Let a and b be positive integers, with a < b < 2a. Then, given more than half of the integers in the set {1, 2, ..., a + b}, some two of the given integers differ by a or by b.

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Copyright © 1996-2018 Alexander Bogomolny
Let a and b be positive integers, with a < b < 2a. Then, given more than half of the integers in the set {1, 2, ..., a + b}, some two of the given integers differ by a or by b.

The proof is by William A McWorter Jr.

Form the array below.

1 | 2 | 3 | ... | b | b+1 | b+2 | b+3 | ... | a+b |

a+1 | a+2 | a+3 | ... | a+b | 1 | 2 | 3 | ... | a |

There are a + b columns and each integer appears exactly twice in the array. Now, given more than half the integers in

The reason for the condition b < 2a is that you need only a little over a third of the integers if

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