From a Voucher Puzzle to Extremal Sums of Adjacent Products

Motivated by a self-referential puzzle, we study sequences of voucher price tags in which each choice multiplies the cost of the following one. We connect the puzzle setting to classical permutation statistics, introducing the $\mathit{\text{voucher}}\mathit{\text{cost}}$ alongside the related $\mathit{\text{pairwise}}$ and $\mathit{\text{loop}}$ costs. This perspective allows us to translate questions about budgeting into extremal problems on permutations. We review known results for permutations of $1,2,\dots ,n$ and extend them to arbitrary sets of distinct non-negative price tags.