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Here's a puzzle game. I call it Reverse the List of Integers

Here's a puzzle game. I call it Reverse the List of Integers. How it works is, you start with a list of positive integers, (e.g. [7, 5, 3]) and your goal is to make the same list, in reverse ([3, 5, 7]). You have two moves you can make: 1) Split an integer into two smaller integers. (e.g. [7, 5, 3] → [6, 1, 5, 3]) 2) Combine (add) two integers into a larger one. (e.g. reverse the last e.g.) There are two restrictions that seem natural for making this into an interesting game: 1) You can never make an integer greater than the largest integer in the original list. 2) You can never make a move that results in the same integer appearing in the list more than once. With these rules, even pretty simple lists can make non-trivial puzzles. (Try it with [7, 5, 3].) Some questions: 1) What are good algorithms, or even general strategies, for solving these? 2) For a given n, there must be some puzzle where n is the largest number in the list, and the number of moves required to solve the puzzle is maximized. What does the sequence of maximal required moves look like as a function of n? What do the "hardest" puzzles look like? Is there a way to determine either without using brute force?

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