board, can a knight visit every square once and finish at the top left? Solution: No. Each knight move switches color. Visiting all squares needs 63 moves. Starting on white ends on black or reverse. Opposite corners share the same color, so starting and ending there is impossible. 3. Pawn return. On a standard chessboard setup, what is the minimum moves for a pawn to leave its start, promote to queen, and return to its original square? Players cooperate. Solution: 6. One sequence starts with the pawn on B2. White advances B2 to B4. Black moves A7 to A5. White captures with B4 to A5. Black advances B7 to B6. White captures with A5 to B6. Black moves knight B8 to A6. The pawn then advances up the B file, promotes, and returns to B2 in six total moves. 4. Four knights. Swap two pairs of knights on a shaped grid. Knights move one at a time. Exchange black and white positions. Solution: Possible moves are limited. Number positions top to bottom, left to right. White knights start at 1 and 5, black at 7 and 9. This reduces to a shunting task: shift black to 8 and 6, move white from 5 to 9, return black to 5 and 7, shift white from 9 to 3, reposition black to 6 and 8, move white from 1 to 9, place black at 1 and 5, and finish by moving white from 3 to 7.
