Inspired by AOTM: Day 1

Discrete Mathematics Level 5

So we went to a lecture yesterday and Sir Bill Chen, a very accomplished quantitative analyst, gave a speech on his life and how mathematics has affected him. During his speech, he talked about one problem in particular which caught my mind.

"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?


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