March 2009 - mathfun@mailman.xmission.com

Re: [math-fun] muffin problem
by James Propp 04 Mar '09

04 Mar '09

David Wilson asks: >Do all these muffin results justify a paper? I think so. >Are they being collected anywhere? I've been forwarding them all to Alan Frank (who asked the question in the first place), and I keep all my outgoing correspondence, so I have a very convenient archive of all muffin-conversations. Michael Kleber wrote: >If there is something coherent to say on the subject, I'd like to suggest >(putting on my editor's hat) the Entertainments part of the Mathematical >Intelligencer... I had this venue in mind from the start. But I don't know how coherent the story is at this stage. David Moulton's proof about m,m+1 is great, but clearly it's just a start, and we don't even know yet whether S(km,kp) = S(m,p) for all k,m,p, which strikes me as a pretty basic thing not to know. Also, I'm worried that some of the arguments for specific pairs m,p are a bit tedious and wouldn't interest most readers once they've gleaned the general idea. At the same time, I've frequently imagined what an Intelligencer article about muffins would be like (though my imaginings have changed form as the group's discussions have evolved). One thing I've pretty consistently leaned towards is including a discussion of the process by which this body of results has been evolving. That is, I think the role of math-fun should be openly acknowledged, and that something about the nature of the group should be said, both because the social process is interesting and because there are probably a fair number of people who should be in math-fun but aren't because they don't know about it; I suspect that the overlap between this body of people and the readers of the Intelligencer is significant. (Or did Michael already "out" us in one of his columns?) Incidentally, I've been sending all the group's muffin-stuff to Alan Frank (who asked the question in the first place), and his response to the latest installment was >Thanks for keeping me posted on the muffin research. Personally, I'm just >going to get my own muffin and eat the whole thing. NOW he tells us... :-) Jim

4 3

[math-fun] MIST input example
by Huddleston, Scott 04 Mar '09

04 Mar '09

Hi, Could y'all show me an example of MIST input files you are using? Thanks, - Scott

1 1

[math-fun] 3/3/09: Math fans to celebrate Square Root Day
by Henry Baker 03 Mar '09

03 Mar '09

FYI -- The Associated Press 3/3/09: Math fans to celebrate Square Root Day March 2, 2009 REDWOOD CITY, Calif. --Dust off the slide rules and recharge the calculators. Square Root Day is upon us. The math-buffs' holiday, which only occurs nine times each century, falls on Tuesday -- 3/3/09 (for the mathematically challenged, three is the square root of nine). "These days are like calendar comets, you wait and wait and wait for them, then they brighten up your day -- and poof -- they're gone," said Ron Gordon, a Redwood City teacher who started a contest meant to get people excited about the event. The winner gets, of course, $339 for having the biggest Square Root Day event. Gordon's daughter even set up a Facebook page -- one of a half-dozen or so dedicated to the holiday -- and hundreds of people had signed up with plans to celebrate in some way. Celebrations are as varied: Some cut root vegetables into squares, others make food in the shape of a square root symbol. The last such day was five years ago, Feb. 2, 2004, which coincided with Groundhog Day. The next is seven years away, on April 4, 2016.

