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coloring 1-8 with black or white, given rules, find each number's color

Source: Greece JBMO TST 2010 p4

April 29, 2019
combinatoricsColoring

Problem Statement

We color one of the numbers 1,...,81,...,8 with white or black according to the following rules: i) number 44 gets colored white and one at lest of the following numbers gets colored black ii) if two numbers a,ba,b are colored in a different color and a+b≤8a+b\le 8, then number a+ba+b gets colored black. iii) if two numbers a,ba,b are colored in a different color and a⋅b≤8a\cdot b\le 8, then number a⋅ba\cdot b gets colored white. If by those rules, all numbers get colored, find the color of each number.
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