July 2011 - mathfun@mailman.xmission.com

[math-fun] Tiny C Code to generate Sierpinsky
by quad 19 Jul '11

19 Jul '11

main(c,r){for(r=32;r;)printf(++c>31?c=!r--,"\n":c<r?" ":~c&r?" `":" #");} Cheers! -Robert

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[math-fun] bridge and randomness
by Bernie Cosell 19 Jul '11

19 Jul '11

The thread on tennis and randomness reminded me of Persi Diaconis's two discoveries about bridge: 1) that the 'standard' method of shuffling wasn't very good and you needed to do additional shuffles to really randomize the deck. 2) bridge masters seemed to occasionally make counter-probabilistic plays [playing finesses and for suit splits, etc], but as a corollary to (1), he discovered that their play was actually correct for the *actual* probabilities [due to the inadequate shuffles]. I found it pretty impressive that the deviation from "true randomness" almost certainly wasn't all that much, but it was enough that the top bridge players could detect the deviation and took advantage of it. /bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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[math-fun] Non-isomorphic finite simple groups of the same order
by Dan Asimov 18 Jul '11

18 Jul '11

According to this website: < http://www.madore.org/~david/math/simplegroups.html >, the smallest order for which there are more than one isomorphism class of finite simple groups is 2160. Question: How many such orders are there? Infinitely many? --Dan Sometimes the brain has a mind of its own.

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[math-fun] OOPS Re: To Fred: Update
by Dan Asimov 17 Jul '11

17 Jul '11

Sorry everyone -- that message to Fred was meant for Fred. --Dan Sometimes the brain has a mind of its own.

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Re: [math-fun] Reverse then Add in base -1+i
by N. J. A. Sloane 17 Jul '11

17 Jul '11

Kerry, yes, that's interesting! Please submit appropraite sequences to the OEIS! Best regards Neil

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[math-fun] Reverse then Add in base -1+i
by Kerry Mitchell 17 Jul '11

17 Jul '11

I've been playing with the "reverse then add" algorithm for complex bases. I did it like this: 1. Begin with a decimal number, xd. 2. Convert xd to a binary string, x$. 3. Convert x$ to a complex number, xc, using the base -1+i (which can be used to represent all Gaussian integers, without needing a sign bit). 4. Reverse x$ to form y$. 5. Convert y$ to a complex number, yc, using the base -1+i. 6. Add xc and yc to form zc. 7. Convert zc to a binary string z$ (this is where I needed help). 8. z$ becomes the new x$. Repeat from step 3 until x$ is a palindrome. The first number that seems to not lead to a palindrome (within a few thousand steps) is 6 decimal, 110 binary, or -1-i. Here are the first 10 iterations: i complex binary decimal 0 -1-i 110 6 1 -1 11101 29 2 -5-i 10110 22 3 -2-i 11101011 235 4 -14-2i 1110100111000 7480 5 -26+5i 1110001101111 7279 6 -82+20i 1100100110101100 51628 7 -78-83i 1110011000111111 58943 8 -170-144i 1100110101111011100 420828 9 -98-139i 1000110010101111 36015 10 -209-239i 1111101001000000010 512514 Decimal 20 (binary 10110, complex -4-2i) also seems to not lead quickly to a palindrome. Its first 10 iterations are: i complex binary decimal 0 -4-2i 10100 20 1 -3-4i 11100001 225 2 -11-13i 11111011010 2010 3 -37+4i 1111110111101 8125 4 -107-23i 1111101110011110 64414 5 -69+74i 111010001110001001 238473 6 -329+441i 110011110000010000010 1695874 7 239+886i 10100101110110101001 679337 8 972+1560i 1110100101000001111001010000 244588112 9 -208-1889i 111010111010100100100000111 123554055 10 -396-6658i 111101010011100000011010100 128565460 If any of this is of interest, I'll submit relevant sequences (e.g., iteration counts or trajectories) to the OEIS. Kerry -- lkmitch(a)gmail.com www.kerrymitchellart.com

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[math-fun] Converting numbers with complex bases
by Kerry Mitchell 17 Jul '11

17 Jul '11

Hi, Using the digits {0, 1} and the complex base -1+i, it's straightforward to show that 1010 = 1+3i. Is there an easy way to go backward? That is, given a complex number in the form a+bi, recover its "binary" representation? I'm immediately interested in the case for bases -1+i and -1-i (digits {0, 1}), but also interested in the general case. Thanks for any help, Kerry Mitchell -- lkmitch(a)gmail.com www.kerrymitchellart.com

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[math-fun] To Fred: Update
by Dan Asimov 17 Jul '11

17 Jul '11

Dear Fred, Just wanted to let you know what I'm up to. After the various pressing chores, etc., that had accumulated over my month away, my monthly crossword editing job came (and went -- it's now over). In all of this I've been studying the 11-cell in my free time. Partly reading Coxeter's detailed paper, partly making a lot of little sketches. Don't know if I'll ever get to the point of being able to visualize it in one mental picture. But at least I have a good sense of what the neighborhood of a vertex looks like. So that when I return to your manuscript, I will feel that I have a good grip on what we're talking about -- so I can see what happens when one icosa/2 has the interiors of all its triangles removed. (Part of my visualization problem is that the 11 (icosa/2)'s fit together like the graph K_11, aka the 1-skeleton of the (say regular) 10-simplex. Which like any regular k-simplex (without its one k-dimensional cell) naturally sits in the (k-1)-sphere . . . so K_11 naturally sits in the 9-sphere. And I'm not that familiar with visualizing the 9-sphere! --Dan Sometimes the brain has a mind of its own.

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[math-fun] Elementary geometry super-theorems
by Fred lunnon 17 Jul '11

17 Jul '11

Several people --- Roger Penrose and Terry Tao for two --- have (independently?) had the idea of proving Euclid-style theorems via special cases of a single result about cubic or quartic curves: see http://terrytao.wordpress.com/2011/07/15/pappuss-theorem-and-elliptic-curve… Does anybody know of an available reference? Fred Lunnon

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Re: [math-fun] Guessing game
by Dave Dyer 17 Jul '11

17 Jul '11

There's an interesting game called "Raj" that is based on this idea. In the game, there is a set of prizes valued at -5 to 10, and each player starts with a set of integer cards valued 1-15. All players select and play a card simultaneously. High card wins positive prizes, low card wins negative prizes and duplicate cards cancel one another, leaving the competition to the non-duplicate players. There are a LOT of potential strategies, which theoretically could take into account the cards each player has remaining, but it's not humanly possible to keep track of 5x15 cards in a fast paced game. http://boardgamegeek.com/boardgame/175/raj

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