1 0

Re: [math-fun] Chocolate bar puzzle
by James Propp 01 Mar '09

01 Mar '09

Dan Asimov writes: >Speaking of muffinsectors, here is a problem about chocosectors: > >You are given a rectangular chocolate bar that is scored in the >usual way into KxL small squares. Repeat this step: Choose one >of the pieces (at first, there's only the whole bar) and break >it all the way along one of its scorings . . . until all the >pieces are the small squares. There is a fun way to present this problem so that it has the surprising property that mathematically sophisticated undergrads take LONGER to see the answer than less sophisticated undergrads. (It'd be fun to make a whole collection of problems that smart and/or knowledgeable people tend to do poorly on: an anti-intelligence or anti-aptitude test if you will.) Specifically, present the candy bar problem as a problem in combinatorial game theory, with players taking turns, and ask WHO WINS?, without revealing that the parity of the duration of the game, and indeed the duration of the game itself, isn't affected by the strategies the players adopt. (This is in fact the way I invented this problem twenty years ago: I like to call the game "Impartial Cutcake" to disguise the fact that there's no real game-theory going on at all.) Lead the students to prove that an even-by-even or even-by-odd bar is a win for the first player (strategy: the first player cuts the bar into two identical pieces), and a fancier argument to prove that an odd-by-odd bar is a win for the second player (imagine that the pieces are still in place, and copy whatever your opponent just did but rotated 180 degrees). Now challenge the students, working alone or in small groups, to figure out who wins under optimal play when the initial configuration is a pair of chocolate bars, one a-by-b and one c-by-d. It is vital to keep a straight face, and not to hint that there's lurking triviality. My experience is that students who know combinatorial game theory jump in and compute Grundy values and get really distracted from the heart of the matter. Students who don't know the theory of impartial games but who have some mathemtical taste and a fondness for proof take longer to find the slick solution than the others. Why? I think it's partly because they value the work that the group has already done in finding strategies for the players, so they emotionally resist making a discovery that would render the earlier arguments pointless. Also, they're used to the idea that in math, you keep building on what you already know, so they want to use what the group has already proved about single rectangles as a tool for analyzing the two-rectangle case. Also, they're inclined to use a trick that's worked before on new problems to see how far the trick can be pushed. The less sophisticated students have a kind of blankness that on the one hand makes it hard for them to get started on the problem but on the other hand keeps them from getting distracted once they do start. I did meet one sophisticate who put his sophistication to good use by imediately asking "What are the invariants here?", but this kind of reaction was not the norm. Jim Propp

6 6

[math-fun] common refinements of partitions (was: muffins)
by Michael Reid 01 Mar '09

