The game of Oxflip is for one player and involves circular counters, which are white on one side and black on the other, placed in a grid. During a game, the counters are flipped over (changing between black and white side uppermost) following certain rules.

Given a particular size of grid and a set starting pattern of whites and blacks, the aim of the game is to reach a certain target pattern. Each “move” of the game is to flip over either a whole row or a whole column of counters (so one whole row or column has all its blacks swapped to whites and vice versa). For example, in a game played in a three-by-three square grid, if you are given the starting and target patterns

Pattern of 9 counters with 5 black counters being the middle counter and the four corner counters.

a sequence of three moves to achieve the target is:

Three moves, flipping over the first row of counters so that now 4 black counters remain, flipping over the middle column so that only the bottom row of counters is now black, and flipping the bottom row to leave all counters white.

There are many other sequences of moves which also have the same result.

  1. Consider the two-by-two version of the game with starting pattern

    Pattern of 4 counters with 2 black; the top-left and the bottom-right.

    Draw the eight different target patterns (including the starting pattern) that it is possible to obtain.

    What are the possible numbers of white counters that may be present in these target patterns?

  1. In the four-by-four version of the game, starting with pattern

    4 by 4 pattern where counting left to right along each row from top to bottom, the second, fourth, sixth, eighth, tenth, twelfth, fourteenth and sixteenth tiles are black.

    explain why it is impossible to reach a pattern with only one white counter.

    [Hint: don’t try to write out every possible combination of moves.]

  2. In the five-by-five game, explain why any sequence of moves which begins

    5 by 5 pattern where counting the same way as in the previous pattern, all odd-numbered tiles are black.

    and ends with an all-white pattern, must involve an odd number of moves. What is the least number of moves needed? Give reasons for your answer.