In this problem, you need to find a set of n positive integers so that

• each pair has a different difference
• each pair has a different sum
• the largest integer is as small as possible.

For instance, if n is 3, the set could be {1, 2, 4}: their differences are 2 – 1 = 1, 4 – 1 = 3 and 4 – 2 = 2: all different and their sums are 2 + 1 = 3, 4 + 1 = 5 and 4 + 2 = 6: all different. The largest integer is 4. In this case, the largest integer could not be 3 as the set would then only be {1, 2 ,3} and the differences are 1, 1 and 2: not all different.

a        Show that the numbers (1, 2, 4, ..., 2^{n – 1}} all have different differences and sums.

b       Show that the largest integer must always be at least 1 + ½n(n – 1).

c        In the case n = 4, show that the solution to the problem has largest integer 7.

d       Solve the problem in the case n = 5.