01 Mar '09

i'm glad to see that there is still some interest in the muffin/common refinement of partitions problem. some time ago, veit elser gave the upper bound T(m, n) <= 1/(3m) for m < n < 3m/2 , and noted that some of the values he found achieved this bound. i looked at this range more closely and found that i could improve the upper bound as follows. here t denotes a positive integer. if ((3t+1)/(3t))m < n <= ((3t)/(3t-1))m , then T(m, n) <= 1/(3n) . if ((3t+2)/(3t+1))m < n <= ((3t+1)/(3t))m , then T(m, n) <= (t+1)/(2n) - t/(2m) . if ((3t+3)/(3t+2))m < n < ((3t+2)/(3t+1))m , then T(m, n) <= (t+1)/(4m) - t/(4n) . if ((3t+3)/(3t+2))m = n , then T(m, n) <= 1/(3m) . (here, as in previous messages, i am using T(m, n) = maximum c such that there is a common refinement of 1/m + 1/m + ... + 1/m = 1 and of 1/n + 1/n + ... + 1/n = 1 , with all parts >= c .) the first three cases are all improvements over veit's bound. in the fourth case, where there is no improvement, equality is always attained by veit's construction for T(3t+2, 3t+3) , and by obvious extension to T((3t+2)k, (3t+3)k) . however, the above shows that this is the exception, rather than the rule. i find the first case interesting, because i have not seen any cases (with m <= n ) where T(m, n) is strictly less than 1/(3n) . if we allow parts of size 1/(3n) , we may split some of the 1/n 's into 3 parts, and that seems to allow a lot of flexibility. in fact, the only occurrences of T(m, n) = 1/(3n) i've seen are those from the first case above. so i conjecture that equality always holds in the first case. on the other hand, the bounds in the other cases cannot always be attained. they can be attained if n = m+1 , by veit's and moulton's constructions. they can also be attained if n = m+2 by similar constructions. however, if n >= m + 3 , there are cases that require more careful analysis and for which the upper bound can be lowered; the first such case is (13, 16) . when n = m + 3 , it seems that the upper bound can be attained in all cases except when m = 4 mod 9 ; in that case the exact value is T(9k+4, 9k+7) = (9k+5)/(3(9k+4)(9k+7)) for k >= 1 . below are some data i calculated, all by pencil and paper, so subject to a small possibility of error. obviously this amount of data starts to become unwieldy; corrections and/or confirmations are welcome! i believe i have a proof for each of these values, but may have given short shrift to some subcases i thought couldn't give optimal partitions. mike T(2, 3) = 1/6 T(2, 4) = 1/4 T(3, 4) = 1/12 T(2, 5) = 1/10 T(3, 5) = 1/12 T(4, 5) = 3/40 T(2, 6) = 1/6 T(3, 6) = 1/6 T(4, 6) = 1/12 T(5, 6) = 1/15 T(2, 7) = 1/14 T(3, 7) = 5/84 T(4, 7) = 5/84 T(5, 7) = 1/21 T(6, 7) = 1/21 T(2, 8) = 1/8 T(3, 8) = 1/18 T(4, 8) = 1/8 T(5, 8) = 1/20 T(6, 8) = 1/24 T(7, 8) = 5/112 T(2, 9) = 1/18 T(3, 9) = 