Sunday, May 29, 2016

A 15-year-old question

Yesterday I asked a question on stackexchange's mathematics subsite. It was about a problem I thought up 15 years ago, and I got the basic stuff done, and I got stuck along the way. I wouldn't have remembered it if I hadn't just signed into one site and poked around.

I wondered when you could make a big n-sided equilateral triangle of pennies together from groups of 3 pennies. I suspected the answer was 12x+(0,2,9,11) and proved it worked for them but couldn't find a parity argument for 3/5/6/8. Maybe if x was large enough, there was something.

It was fun to mess with and I even wrote a brute force script to try things out. But I put the problem aside after a few unsuccesssful googles (I got the puzzle where you have 10 pennies and need to flip the triangle moving 3, and so forth) I figured why not.

And got an answer in an hour. Not only an answer, but one that pointed me to John Conway's work. I'd read about him and how important he was, etc., but my eyes glossed over. Here, though, there was a paper based on his work that solved a neat problem I'd wondered about.

I haven't processed it totally. But it's nice to know my intuition was right even if my technique never stood a chance. The basic idea is, you can build the triangle with 3-rows of pennies or triangles of pennies. If you build it with an odd number of 3-rows one way, that is the only way to build it, and you can't build it from triangles. But with an even number, you can. I'd had that idea but had no way to prove it because standard tilings like the checkerboard problem fell short.

There's a small moral here about asking questions and not waiting too long. I can box myself in with the "I should know this" and I've even become more comfortable asking those I should've known. Because in some ways it's not me asking the question but someone trollish from years ago. And once I asked the question I thought, boy I'll feel silly if someone responds quickly, because time wasted, and boy I'll feel silly if nobody responds.

So, on to the next question I've had for a while. Or maybe even one(s) I've forgotten.

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