# Looping Chips Puzzle

In this puzzle, 12 squares are arranged in a chain pattern with four chips of different colors covering the first four squares. You are allowed to drag a chip to a square 5 places from its current location, either clockwise or counterclockwise. The purpose of the puzzle is to have the chips in the originally occupied four squares but in the reverse order, or to determine that this is not possible.

You always play on the left board. The right one is very helpful and it may be worth your while to figure out its relationship to the left board.

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Explanation

## Explanation

The same logic that worked for 4 Knights puzzle works here as well. From every square their are only two possible moves. If we count squares according to a sequence of eligible moves then the right board represents these moves sequentially. On the right board the chips can't jump over each other. Therefore, their order on the right is unchangeable. The same is true with regard to any four fixed squares on the left.

The puzzle suggests several variations. There are three numeric parameters:

1. N, the number of squares.
2. M, the number of chips.
3. L, the length of a move.

Above, N = 12, M = 4, L = 5. The number of chips M is only important to limit L from below. For example, with M = 4 the least meaningful move should unable to skip over 3 occupied squares and onto the next one. So L could not be less than M. (The size of the puzzle, N obviously imposes an upper bound as well.) When N and L are coprime any chip may eventually reach any square. If gcd(N,L) > 1 the board on the right will have to be split into separate and non-intersecting loops. For example, for N = 12 and L = 4, there will be four loops:

1 > 5 > 9 > 1,
2 > 6 > 10 > 2,
3 > 7 > 11 > 3,
and 4 > 8 > 12 > 4

which makes it even more obvious that the chips can't swap places.

It follows then that however we change the three parameters of the puzzle, the puzzle is, in principle, insolvable. Perhaps, we can change the basic rules? Right now, the set of eligible moves depends on the location of a square from where a move is executed. What if we make the set of eligible moves depend on the chip that makes a move. Say, chips 1 and 2 are allowed to skip 4 places whereas chips 3 and 4 skip 5 places as before. Will this change anything?