My solution sounds different in my head, but is possibly the same on some level I don't understand. The set of toy counts contains positive and distinct integers. One of those integers, N, is the largest. In order to form N unique pairs, there must be N other distinct positive integers in the set. However, there are only N - 1 positive integers lower than N.
I feel like the statement of requirements does not match the limitations presumed by the solution, and that the whole problem is kind of busted.
For any number of children `n`, Santa can just produce `n+1` different toys for a trivial solution. If this isn't allowed because Santa must decide how many different toys to manufacture in advanced, then it's easy to see that any fixed number of `m` toys has a finite number of unique combinations, so cannot generalize to any number of children.
My solution sounds different in my head, but is possibly the same on some level I don't understand. The set of toy counts contains positive and distinct integers. One of those integers, N, is the largest. In order to form N unique pairs, there must be N other distinct positive integers in the set. However, there are only N - 1 positive integers lower than N.
I feel like the statement of requirements does not match the limitations presumed by the solution, and that the whole problem is kind of busted.
For any number of children `n`, Santa can just produce `n+1` different toys for a trivial solution. If this isn't allowed because Santa must decide how many different toys to manufacture in advanced, then it's easy to see that any fixed number of `m` toys has a finite number of unique combinations, so cannot generalize to any number of children.