I happened on the most interesting deck of cards the other day. Each card in the deck shares one, and only one, symbol with every other card. The game entails a race to see who can find the shared symbol on any two cards first. It got me thinking about how they are made.
So, as a puzzler, I asked Z to think about the problem. How many unique symbols does a deck of ten cards have to have for each card to share one and only one symbol with every other card in the set? I said, don't worry about solving this immediately - but, I'll be interested to hear if you can solve it.
Without a pause, he starting talking. Here are his words, as closely as I can recall...
"Oh, I've solved this before. I was thinking about this once while we were driving somewhere. The problem isn't how many symbols... it's the summing them up. You see, for ten cards, you'd need nine plus eight plus seven and so on to one symbols. The tenth card would share one of nine symbols with each card. Then, the ninth card would share eight unique symbols with the remaining cards and so on. It's really the summing that is a problem. Oh, wait, you don't have to sum them. Nine plus one is ten, eight plus two is ten, and so on until we get to five. That's, uh, four times ten, leaving the five. So, you need 45 symbols for ten cards."
I can't decide which is more impressive... Is it that he has thought about this while we were driving his brother to practice? Is it that he could just spew the answer off the top of his head? Is it that he found summation harder than the problem? That he found the alternate work-around from summation? Or, is it that he gave me the answer in about thirty seconds of talking it through out loud?
Now, when he doesn't chit-chat with me while we drive around town, I'll be wondering what algorithms are kicking around in his head.