KA-0028●●●5 points, difficulty 3 of 35 ptLevel 5–6not yet calibratedSeven cups all stand upside down. In one move you must turn over exactly two cups. Can all seven ever stand the right way up?LOG-05Parity argumentsalso NUM-03, LOG-09
KA-0034●●●5 points, difficulty 3 of 35 ptLevel 5–6has a figurenot yet calibratedA 4 by 4 board has two opposite corner squares removed, leaving 14 squares. Each domino covers exactly two squares that share an edge. Can 7 dominoes cover the board?LOG-07Coloring argumentsalso LOG-05, GEO-14
KA-0043●●●5 points, difficulty 3 of 35 ptLevel 5–6not yet calibratedA bag holds 5 black and 6 white stones. You repeatedly remove two stones: if they match you put a black one in, if they differ you put a white one in. What colour is the last stone?LOG-06Invariants and monovariantsalso LOG-05, NUM-03
KA-0094●●●5 points, difficulty 3 of 35 ptLevel 5–6not yet calibratedOne of nine identical-looking coins is slightly heavier. Using only a balance, what is the smallest number of weighings that is certain to find it?LOG-10Weighing and balance puzzlesalso CNT-04, LOG-11
KA-0095●●●5 points, difficulty 3 of 35 ptLevel 5–6not yet calibratedOne hundred apples are packed into twelve boxes. What is the largest number n for which you can always be sure that some box holds at least n apples?LOG-11The extremal principlealso CNT-17, NUM-15
KA-0097●●●5 points, difficulty 3 of 35 ptLevel 5–6not yet calibratedThe numbers 1 to 10 stand in a row. A move swaps two neighbours. After exactly 45 moves, can the row be back in its starting order?LOG-05Parity argumentsalso NUM-03, LOG-06
KA-0098●●●5 points, difficulty 3 of 35 ptLevel 5–6not yet calibratedThe numbers 1 to 8 are on a board. A move rubs out two of them and writes their difference, larger minus smaller. After seven moves one number is left. Can it be 1?LOG-06Invariants and monovariantsalso LOG-05, NUM-03
KA-0101●●●5 points, difficulty 3 of 35 ptLevel 5–6not yet calibratedCan a 10 by 10 board be covered exactly by T-shaped tiles of four squares each, with no gaps and no overlaps?LOG-07Coloring argumentsalso GEO-14, LOG-09