What is the most discs you can flip in one move?

This is a favourite Reversi puzzle, and the answer people remember is often 19. That number is real, but it belongs to squares you can never play on. The rest of this page shows the 18-disc position, proves nothing higher is possible, and gives the counting method in code.

The position

After the 42 moves listed below, it is Black's turn. Black has 8 discs and White has 38. The corner a1 is empty, and three full lines run out from it: b1 to g1 along the top edge, a2 to a7 down the left edge, and b2 to g7 along the diagonal. All eighteen of those squares are white, and each line ends in a black disc on h1, a8 or h8.

abcdefgh

12345678

Black plays a1. All three lines close at once, and every disc in them turns over. Black goes from 8 discs to 27, and White drops from 38 to 20. For comparison, the best of Black's other legal moves here flips 4.

abcdefgh

12345678

The game that leads here, in standard notation:

c4 c5 f6 c3 b4 a3 b5 g7 a4 a5 a6 a7 c2 b2 b3 c1 d2 e1 d1 a2 f1 g1 d3 f3 e6 b1 g2 g3 h1 f4 c6 b7 a8 f7 f5 g5 g6 h7 h6 h4 h8 c7

It is a strange game, found by a computer search rather than played by people, and no strong player would let it happen. It is also entirely legal. The rules engine that runs this site replays every move and counts the flips each time this page is built, so the diagrams and numbers above cannot drift from the rules. If you want to read the moves yourself, notation explained.

Why nothing can flip more than 18

A move flips discs along straight lines, in up to eight directions: left, right, up, down and the four diagonals. In each direction, the most it can flip is every square between the new disc and the edge, minus one. The last square has to hold one of your own discs to close the line, so it cannot be one of the flipped discs.

From a corner, three directions point into the board, and each has seven squares before the edge. Six flips in each gives 6 + 6 + 6 = 18. The other five directions point off the board and flip nothing.

Do that count for every square and you get the grid below. Each number is the most a move on that square could ever flip.

The four centre squares top the grid at 19. From d4, for example, there are three squares to the left and four to the right, so at most 2 and 3 flips that way, and the same shape repeats up, down and along the diagonals: 2 + 3 + 2 + 3 + 2 + 2 + 2 + 3 = 19.

Some websites give 24 as the maximum. That cannot be right: no square on the grid above goes past 19, and the grid only counts the space available, before any question of whether the position can be reached.

How to count the flips for any move

To count by hand, stand on the empty square and look along each of the eight lines in turn.

1. Step to the next square in that direction.
2. While it holds an opposing disc, remember it and step again.
3. If you stop on one of your own discs, every disc you remembered flips. If you stop on an empty square or run off the board, nothing in that direction flips.
4. Add the eight directions together.

A total of zero means the square is not a legal move at all. Legal moves walks through the same test on the starting position.

For programmers, here is the same scan in plain JavaScript. It returns the list of squares that would flip, so its length is the count.

const DIRECTIONS = [
  [-1, -1], [-1, 0], [-1, 1],
  [0, -1],           [0, 1],
  [1, -1],  [1, 0],  [1, 1],
];
const onBoard = (r, c) => r >= 0 && r < 8 && c >= 0 && c < 8;

// board[row][col] is 'B', 'W' or null. Returns every square that flips.
function flipsFor(board, row, col, me) {
  if (board[row][col] !== null) return [];
  const them = me === 'B' ? 'W' : 'B';
  const flips = [];
  for (const [dr, dc] of DIRECTIONS) {
    const run = [];
    let r = row + dr;
    let c = col + dc;
    while (onBoard(r, c) && board[r][c] === them) {
      run.push([r, c]);
      r += dr;
      c += dc;
    }
    if (run.length > 0 && onBoard(r, c) && board[r][c] === me) {
      flips.push(...run);
    }
  }
  return flips;
}

To find the move that flips the most, call it for every empty square and keep the longest result. That is the whole of a "greedy" computer player, one that always takes the most discs, and it is also why greedy players are easy to beat. Serious engines do not loop like this. They store each colour as a 64-bit number and find every legal move with a few shifts and masks. How Reversi AI works covers that approach.

Bigger and smaller boards

The same counting works on any square board. The table assumes the four centre squares start filled, as in Othello, so they are listed separately.

From 10x10 upward the corners stop being the best squares. The ring of squares just outside the centre can reach further in all eight directions, so on a 20x20 board the ceiling is 65, not 54. These are ceilings from counting squares only. We have checked that the 8x8 figure can be reached in a legal game, and we have not checked the others. Other board sizes.

Should you go for the big flip?

Almost never before the last few moves. Discs in the middle of a game change hands constantly, and a player with lots of them usually has fewer safe moves left. The position above is a good example: White has 38 discs to Black's 8 and loses 18 of them to a single corner move. Corners matter because discs on them can never be flipped back. Corner strategy and mobility explain why.

Sources

• World Othello Federation, official rules, for the rule that every outflanked disc must be flipped.
• The position, the flip counts and the grid of ceilings are computed by this site's own rules engine when the page is built, and checked again by its test suite.

Try it yourself: set up a two player game and see how close you can get, or play the computer, which will not fall for it.