For an arbitrary set of numbers S, define S+S to be the set of all sums x+y where x and y are in S. Let A, B, C, D be sets of integers such that (A∪B) +(A∪B) =(C∪D) +(C∪D) . For example, if A−B−C−D−{1,2} then (A∪B) +(A∪B) =(C∪D) +(C∪D') ={1+1,1+2,2+1,2+2}={2,3,4}. Can you find four sets of integers A, B, C, D so that (A∪B) +(A∪B) =(C∪D) +(C∪D) =(A∪C) +(A∪C) =(B∪D) +(B∪D) ?
