Inspired by AOTM: Day 1Discrete Mathematics Level 5
"An 8x8 chessboard has two of its opposite corners removed. For example, A1 and H8. Can this board be tiled completely by a sufficient number of 2x1 rectangles with no overlap?"
After thinking for a while, I said "yes" (and by thinking I mean guessing at a 50/50 answer).
Also after thinking for a while, Kishlaya said "no" (and by thinking I mean using Riemann sums, integrals, and eigenvalues).
Of course, the answer was no. So I decided to give the problem some actual thought and I came up with an over complicated proof using modular arithmetic.
This of course inspired me to make a new problem:
How many ways can you remove two tiles from an 8x8 chess board such that the board cannot be completely tiled by 31, 2x1 rectangles without overlap?