1/9 T(4, 9) = 7/144 T(5, 9) = 2/45 T(6, 9) = 1/18 T(7, 9) = 5/126 T(8, 9) = 1/24 T(2, 10) = 1/10 T(3, 10) = 2/45 T(4, 10) = 1/20 T(5, 10) = 1/10 T(6, 10) = 1/24 T(7, 10) = 1/30 T(8, 10) = 3/80 T(9, 10) = 1/30 T(2, 11) = 1/22 T(3, 11) = 1/24 T(4, 11) = 9/220 T(5, 11) = 13/330 T(6, 11) = 7/198 T(7, 11) = 1/28 T(8, 11) = 1/33 T(9, 11) = 13/396 T(10, 11) = 7/220 T(2, 12) = 1/12 T(3, 12) = 1/12 T(4, 12) = 1/12 T(5, 12) = 1/30 T(6, 12) = 1/12 T(7, 12) = 1/28 T(8, 12) = 1/24 T(9, 12) = 1/36 T(10, 12) = 1/30 T(11, 12) = 1/33 T(2, 13) = 1/26 T(3, 13) = 11/312 T(4, 13) = 11/312 T(5, 13) = 1/30 T(6, 13) = 1/30 T(7, 13) = 8/273 T(8, 13) = 1/32 T(9, 13) = 1/39 T(10, 13) = 7/260 T(11, 13) = 1/39 T(12, 13) = 1/39 T(2, 14) = 1/14 T(3, 14) = 1/30 T(4, 14) = 1/28 T(5, 14) = 11/350 T(6, 14) = 5/168 T(7, 14) = 1/14 T(8, 14) = 5/168 T(9, 14) = 1/36 T(10, 14) = 1/42 T(11, 14) = 2/77 T(12, 14) = 1/42 T(13, 14) = 9/364 T(2, 15) = 1/30 T(3, 15) = 1/15 T(4, 15) = 13/420 T(5, 15) = 1/15 T(6, 15) = 1/30 T(7, 15) = 1/35 T(8, 15) = 1/40 T(9, 15) = 1/36 T(10, 15) = 1/30 T(11, 15) = 1/45 T(12, 15) = 1/40 T(13, 15) = 3/130 T(14, 15) = 1/42 T(2, 16) = 1/16 T(3, 16) = 7/240 T(4, 16) = 1/16 T(5, 16) = 1/35 T(6, 16) = 1/36 T(7, 16) = 3/112 T(8, 16) = 1/16 T(9, 16) = 11/432 T(10, 16) = 1/40 T(11, 16) = 1/48 T(12, 16) = 1/48 T(13, 16) = 7/312 T(14, 16) = 5/224 T(15, 16) = 1/48 T(2, 17) = 1/34 T(3, 17) = 1/36 T(4, 17) = 15/544 T(5, 17) = 13/510 T(6, 17) = 13/510 T(7, 17) = 11/476 T(8, 17) = 1/40 T(9, 17) = 10/459 T(10, 17) = 2/85 T(11, 17) = 1/44 T(12, 17) = 1/51 T(13, 17) = 9/442 T(14, 17) = 5/238 T(15, 17) = 7/340 T(16, 17) = 11/544 T(2, 18) = 1/18 T(3, 18) = 1/18 T(4, 18) = 1/36 T(5, 18) = 1/40 T(6, 18) = 1/18 T(7, 18) = 1/42 T(8, 18) = 7/288 T(9, 18) = 1/18 T(10, 18) = 1/45 T(11, 18) = 1/44 T(12, 18) = 1/36 T(13, 18) = 1/54 T(14, 18) = 5/252 T(15, 18) = 1/45 T(16, 18) = 1/48 T(17, 18) = 1/51 T(2, 19) = 1/38 T(3, 19) = 17/684 T(4, 19) = 17/684 T(5, 19) = 16/665 T(6, 19) = 1/42 T(7, 19) = 25/1064 T(8, 19) = 13/608 T(9, 19) = 1/45 T(10, 19) = 11/570 T(11, 19) = 9/418 T(12, 19) = 1/48 T(13, 19) = 1/57 T(14, 19) = 1/57 T(15, 19) = 7/380 T(16, 19) = 1/57 T(17, 19) = 1/57 T(18, 19) = 1/57 T(2, 20) = 1/20 T(3, 20) = 1/42 T(4, 20) = 1/20 T(5, 20) = 1/20 T(6, 20) = 1/45 T(7, 20) = 3/140 T(8, 20) = 1/40 T(9, 20) = 1/45 T(10, 20) = 1/20 T(11, 20) = 13/660 T(12, 20) = 1/48 T(13, 20) = 1/52 T(14, 20) = 1/60 T(15, 20) = 1/60 T(16, 20) = 3/160 T(17, 20) = 1/60 T(18, 20) = 1/60 T(19, 20) = 13/760 T(2, 21) = 1/42 T(3, 21) = 1/21 T(4, 21) = 19/840 T(5, 21) = 1/45 T(6, 21) = 1/42 T(7, 21) = 1/21 T(8, 21) = 1/48 T(9, 21) = 5/252 T(10, 21) = 1/50 T(11, 21) = 4/231 T(12, 21) = 5/252 T(13, 21) = 1/52 T(14, 21) = 1/42 T(15, 21) = 1/63 T(16, 21) = 11/672 T(17, 21) = 2/119 T(18, 21) = 1/63 T(19, 21) = 13/798 T(20, 21) = 1/60 T(2, 22) = 1/22 T(3, 