May 2003 - mathfun@mailman.xmission.com

[math-fun] problem with prime gap proof
by Thane Plambeck 07 May '03

07 May '03

http://www.siliconvalley.com/mld/siliconvalley/5804885.htm Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm

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Re: [math-fun] m & m's
by Henry Baker 06 May '03

06 May '03

Hersey's is obviously trying to rediscover Zipf's Law: BRN YEL RED ORG BLU GRN 30% 20% 20% 10% 10% 10% 31% 21% 16% 13% 10% 9% True Zipf's Law %'s Zipf's Law is "satisfying" to humans, because it has just the right combination of comfort and surprise -- the words of languages occur in an approximation to Zipf's Law. The popularity of TV channels is an approximation to Zipf's Law. The popularity of web pages is an approximation to Zipf's Law. At 04:19 PM 5/6/03 -0400, Thomas Colthurst wrote: >According to >http://global.mms.com/us/about/products/milkchocolate.jsp, >30% of M&M's are brown, >20% are yellow, >20% are red, >and orange, blue, and green all account for 10% each.

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Re: [math-fun] A proposed new chess rule
by Dan Hoey 06 May '03

06 May '03

Of course pinned pieces can check just like any others; I can't imagine players of real chess wanting it different. Still, a variant chess without check-pins might be fun. But then can a pinned piece pin another piece? If A pins B pins C pins D pins A, then I guess any one piece could move, but its ally would then be "really" pinned. I seem to recall a problem where there is a mate in N moves after 2N+2 checks, all by pinned pieces. Or maybe it was 2N+3 checks. Dan

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[math-fun] A proposed new chess rule
by asimovd＠aol.com 06 May '03

06 May '03

I don't think this comes up often in practice, but unless I'm mistaken one's king may not be on a square that is attacked by any of the opponent's pieces. AFAIK, this even includes the situation when the said opponent's piece is immobilized due to its being pinned against its own king. Thus a checkmate could occur even when the checkmate hinges on a king's being forbidden to escape to a square "attacked" by an opponent's piece that is at the time immobilized by a king pin. Is this correct? It won't change the course of chess very much, but if this is possible I think the rule should be changed so that the king is allowed to move to any unoccupied square unless the opponent could conceivably "take" the king on that square on the very next move. Comments? --Dan

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Re: [math-fun] m & m's
by Mike Stay 06 May '03

06 May '03

>From the FAQ on m-ms.com: Question What is the percent of each colour in M&M's® packs? Answer Hello, M&M's® Milk Chocolates are made up as follows: 31% Brown, 19% Red and 12.5% each of yellow, orange, green and blue. M&M's® Peanut and Crispy are made up as follows: 16.6% each of brown, yellow, red, blue, green or orange. Because we make so many at any given time, it is not likely that you will open a pack and get this exact percentage split of colours. To test if this split actually works you'd need about 5,000 M&M's® and that's a lot of M&M's® to eat ! Regards M&M's® -- Mike Stay staym(a)clear.net.nz http://www.xaim.com/staym

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Re: [math-fun] m & m's
by Thomas Colthurst 06 May '03

06 May '03

According to http://global.mms.com/us/about/products/milkchocolate.jsp, 30% of M&M's are brown, 20% are yellow, 20% are red, and orange, blue, and green all account for 10% each. I've seen more than one elementary school teacher assign the problem of estimating these proportions from empirical data (such as thane(a)best.com presents), but not of course the problem of whether the differences are statistically significant. I would guess that the color proportions are primarily determined by aesthetics (single blue or green splotches on a brown background look more appealing to me than brown splotches on a blue or green background). Also, history plays a role: the original M&Ms were apparently all brown, with red, green and yellow all added in 1960. [Orange was introduced in 1976, blue in 1995. Red was yanked in 1976 and reinstated in 1987. Purple is scheduled to be added to the standard mix sometime soon.] -Thomas C

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[math-fun] m & m's
by Thane Plambeck 06 May '03

06 May '03

My son complained that his bag of m&m's contained "all browns, and no blues." His evidence being unavailable (he ate them all), and suspecting exaggeration, I bought three bags at two locations with these results Milk Chocolate M&M's 1.69oz (47.9g) bag, 240 calories Package shows all colors but brown. BAG 1 6 May 2003 10:30am - Safeway, Middlefield Road, Palo Alto 20 red 16 brown 5 orange 7 yellow 6 green 4 blue 58 Total BAG 2A 6 May 2003, 11am -- Long's Drugs, downtown Palo Alto 21 brown 12 green 11 orange 6 yellow 4 blue 3 red 57 Total BAG 2B 6 May 2003, 11am - Long's Drugs, downtown Palo Alto 20 brown 9 yellow 9 blue 9 red 7 orange 3 green 57 Total Brown does seem to be overrepresented. Statistically significant, or merely tasty? Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm

