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 [Figure][Figure] a sequences of three moves to achieve the target is: [Figure] There are many other sequences of moves which also have the same result. (i) Consider the two-by-two version of the game with starting pattern [Figure] Draw, in the blank patterns below, the eight different target patterns (including the starting pattern) that it is possible to obtain. [Figure] What are the possible numbers of white counters that may be present in these target patterns? (ii) In the four-by-four version of the game, starting with pattern [Figure] explain why it is impossible to reach a pattern with only one white counter. [0pt] [Hint: don't try to write out every possible combination of moves.] (iii) In the five-by-five game, explain why any sequence of moves which begins [Figure] 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.
\section*{7. For APPLICANTS IN COMPUTER SCIENCE ONLY.}
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\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-18_159_279_863_715}\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-18_159_307_863_1032}\\
a sequences of three moves to achieve the target is:\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-18_202_1237_1155_406}
There are many other sequences of moves which also have the same result.\\
(i) Consider the two-by-two version of the game with starting pattern\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-18_108_113_1576_968}
Draw, in the blank patterns below, the eight different target patterns (including the starting pattern) that it is possible to obtain.\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-18_246_633_1868_708}
What are the possible numbers of white counters that may be present in these target patterns?\\
(ii) In the four-by-four version of the game, starting with pattern\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-19_213_213_379_918}\\
explain why it is impossible to reach a pattern with only one white counter.\\[0pt]
[Hint: don't try to write out every possible combination of moves.]\\
(iii) In the five-by-five game, explain why any sequence of moves which begins\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-19_259_266_1361_890}\\
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.