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Problem 2 of First round - Chessboard and lines

Source: VII International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

August 29, 2019
combinatoricsChessboard

Problem Statement

A cell is cut from a chessboard 8x88\, x\, 8, after which an open broken line was built, which vertices are the centers of the remaining cells. Each segment of the broken line has a length 17\sqrt{17} or 65\sqrt{65}. When is the number of such broken lines bigger – when the cut cell is (1,2)(1,2) or (3,6)(3,6)? (The rows and columns on the board are numerated consecutively from 1 to 8.)