A Voucher Puzzle

Vouchers are supposed to make shopping cheaper, right? Apparently, the shopkeeper in the following puzzle didn’t know. By the way, the puzzle is from Mathematical Puzzles and Curiosities, my book with Ivo David and Yogev Shpilman. Puzzle. A shop sells vouchers with price tags of 1, 2, 3, … dollars. There is exactly one voucher of each price. Buying the voucher tagged N makes your very next voucher cost N times its own price tag. Each multiplier applies only to the next purchase; the multipliers do not accumulate. Your first voucher costs exactly its price tag. You have 26 dollars. What is the largest number of vouchers you can buy? For example, buying the vouchers tagged 1, 2, 3, and 4, in that order, costs 1 + 1 · 2 + 2 · 3 + 3 · 4 = 21 dollars. I will leave the 26-dollar puzzle to you. Meanwhile, suppose you have already chosen a finite set of vouchers and must buy all of them. In what order should you buy them to spend as little as possible? And what order makes you spend as much…