Reversi (Othello) — Strategy
VOG — Vinco Online Games
Parity
This extremely important notion is an ideal completion to our discussion of mobility. If neither player passes his turn during the game, there will be an even number of empty squares whenever black moves, and an odd number of empty squares whenever white moves. From this we could conclude that white will play the final move of the game and may possess a slight advantage, since the disc which he places and those which he flips are clearly stable. In Diag. 29, black must play g8 (the only possible move), white takes h8 and wins. But if it were white to move, he would play his disc in one of the two squares g8 or h8, black would play to the other square and win !
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This advantage can become even more important if white plays the last move in several even regions (regions with an even number of empty squares). Consider Diag. 30: there are four regions of two squares each. Black must play first into each one of these, and white may respond by playing to the same region. The game could continue for example with the moves g2-h1-g7-h8-b7-a8-b1-a1, where white wins 24-40. Parity gives an intrinsic advantage to white. However, black has a method to turn it to his advantage: if one of the players passes his turn, parity reverses; but if there is a second pass, the situation returns to its initial state. Black therefore would like to force an odd number of passes in the game. One effective method for black to gain parity is to force white to create an odd region into which he cannot play. In the situation of Diag. 31, white cannot play to the singleton region g8. Black should not play there either! Besides g8, there are an odd number of squares into which black may move. He must play in such a fashion that after each move, there are only even regions (excepting of course g8), that is, here, play g2. |
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White is then obliged to play first into each of the two even regions on the north-west and north-each. Parity ends with g2-h1-g1-a1-a2, white passes, and black finishes with the devastating move g8, winning 37-27. And if black had begun with g8? White would have responded g1 (leaving two even regions), and would have won 26-38 after the sequence g2-h1-a2-a1 ! |
