All my bags are packed, I'm ready to go

Discrete Mathematics Level 4

A subset \(S\) of \(\{1,2,\ldots,n\}\) is said to be packed if whenever \(i, j \in S\) the number \(\left\lfloor \frac{i+j}{2} \right\rfloor\) is also in \(S.\) Determine how many subsets of \(\{1,2,\ldots, 25\}\) are packed.

Details and assumptions

\(i\) and \(j\) need not be distinct. If \(i= j\) is in the set, then clearly so is \( \left\lfloor \frac{i+j}{2} \right\rfloor\).

The sets \(S\) and the empty set clearly satisfy the conditions of the question, and should be included in your count.


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