Prove that if 11 integers are selected from among {1, 2, ..., 20}, Then the selection includes two integers a and b such that a - b = 2. Hint: Consider the pairs (1,3),(2,4),(3,5),etc. Solution Consider the sets of pairs which have a difference of two between the numbers: (1,3)(2,4)(3,5)(4,6)(5,7)(6,8)(7,9)(8,10)(9,11)(10,12)(11,13)(12,14)(13,15)(14,16)(15,17)(16,18)( 17,19)(18,20) Now consider the sets of pairs when using each number only once: (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20) There are only 10 choices of pairs when you use each number only once, but you have 11 integers you need to select. Therefore, due to the pigeonhole principle, two of the numbers you pick must belong to the same set...thus two of the numbers will have a difference of 2..