Mark Gritter (markgritter) wrote,
Mark Gritter
markgritter

Classic Math Geek Sunday

...will not become a regular feature, unlike timprov's classic photos.

But I did get a question from a high school student about a calculation I'd done about the game Tonk, five years ago on 2+2: http://forumserver.twoplustwo.com/21/draw-other-poker/profitably-dropping-tonk-315444/

I didn't mention the technique in the forum post, but I used a generating function to count the possibilities. If you multiply (1+yxk) together for each value of "k" in the deck (as a multiset, not a set), you get a polynomial whose coefficient on yixj is the number of ways to make a "j"-point hand with "i" cards.

Back in 2008 there wasn't a readily available symbolic manipulation tool available on the web, so I hacked some python together to do the math. Today there's Wolfram Alpha, whose limited-duration (free) computation is powerful enough to give:

simplify | Coefficient[(1+y x)^4 (1+y x^2)^4 (1+y x^3)^4 (1+y x^4)^4 (1+y x^5)^4 (1+y x^6)^4 (1+y x^7)^4 (1+y x^8)^4 (1+y x^9)^4 (1+y x^10)^16, y, 5] as

4368 x^50+7280 x^49+10640 x^48+16720 x^47+22496 x^46+31216 x^45+40436 x^44+52556 x^43+65532 x^42+82176 x^41+92548 x^40+105176 x^39+116832 x^38+127484 x^37+136344 x^36+143676 x^35+147784 x^34+149268 x^33+146936 x^32+140224 x^31+134052 x^30+125188 x^29+115520 x^28+103808 x^27+92416 x^26+79416 x^25+67600 x^24+55712 x^23+45584 x^22+36708 x^21+28948 x^20+22144 x^19+16520 x^18+11988 x^17+8344 x^16+5724 x^15+3784 x^14+2492 x^13+1552 x^12+920 x^11+484 x^10+240 x^9+92 x^8+28 x^7+4 x^6


which (fortunately) seem to be the numbers I came up with in October 2008.

I wonder how well Wolfram Alpha's freemium model is working for them. :)
Tags: combinatorics, games, geek, mathematics
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