7th Puzzle Answer
Roger Penrose would have liked this one (then again maybe he invented it). The answer is only 30 dominoes can be fitted. No matter how hard you try it is impossible to fit 31.
The reason why is obvious once you think colours. Each dominoe, no matter where they are always cover a white AND a black square. On the other hand, the squares on a chess board along a diagonal are all the same colour (either all white or all black) so the two squares which were cut off are the same colour. The reduced board now consists of 30 white and black pairs of squares plus 2 squares of identical colour. Since only white and black pairs can be covered by dominoes, only 30 dominoes can now fit.
(Roger Penrose is the famous Cambridge mathematician who, among other things, was interested in colour mapping, including the famous 5 colour theorem which states that only 5 colours are required for map makers so that the colours of all adjacent countries are different. Something that map makers knew very well but nobody had proven.)