Every so often we get hung up on important questions like these...
What's the most number of Sets you can have with 12 cards?
We thought about this one day after doing a SET® puzzle, where we were presented with 12 cards and asked to find the 6 Sets that were present. (We did.)
Our first thought was that this would make a nifty CGI script. Just hit "reload" on your browser and get a new and different SET® puzzle every time. So we did that, too.
Along the way, we started wondering, if you pick 12 SET® cards at random, how many Sets will you usually get? So we made a little program that would pick 12 cards and count the Sets, over and over again. Here are the results for a million trials:
| # of Sets | # of Times |
|---|---|
| 0 | 32430 |
| 1 | 145770 |
| 2 | 261464 |
| 3 | 272410 |
| 4 | 179171 |
| 5 | 79899 |
| 6 | 23333 |
| 7 | 4683 |
| 8 | 693 |
| 9 | 114 |
| 10 | 30 |
| 11 | 2 |
| 12 | 1 |
| 13+ | 0 |
We briefly wondered if 12 were maximal, but it turns out it isn't. After a little more thought (by our own brain this time, instead of the computer's), we figured out you can get at least 14. But we haven't proved that this is maximal. Can you find 12 cards with 15 or more Sets, or prove that it can't be done?