There are actually 55 cards and 57 objects altogether, but not all objects have the same number of occurrences. I thought it would have something to do with this, but I didn't want to review 50+ words each time. My idea was to have a folder full of pictures of vocabulary we had studied in the last week or so, just named "01, 02, 03..." I could then make the game on ppt with the pictures as links rather than importing them, then repeat the game a few weeks later with new vocabulary, without changing the "one and only one match" rule.
This site has a discussion of the maths, and one of the explanations arrives at n^2 - n + 1.
I derived this formula logically but not necessarily mathematically as follows:
I picked a random card and focused on a single image. Assuming eight images per card as are found in this game, this image can only be found 8 times, once on the card you're holding and 7 more times.
The same holds true for the next image. It can only appear 8 times if it to remain unique - once on the card you are holding and once over each of 7 more cards.
I noticed the trend. Each image appears once on the card you're holding and requires 7 more cards. So, you need the 1 card you are holding and 7 more per image. Mathematically, I guess that's: 1card+(7cards×8images). That's 1+(7×8) or 1+56=57.