Introductory Olympiad level question for high school
Twelve squares are laid out in a circular pattern [as on the circumference of a circle]. Four different coloured chips, red, yellow, green, blue, are placed on four consecutive squares. A chip may be moved in either a clockwise or a counterclockwise direction over four other squares to a fifth square, provided that the fifth square is not occupied by a chip. After a certain number of moves the same four squares will again be occupied by chips. How many permutations (rearrangements) of the four chips are possible as a result of this process?
The answer is 4.
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1 solution
The plan is to number the squares consecutively from 1 to 12, starting, say, from the red chip, and continuing on through the chips and around the circle, and then to rearrange the squares by putting them in an order in which it is possible to move a chip from one square to the next. That is, after square 1 we place square 6 (since by the conditions of the problem it is possible for a chip to move from square 1 to square 6, and vice-versa), then after square 6 we place square 11 (since a chip may move from square 6 to 11), and after square 11 we place square 4, and so on. After this rearrangement we have the following order of squares: The chips (red, yellow, green, blue, designated R, Y, G, B) are shown adjacent to their original positions; R on square 1, Y on 2, G on 3, and B on 4. The rule by which the chips may now move is simple: A chip may move one square in either direction, provided that square is unoccupied. Clearly, the only way in which a chip can change places with another is for it to move around the rectangle in either direction, but now a chip can neither jump over nor occupy the position of another chip. Thus, if chip R moves to occupy square 4, then B must occupy square 2, Y must occupy square 3, and G must move to square 1. If R occupies square 2 then B must occupy square 3, Y must move to square 1, and G will occupy square 4. If R occupies square 3, then B occupies square 1, Y moves to 4 and G moves to square 2. Other than these three rearrangements, no arrangement differing from the initial one is possible.