discussions or where a linear representation is desirable in print, but may also be permissible as during-game transcription by either or both players. Tournament play using ordinary sets rather than a computer interface—where this cannot be an issue—have various ways of handling illegal moves and over- or underflipping (flips that should not be made but are or should be but are not). For example, permitting either player (perpetrator or its opponent) to make a correction going back some fixed number of moves (after which no remedy is available) is one procedure that has been used. Significant variants of the game, such as where the starting position differs from standard or the objective is to have the fewest pieces one's color at the end, are sometimes—but rarely—played. Brightwell Quotient Invented by the British mathematician and three times runner-up at the World Championship and five times British Champion Graham Brightwell, this is the tie-breaker that is now used in many tournaments including the W.O.C. If two players have the same number of points in the thirteen-round W.O.C. Swiss, the tie is resolved in favour of the player with the higher Brightwell Quotient. The Brightwell Quotient (BQ) is calculated as follows: A constant c is calculated. It is the integer nearest to (number of squares on the board) divided by (number of rounds in the tournament). If any of the player's opponents have withdrawn in the course of the tournament, or if a player has been paired against bye, ignore such games for the moment. Calculate the total number of disks scored by the player in all games not covered by step 2 and add c times the sum of points scored by all of the player's opponents, except those who have withdrawn. For each game against an opponent who has withdrawn, and each bye received, add half the number of squares on the board plus (c times the player's own tournament score) to the result calculated in step 3. The number resulting is the player's BQ. Computer opponents and research Good Othello computer programs play very strongly against human opponents. This is mostly due to difficulties in human look-ahead peculiar to Othello: The interchangeability of the disks and therefore apparent strategic meaninglessness (as opposed to chess pieces for example) makes an evaluation of different moves much harder. This can be demonstrated with blindfold games, as the memorization of the board demands much