If you haven't thought of it already, the easiest way to solve this chess puzzle besides being good at Chess is to simply use a chess analyzer program to show you the correct moves in order to win so you can get the inventory size expansion.

There are two different methods I would recommend to doing this --

Method 1: If you somewhat understand the rules of chess already, I would just continue playing chess until it seems you are in a decent position and then use the analyzer tool to give you the last 5-10 turns of the game so it doesn't take too long.

Method 2: If you have no clue how to play Chess and still want that upgrade, then you will need to just suffer through going turn by turn in the analyzer to beat the computer. This will probably take a little while, but you will win eventually.

I recommend using this tool: https://chesshub.com/analysis/3557888/edit