Mark Gritter (markgritter) wrote,
Mark Gritter

Maximum size of a Naturalchemist game

Naturalchemist does not remove groups of three gold:

This means that we can bound the length of a game. Each gold must be made up of three shiny crystals, which must be made up of three blue crystals, etc. There are 11 different ingredients in the hierarchy. Thus each gold piece "represents" 3^11 green potions. The board is 49 squares large. So the first estimate for the maximum length of a game = 49 gold built from 3^11 green potions dropped two per move = 4.3 million moves.

But, even with this "drop only green" bound you cannot fill up all 49 squares with gold; there is not enough room to put three crystals together to make the final gold. The best you can do is fill 47 squares with gold that way. By getting non-green potions you can fill one of the remaining spots with gold but this leaves only one empty square which ends the game. So a better bound is 4.16 million moves.

Further thought will show that you need considerable space to work from green up to gold, so the number of spaces that can be filled using the maximum number of drops is considerably smaller.
Tags: games, geek, math
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