# Inspired by AOTM: Day 1

**Discrete 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?

**Your answer seems reasonable.**Find out if you're right!

**That seems reasonable.**Find out if you're right!

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