22) = 5/231 T(4, 22) = 1/44 T(5, 22) = 9/440 T(6, 22) = 1/48 T(7, 22) = 1/49 T(8, 22) = 9/440 T(9, 22) = 7/396 T(10, 22) = 13/660 T(11, 22) = 1/22 T(12, 22) = 7/396 T(13, 22) = 21/1144 T(14, 22) = 1/56 T(15, 22) = 1/66 T(16, 22) = 1/66 T(17, 22) = 3/187 T(18, 22) = 13/792 T(19, 22) = 13/836 T(20, 22) = 7/440 T(21, 22) = 1/66 T(2, 23) = 1/46 T(3, 23) = 1/48 T(4, 23) = 21/1012 T(5, 23) = 1/50 T(6, 23) = 19/966 T(7, 23) = 19/966 T(8, 23) = 17/920 T(9, 23) = 1/54 T(10, 23) = 17/920 T(11, 23) = 1/55 T(12, 23) = 13/828 T(13, 23) = 53/2990 T(14, 23) = 17/966 T(15, 23) = 1/60 T(16, 23) = 1/69 T(17, 23) = 1/69 T(18, 23) = 19/1242 T(19, 23) = 27/1748 T(20, 23) = 7/460 T(21, 23) = 29/1932 T(22, 23) = 15/1012 T(2, 24) = 1/24 T(3, 24) = 1/24 T(4, 24) = 1/24 T(5, 24) = 7/360 T(6, 24) = 1/24 T(7, 24) = 1/56 T(8, 24) = 1/24 T(9, 24) = 1/54 T(10, 24) = 1/60 T(11, 24) = 19/1056 T(12, 24) = 1/24 T(13, 24) = 5/312 T(14, 24) = 1/56 T(15, 24) = 1/60 T(16, 24) = 1/48 T(17, 24) = 1/72 T(18, 24) = 1/72 T(19, 24) = 9/608 T(20, 24) = 1/60 T(21, 24) = 5/336 T(22, 24) = 1/66 T(23, 24) = 1/69 T(2, 25) = 1/50 T(3, 25) = 23/1200 T(4, 25) = 23/1200 T(5, 25) = 1/25 T(6, 25) = 1/54 T(7, 25) = 1/56 T(8, 25) = 1/56 T(9, 25) = 4/225 T(10, 25) = 1/50 T(11, 25) = 19/1100 T(12, 25) = 1/60 T(13, 25) = 14/975 T(14, 25) = 17/1050 T(15, 25) = 1/60 T(16, 25) = 1/64 T(17, 25) = 1/75 T(18, 25) = 1/75 T(19, 25) = 13/950 T(20, 25) = 3/200 T(21, 25) = 1/75 T(22, 25) = 23/1650 T(23, 25) = 1/75 T(24, 25) = 1/75 T(2, 26) = 1/26 T(3, 26) = 1/54 T(4, 26) = 1/52 T(5, 26) = 1/55 T(6, 26) = 11/624 T(7, 26) = 1/56 T(8, 26) = 11/624 T(9, 26) = 19/1170 T(10, 26) = 1/60 T(11, 26) = 9/572 T(12, 26) = 1/60 T(13, 26) = 1/26 T(14, 26) = 4/273 T(15, 26) = 37/2340 T(16, 26) = 1/64 T(17, 26) = 1/68 T(18, 26) = 1/78 T(19, 26) = 1/78 T(20, 26) = 7/520 T(21, 26) = 11/819 T(22, 26) = 1/78 T(23, 26) = 4/299 T(24, 26) = 1/78 T(25, 26) = 17/1300 T(2, 27) = 1/54 T(3, 27) = 1/27 T(4, 27) = 25/1404 T(5, 27) = 23/1350 T(6, 27) = 1/54 T(7, 27) = 22/1323 T(8, 27) = 7/432 T(9, 27) = 1/27 T(10, 27) = 1/60 T(11, 27) = 17/1188 T(12, 27) = 7/432 T(13, 27) = 1/65 T(14, 27) = 5/378 T(15, 27) = 2/135 T(16, 27) = 13/864 T(17, 27) = 1/68 T(18, 27) = 1/54 T(19, 27) = 1/81 T(20, 27) = 1/81 T(21, 27) = 5/378 T(22, 27) = 4/297 T(23, 27) = 1/81 T(24, 27) = 1/72 T(25, 27) = 17/1350 T(26, 27) = 1/78 T(2, 28) = 1/28 T(3, 28) = 13/756 T(4, 28) = 1/28 T(5, 28) = 1/60 T(6, 28) = 1/60 T(7, 28) = 1/28 T(8, 28) = 1/56 T(9, 28) = 1/63 T(10, 28) = 11/700 T(11, 28) = 1/66 T(12, 28) = 5/336 T(13, 28) = 1/65 T(14, 28) = 1/28 T(15, 28) = 17/1260 T(16, 28) = 5/336 T(17, 28) = 1/68 T(18, 28) = 1/72 T(19, 28) = 1/84 T(20, 28) = 1/84 T(21, 28) = 1/84 T(22, 28) = 1/77 T(23, 28) = 33/2576 T(24, 28) = 1/84 T(25, 28) = 1/84 T(26, 28) = 9/728 T(27, 28) = 1/84 T(2, 29) = 1/58 T(3, 29) = 1/60 T(4, 29) = 