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I've been trying to find an optimal algorithm for a problem inspired by the game 'triple town'. The game goes as such: You place objects in a grid, and each time you make a set of three, they condense into one object of higher level at the position of the last object placed . Furthermore, if you place three of these b objects together they again compress to form an even higher level object. Note: in these diagrams the level of an object is expressed as a i , b i , and c i and the subscript denotes the objects number in the set of three. To simplify things, I am only considering when every object you have to place is of the lowest level. Now my questions are: 1: Is there an algorithm to determine the least amount of grid area needed to make an object of level x, given x? For example, for level a you need 1x1, for level b you need 1x3, for level c you need 1x5. 2: Given the dimensions of a grid, can we find the highest level and number of objects achievable? For example, for a 2x2 you can get 2 level 'a's and 2 level 'b's 3: Is there an algorithm to find the optimal order and position of objects to get the highest possible level, given a fixed grid? For example, for a 2x2 you can get (1,1),(1,2),(2,2) 4: Given a position of an intended level x object, what set of moves minimize the amount of space needed to make this object? 5:What are the optimal complexities of these algorithms? Update: One thing that I think is prominent in the finding of solutions is that the getting an item of level x can't be done in any arbitrary p

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