It is known (see this other answerthis other answer) that an 8x8 board in which squares at opposite corners have been removed cannot be tiled with dominoes, as the removed squares are of the same "colour". But what if two squares of different colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and this is his proof:
(Imagine A and B are the squares removed.) The image is from Mathematical Gems I by Ross Honsberger.
It is known (see this other answer) that an 8x8 board in which squares at opposite corners have been removed cannot be tiled with dominoes, as the removed squares are of the same "colour". But what if two squares of different colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and this is his proof:
(Imagine A and B are the squares removed.) The image is from Mathematical Gems I by Ross Honsberger.
It is known (see this other answer) that an 8x8 board in which squares at opposite corners have been removed cannot be tiled with dominoes, as the removed squares are of the same "colour". But what if two squares of different colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and this is his proof:
(Imagine A and B are the squares removed.) The image is from Mathematical Gems I by Ross Honsberger.
It is known (see Thisthis other answer shows) that an 8x8 board with oppositein which squares at opposite corners have been removed cannot be tiled with dominoes, as theythe removed squares are of the same "colour". But what if two squares of oppositedifferent colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and this is his proof.this is his proof:
(Imagine A and B are the squares removed.) The image is from Honsberger's Mathematical Gems I by Ross Honsberger.
This other answer shows that an 8x8 board with opposite squares removed cannot be tiled with dominoes, as they are of the same "colour". But what if two squares of opposite colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and this is his proof.
(Imagine A and B are the squares removed.) The image is from Honsberger's Mathematical Gems I.
It is known (see this other answer) that an 8x8 board in which squares at opposite corners have been removed cannot be tiled with dominoes, as the removed squares are of the same "colour". But what if two squares of different colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and this is his proof:
(Imagine A and B are the squares removed.) The image is from Mathematical Gems I by Ross Honsberger.
This other answerThis other answer shows that an 8x8 board with opposite squares removed cannot be tiled with dominoes, as they are of the same "colour". But what if two squares of opposite colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and this is his proof.
alt text http://shreevatsa.files.wordpress.com/2010/03/tiling-gomory.png
(Imagine A and B are the squares removed.) The image is from Honsberger's Mathematical Gems I.
This other answer shows that an 8x8 board with opposite squares removed cannot be tiled with dominoes, as they are of the same "colour". But what if two squares of opposite colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and this is his proof.
alt text http://shreevatsa.files.wordpress.com/2010/03/tiling-gomory.png
(Imagine A and B are the squares removed.) The image is from Honsberger's Mathematical Gems I.
This other answer shows that an 8x8 board with opposite squares removed cannot be tiled with dominoes, as they are of the same "colour". But what if two squares of opposite colours are removed? Ralph E. Gomory showed that it is always possible, no matter where the two removed squares are, and this is his proof.
(Imagine A and B are the squares removed.) The image is from Honsberger's Mathematical Gems I.