27/1624 T(5, 29) = 26/1595 T(6, 29) = 25/1566 T(7, 29) = 1/63 T(8, 29) = 1/64 T(9, 29) = 41/2610 T(10, 29) = 21/1450 T(11, 29) = 1/66 T(12, 29) = 19/1392 T(13, 29) = 45/3016 T(14, 29) = 1/70 T(15, 29) = 16/1305 T(16, 29) = 19/1392 T(17, 29) = 27/1972 T(18, 29) = 1/72 T(19, 29) = 1/76 T(20, 29) = 1/87 T(21, 29) = 1/87 T(22, 29) = 15/1276 T(23, 29) = 49/4002 T(24, 29) = 17/1392 T(25, 29) = 17/1450 T(26, 29) = 1/87 T(27, 29) = 37/3132 T(28, 29) = 19/1624 T(2, 30) = 1/30 T(3, 30) = 1/30 T(4, 30) = 1/60 T(5, 30) = 1/30 T(6, 30) = 1/30 T(7, 30) = 13/840 T(8, 30) = 13/840 T(9, 30) = 2/135 T(10, 30) = 1/30 T(11, 30) = 1/66 T(12, 30) = 1/60 T(13, 30) = 11/780 T(14, 30) = 1/70 T(15, 30) = 1/30 T(16, 30) = 1/80 T(17, 30) = 7/510 T(18, 30) = 1/72 T(19, 30) = 1/76 T(20, 30) = 1/60 T(21, 30) = 1/90 T(22, 30) = 1/90 T(23, 30) = 4/345 T(24, 30) = 1/80 T(25, 30) = 1/75 T(26, 30) = 3/260 T(27, 30) = 1/90 T(28, 30) = 1/84 T(29, 30) = 1/87 T(2, 31) = 1/62 T(3, 31) = 29/1860 T(4, 31) = 29/1860 T(5, 31) = 1/65 T(6, 31) = 1/66 T(7, 31) = 25/1736 T(8, 31) = 25/1736 T(9, 31) = 23/1674 T(10, 31) = 1/70 T(11, 31) = 24/1705 T(12, 31) = 1/72 T(13, 31) = 21/1612 T(14, 31) = 3/217 T(15, 31) = 1/75 T(16, 31) = 17/1488 T(17, 31) = 20/1581 T(18, 31) = 11/837 T(19, 31) = 54/4123 T(20, 31) = 1/80 T(21, 31) = 1/93 T(22, 31) = 1/93 T(23, 31) = 1/93 T(24, 31) = 25/2232 T(25, 31) = 26/2325 T(26, 31) = 1/93 T(27, 31) = 37/3348 T(28, 31) = 19/1736 T(29, 31) = 1/93 T(30, 31) = 1/93 T(2, 32) = 1/32 T(3, 32) = 1/66 T(4, 32) = 1/32 T(5, 32) = 7/480 T(6, 32) = 7/480 T(7, 32) = 1/70 T(8, 32) = 1/32 T(9, 32) = 1/72 T(10, 32) = 1/70 T(11, 32) = 23/1760 T(12, 32) = 1/72 T(13, 32) = 5/416 T(14, 32) = 3/224 T(15, 32) = 1/75 T(16, 32) = 1/32 T(17, 32) = 19/1632 T(18, 32) = 11/864 T(19, 32) = 31/2432 T(20, 32) = 1/80 T(21, 32) = 1/84 T(22, 32) = 1/96 T(23, 32) = 1/96 T(24, 32) = 1/96 T(25, 32) = 11/1000 T(26, 32) = 7/624 T(27, 32) = 1/96 T(28, 32) = 5/448 T(29, 32) = 5/464 T(30, 32) = 1/96 T(31, 32) = 21/1984 T(2, 33) = 1/66 T(3, 33) = 1/33 T(4, 33) = 31/2112 T(5, 33) = 1/70 T(6, 33) = 1/66 T(7, 33) = 1/70 T(8, 33) = 1/72 T(9, 33) = 1/72 T(10, 33) = 3/220 T(11, 33) = 1/33 T(12, 33) = 3/220 T(13, 33) = 1/78 T(14, 33) = 23/1848 T(15, 33) = 13/990 T(16, 33) = 1/80 T(17, 33) = 2/187 T(18, 33) = 7/594 T(19, 33) = 47/3762 T(20, 33) = 49/3960 T(21, 33) = 1/84 T(22, 33) = 1/66 T(23, 33) = 1/99 T(24, 33) = 1/99 T(25, 33) = 17/1650 T(26, 33) = 3/286 T(27, 33) = 13/1188 T(28, 33) = 1/99 T(29, 33) = 10/957 T(30, 33) = 7/660 T(31, 33) = 7/682 T(32, 33) = 1/96 T(2, 34) = 1/34 T(3, 34) = 8/561 T(4, 34) = 1/68 T(5, 34) = 31/2210 T(6, 34) = 1/72 T(7, 34) = 29/2142 T(8, 34) = 15/1088 T(9, 34) = 29/2142 T(10, 34) = 13/1020 T(11, 34) = 1/77 T(12, 34) = 13/1020 T(13, 34) = 1/78 T(14, 34) = 11/952 T(15, 34) = 13/1020 T(16, 34) = 1/80 T(17, 34) = 1/34 T(18, 