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# Prove that however one selects 55 integers 1 ≤ x1 < x2 < x3 < ... < x55 ≤ 100, there will be some two that differ by 9, some two that differ by 10, a pair that differ by 12, and a pair that differ

Prove that however one selects 55 integers 1 ≤ x1 < x2 < x3 < ... < x55 ≤ 100, there will be some two that differ by 9, some two that differ by 10, a pair that differ by 12, and a pair that differ by 13. surprisingly, there need not be a pair of numbers that differ by 11.