The set {1, 2, 3, 4, 5} is divided into two parts. Show that one part must contain two numbers and their difference.
Solution
Suppose not. Let the two parts be A, B. wlog 2 ∈ A. Hence 1 ∈ B (or 2-1=1 in A) and 4 ∈ B (or 4-2=2 in A). Hence 5 ∈ A (or 5-1=4 in B) and 3 ∈ A (or 4-1=3) in B. But now 5-3=2 in A. Contradiction.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03