Chapter 7 · Endgame

Parity: Even Regions and the Last Move

In the endgame, look at the empty squares as regions connected horizontally and vertically. Without passes, there is an even number of empty squares on Black's turn and an odd number on White's turn, so White plays the last move. That is why White usually holds parity.

The disc placed with the last move and the discs it flips cannot be flipped back, which is an advantage. The same holds inside a region: whoever enters an even region first gives away that region's last move if the opponent answers in the same region. Empty squares that touch only diagonally count as separate regions.

So White plays so that after the move, every region is even. Black must then enter an even region first, and White answers right away in the region Black entered, taking the last move of each region. From Black's side, the basic rule is to play odd regions first and enter even regions as late as possible (the parity rule).

Black to move. The 8 empty squares split into two 2-square regions and one 4-square region, all even. Whichever region Black enters, White answers in the same region and plays its last move; played to the end, White wins by 6 discs (Black A1 → White B1 → Black A7 → White A8 → Black F8 → White G7 → Black G8 → White H8).

Exercise

White to move. Find the move that enters the odd region so that every region is even afterwards.

Exercise

White to move. Find the move that enters the odd region so that every region is even afterwards.

Exercise

White to move. Find the move that enters the odd region so that every region is even afterwards.

Exercise

White to move. Find the move that enters the odd region so that every region is even afterwards.

Exercise

White to move. Find the move that enters the odd region so that every region is even afterwards.

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