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Making the reverses face upward

Source: Japan Mathematical Olympiad Finals 2005, Problem 1

October 31, 2005
linear algebramatrixmodular arithmeticcombinatorics proposedcombinatorics

Problem Statement

Double-faced coins are arranged with all the heads facing upward in the shape of 17×17.17\times17.At one operation, you turn over 55 consecutive coins in longitudinal or 55 ones in transversal or 55 ones in oblique at the same time.Now can you make all those reverses face upward when you repeat this operation?
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