34) = 5/459 T(19, 34) = 23/1938 T(20, 34) = 1/85 T(21, 34) = 1/84 T(22, 34) = 1/88 T(23, 34) = 1/102 T(24, 34) = 1/102 T(25, 34) = 1/102 T(26, 34) = 9/884 T(27, 34) = 8/765 T(28, 34) = 5/476 T(29, 34) = 1/102 T(30, 34) = 7/680 T(31, 34) = 16/1581 T(32, 34) = 11/1088 T(33, 34) = 1/102 T(2, 35) = 1/70 T(3, 35) = 1/72 T(4, 35) = 33/2380 T(5, 35) = 1/35 T(6, 35) = 31/2310 T(7, 35) = 1/35 T(8, 35) = 29/2240 T(9, 35) = 4/315 T(10, 35) = 1/70 T(11, 35) = 1/77 T(12, 35) = 1/84 T(13, 35) = 64/5005 T(14, 35) = 1/70 T(15, 35) = 1/84 T(16, 35) = 1/80 T(17, 35) = 1/85 T(18, 35) = 19/1890 T(19, 35) = 22/1995 T(20, 35) = 1/84 T(21, 35) = 1/84 T(22, 35) = 1/88 T(23, 35) = 1/92 T(24, 35) = 1/105 T(25, 35) = 1/105 T(26, 35) = 1/105 T(27, 35) = 5/504 T(28, 35) = 3/280 T(29, 35) = 41/4060 T(30, 35) = 1/105 T(31, 35) = 43/4340 T(32, 35) = 11/1120 T(33, 35) = 3/308 T(34, 35) = 23/2380 T(2, 36) = 1/36 T(3, 36) = 1/36 T(4, 36) = 1/36 T(5, 36) = 1/75 T(6, 36) = 1/36 T(7, 36) = 1/77 T(8, 36) = 1/72 T(9, 36) = 1/36 T(10, 36) = 1/80 T(11, 36) = 5/396 T(12, 36) = 1/36 T(13, 36) = 29/2340 T(14, 36) = 1/84 T(15, 36) = 1/90 T(16, 36) = 7/576 T(17, 36) = 1/85 T(18, 36) = 1/36 T(19, 36) = 7/684 T(20, 36) = 1/90 T(21, 36) = 1/84 T(22, 36) = 1/88 T(23, 36) = 1/92 T(24, 36) = 1/72 T(25, 36) = 1/108 T(26, 36) = 1/108 T(27, 36) = 1/108 T(28, 36) = 5/504 T(29, 36) = 5/522 T(30, 36) = 1/90 T(31, 36) = 7/744 T(32, 36) = 1/96 T(33, 36) = 1/99 T(34, 36) = 1/102 T(35, 36) = 1/105 T(2, 37) = 1/74 T(3, 37) = 35/2664 T(4, 37) = 35/2664 T(5, 37) = 33/2590 T(6, 37) = 1/78 T(7, 37) = 33/2590 T(8, 37) = 1/80 T(9, 37) = 1/81 T(10, 37) = 1/80 T(11, 37) = 29/2442 T(12, 37) = 1/84 T(13, 37) = 28/2405 T(14, 37) = 1/84 T(15, 37) = 23/2220 T(16, 37) = 27/2368 T(17, 37) = 59/5032 T(18, 37) = 1/90 T(19, 37) = 20/2109 T(20, 37) = 23/2220 T(21, 37) = 103/9324 T(22, 37) = 9/814 T(23, 37) = 1/92 T(24, 37) = 1/96 T(25, 37) = 1/111 T(26, 37) = 1/111 T(27, 37) = 1/111 T(28, 37) = 19/2072 T(29, 37) = 61/6438 T(30, 37) = 16/1665 T(31, 37) = 1/111 T(32, 37) = 11/1184 T(33, 37) = 1/111 T(34, 37) = 1/111 T(35, 37) = 1/111 T(36, 37) = 1/111

2 1

Re: [math-fun] Chocolate bar puzzle
by Dan Asimov 01 Mar '09

01 Mar '09

No apology necessary. (It's actually a quote I'd like to take credit for, but ethically I cannot.) --Dan << On Saturday 28 February 2009, Dan Asimov wrote: > This passage > > << > (It'd be fun to make a whole collection of problems that smart and/or knowledgeable people tend to do poorly on: an anti-intelligence or anti-aptitude test if you will.) > >> > > was erroneously attributed to me. (Possibly because in the previous post where it appeared, it was hard to tell who wrote it.) It was I who made that mistake. Oops. Sorry, Dan; and sorry, Jim Propp (who actually made the suggestion). >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

1 0