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Re: [math-fun] m & m's
by asimovd＠aol.com 06 May '03

06 May '03

Thane writes: << My son complained that his bag of m&m's contained "all browns, and no blues." I don't think blue used to be an M&M color when I was a kid. Brown, green, orange and red, yes. It's hard for me to remember yellows. Maybe just a few. Maybe the company tries to instill a sense of mystery about its identical-tasting candies by making one color "rarer" than another. Then again, maybe blue food coloring is more expensive than the others. --Dan

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[math-fun] Sequences -> "models"; the automated physicist
by Henry Baker 05 May '03

05 May '03

After seeing the success of the integer sequence and the ability of systems to "guess" the function, I would propose a new system that would analogously "guess" one or more "models" of how to produce that sequence, given a "noisy" set of data points. [If this has already been done before, then I'd be interested in links/pointers/references.] Take the following example. A particular human has the capability to run fast for a short while, run further if he/she slows down some, and run even further if he/she slows down a lot. For example, a human might be able to run a 9 second 100 yard dash, or a 3:40 mile, or a 2:10:00 marathon (26 miles). (I apologize for the use of English units -- we're only using units of the "willing", right now!) There might be a variety of physiologic reasons why longer distances require slowing down -- e.g., a variety of energy storage and release mechanisms, where the fastest methods get depleted quickly. One might conceive of a mathematical system that was given various datapoints such as the ones given above, and the system would hypothesize a family of models that "fit" those data points, and might then ask for additional datapoints in order to distinguish the models. Thus, for example, the system might be given one datapoint -- a 9 second 100 yard point. The system would hypothesize that the person runs at a constant speed. You then tell the system that the person runs a 3:40 mile, which is 8 yds/sec versus 11.11 yds/sec for the 100 yard dash. The system then posits some two-level component (smooth) system wherein there are two energy-releasing mechanisms, one fast and one slow, and the fast one "runs out of gas" before the distance of a mile is reached. The system then might ask one or more questions regarding datapoints shorter than 100 yards, intermediate points, or points for distances longer than one mile. If the system is given the marathon datapoint (46,145 yards in 7800 seconds, or 5.92 yards/sec), this point might not introduce more complexity into the system in terms of additional energy systems, but might simply bound the various coefficients more tightly. The idea of this system is not merely to interpolate values, for which there are thousands of different methods, but to come up with one or more "natural" models, which could eventually be used (with much additional research) to start to propose hypotheses about how the human system might actually work, at least in terms of the most important factors for these distances and times. Obvious interpolation schemes would include polynomial and rational functions, sums of exponentials, sums of exponentials times polynomials, etc. I would imagine that such a system ought to be smart enough to "rediscover" a significant number of the simpler "laws of nature" -- e.g., Ohm's law, Colomb's law, etc. Henry

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[math-fun] Re: Consecutive Integers [was Quadratic Question]
by Dean Hickerson 05 May '03

05 May '03

Edwin Clark (eclark(a)math.usf.edu) asked: > Let E=[e_1,e_2,...,e_k] be a list of integers > satisfying 1 <= e_1 <= e_2 <= . . . <= e_k. > > Let S(E) be the set of all positive integers n such that n = Prod{p_i^e_i} > where the p_i's are distinct primes. > > Say that the integers in S(E) have factorization pattern E. I believe this > is a known concept. ... > Is it the case that if 1 is not in E then S(E) contains no consecutive > pair of integers? No. There are at least two counterexamples among the solutions of 23 x^2 + 1 = 8 y^2. The positive solutions of this are given by x(0) = 23; y(0) = 39; x(n) = 24335 x(n-1) + 14352 y(n-1); y(n) = 41262 x(n-1) + 24335 y(n-1). (Note that x(n) is always divisible by 23, so 23 x(n)^2 and 8 y(n)^2 are consecutive powerful numbers.) For n=8, 23 x(n)^2 and 8 y(n)^2 are in S([2,2,2,2,3]); their factorizations are 23^3 1361^2 39157289^2 13475265289^2 43842128059343041^2 and 2^3 3^2 13^2 509^2 61853469184829429986091801293807909^2 For n=9, they're in S([2,2,2,2,2,3]), with factorizations 23^3 151^2 2281^2 23447^2 6461369^2 29366147657538184938924047^2 and 2^3 3^2 13^2 35872892507^2 5281732523824487^2 8087241110442161^2. Dean Hickerson dean(a)math.ucdavis.edu

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