[ Processes and operations ]

12 fields are arranged in a circle: on four adjacent fields, there are four differently colored chips: red, yellow, green, and blue. In one move, any chip can be moved from the field it occupies, over four fields to the fifth (if it is free) in either of the two possible directions. After several moves, the chips are again on the same four fields. How could they have been rearranged?
Answer: KZHS, SHKZ, ZKHS, HZSK. Let's make copies of our 12 fields and arrange them in a circle in the following order: $1,6,11,4,9,2,7,12,5,10,3,8$. On the new circle, the tokens move simply to the adjacent field (to the right or left). Therefore, any move in which the tokens swap places represents a movement around the new circle in one direction. On the circle with four fields numbered 1, 4, 2, 3, a cyclic permutation occurs, so from the set KSHZ we get the sets SHKZ, HZKS, ZKSH. Returning to the original numbering $1,2,3,4$, we get the sets SHKZ, HKSZ, ZSHK.

Send a comment