# Cover 63 squares of a chess board

Can you find 3 similar geometrical figures (common shape but can be different sizes) A, B and C with the following property: If you remove any square from a 8x8 chess board, then the remaining area can be exactly covered with A, B and C.

How do A, B and C look like?

• Maybe add the constraint that the covers are not allowed to stick out over the sides of the board? – jochen Jan 28 at 9:08
• I'd be interested to see a proof, or counterproof, that this set of 3 figures is the only set of 3 figures (similar or otherwise) that has this interesting property with respect to an 8x8 chessboard. Anyone feel like tackling this bonus, or should I post a new puzzle? – Quuxplusone Jan 28 at 21:15

These should do it: For example, like so: Just orient the large block so that the missing piece is in the missing corner of the block, then do the same for each successively smaller block.

Just to show another example:

• Perfect explanation! – z100 Jan 27 at 16:25
• So it's basically a bpgerr (rot13)? – QBrute Jan 27 at 17:23
• Yes, although the 2-d version is called a dhnqgerr I think. – hdsdv Jan 27 at 17:58
• @Fogmeister yeah, I think that was the requirement for this puzzle :) – hdsdv Jan 29 at 0:58
• @hdsdv oh. 😂 I misread the question to begin with. – Fogmeister Jan 29 at 8:56

I think this is the answer.

Three L's. Each L is made up of 3 squares. The first L is 3 $$4\times 4$$ squares, the second is 3 $$2 \times 2$$, the third is 3 $$1\times 1$$ squares. Any one square of the chess board is in one of four quadrants, so you align the L so that the empty section matches with that quadrant. Repeat until only the taken square isn't covered.

This is a generalisation:

For any $$N \times N$$ board with one square missing and $$N=2^k$$ (some power of two), we can solve the puzzle with $$k$$ L-shaped pieces as described in the accepted answer. The first piece has width 1, the next has width 2, each piece doubling in width until we reach the largest piece with width $$2^{k-1}$$.