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Hello! I’m new to the AoPS forums, so apologies in advance if I’ve posted in the wrong place 
Here’s an interesting counting problem I’ve come up with that’s been bothering me for a while:
Say we have a tower of cups such that the top row has one cup, the row directly under that has two cups, the row under that has 3 cups, and so on such that every cup is supported by exactly two cups in the row under it. Each cup in the tower is labelled with a number or letter so that every cup is distinct. If a cup tower has
rows, how many different ways can we pick up all the cups, starting at the top? (Assume that you can’t pick up a cup that’s under another cup, the tower would fall over.)

Here’s an interesting counting problem I’ve come up with that’s been bothering me for a while:
Say we have a tower of cups such that the top row has one cup, the row directly under that has two cups, the row under that has 3 cups, and so on such that every cup is supported by exactly two cups in the row under it. Each cup in the tower is labelled with a number or letter so that every cup is distinct. If a cup tower has

This post has been edited 1 time. Last edited by pzzd, Jun 3, 2025, 7:14 PM
Reason: spelling mistake
Reason: spelling mistake