## Distinct Sums

Let \(S\) be a set of \(n\) different positive integers such that every pair of integers in \(S\) have a different sum. We are looking for the minimum value of the sum of all elements in \(S\). For a fixed integer \(n\), let \(M(n)\) denote the minimum possible sum of the elements of \(S\).

For example \(M(4)=11\), because the minimal set \(S\) with \(4\) elements is \(S = \{ 1,2,3,5 \}\) and \(1+2+3+5=11\).

Part A | What is \(M(8)\)? | |
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Part B | What is \(M(12)\)? |