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2^k grains of oat on a 8x8 chessboard, no that a knight eats is divisible by 3

Source: 1999 Estonia National Olympiad Final Round grade 11 p5

March 11, 2020
Chessboardcombinatoricspower of 2divisibledividesnumber theory

Problem Statement

On the squares a1,a2,...,a8a1, a2,... , a8 of a chessboard there are respectively 20,21,...,272^0, 2^1, ..., 2^7 grains of oat, on the squares b8,b7,...,b1b8, b7,..., b1 respectively 28,29,...,2152^8, 2^9, ..., 2^{15} grains of oat, on the squares c1,c2,...,c8c1, c2,..., c8 respectively 216,217,...,2232^{16}, 2^{17}, ..., 2^{23} grains of oat etc. (so there are 2632^{63} grains of oat on the square h1h1). A knight starts moving from some square and eats after each move all the grains of oat on the square to which it had jumped, but immediately after the knight leaves the square the same number of grains of oat reappear. With the last move the knight arrives to the same square from which it started moving. Prove that the number of grains of oat eaten by the knight is divisible by 33.
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