June 2018 - mathfun@mailman.xmission.com

[math-fun] Constructing a set that isn't a congruence class
by James Propp 22 Jun '18

22 Jun '18

There are only countably many sets of natural numbers that form (the nonnegative part of) a congruence classes. Here's a natural way to list them as bit strings: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ... 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ... 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ... 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 ... 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 ... 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 ... 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 ... 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 ... 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 ... 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 ... 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ... 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ... 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 ... 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 ... 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 ... 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 ... ... We can use diagonalization to construct a set that isn't a congruence class: 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 ... Does this pattern continue? Jim Propp

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[math-fun] Delian Brick, a 3D 2-rep-tile
by Ed Pegg Jr 21 Jun '18

21 Jun '18

A cuboid with sides 2^(1/3) to the powers of 0,1,2 can make a larger copy of itself. Delian Brick seems like a great name for it. It is a 2-reptile. https://math.stackexchange.com/questions/2822566/ Graphics3D[{Cuboid[{0, 0, 0}, {2^(0/3), 2^(1/3), 2^(2/3)}], Cuboid[{1, 0, 0}, {1 + 2^(0/3), 2^(1/3), 2^(2/3)}]}] 3D rep-tiles that are not derived from 2D rep-tiles are currently quite rare. I would not be surprised if this brick was known by the ancient greeks. Has anyone seen it before? --Ed Pegg Jr

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Re: [math-fun] Delian Brick, a 3D 2-rep-tile
by William Marshall 20 Jun '18

20 Jun '18

I've known about d-dimensional rep-k bricks since high school (but never published it): 1 X k^(1/d) X k^(2/d) X k^(3/d) X ... X k^((d-1)/d).

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[math-fun] Rectangle into odd equal perimeter triangles.
by Ed Pegg Jr 19 Jun '18

19 Jun '18

I recently divided a rectangle into seven triangles with the same perimeter. https://math.stackexchange.com/questions/2822589/ Has that been done before? --Ed Pegg Jr

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[math-fun] Cliff College
by James Propp 18 Jun '18

18 Jun '18

Does anyone remember a fictional institution called Cliff College that was part of some web-based game back in the 1990s? I recall being surprised to learn that I was listed as a member of the Cliff College math faculty. (I may have the name of the institution wrong.) Jim Propp P.S. I am still hoping to find the original "Poke the Bunny" web-page, or a recreation of it, from the same period. You use your mouse to move a picture of a virtual finger that pokes a virtual bunny a whole lot of times, seemingly without effect, until after some threshold has been crossed the bunny bites the virtual finger and thereafter won't let go.

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Re: [math-fun] A prime factor factory
by Dan Asimov 17 Jun '18

17 Jun '18

Heuristically, the probability of such a chain starting from a given prime p is of the form C_1 * C_2 * C_3 ... where C_k —> 0 as k —> oo. If we then *add* these probabilities over all primes p, we get an infinitesimal that is countable * 1/continuum that is heuristically *greater than* the chance of an infinite Cunningham chain, suggesting it's pretty unlikely. —Dan W. Edwin Clark wrote: ----- This matter seems to be related to Cunningham chains <https://en.wikipedia.org/wiki/Cunningham_chain>--sequences of primes p_i such that p_{i+1} = 2p_{i} + 1. ... ... It is conjectured that there are Cunningham chains of length k for all positive integers k. Is there a reason that one could not have a Cunningham chain of infinite length? -----

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[math-fun] A prime factor factory
by James Propp 17 Jun '18

17 Jun '18

Given any prime p, we can look at all the prime factors of 2p+1, and then perform the same process (double, add 1, find all prime factors) with those primes, and so on; must the process eventually stabilize? E.g.: 2 -> 5 -> 11 -> 23 -> 47 -> 95=5*19 19 -> 39=3*13 3 -> 7 -> 15=3*5 13 -> 27=3*3*3 Done. I looked at a variant where one uses 2p-1 instead of 2p+1; it appears to lead to closure a lot more rapidly. For the 2p-1 version, I was able to check in my head that all primes under 100 lead to closure; for the 2p+1 version, I wasn’t able to do this. Jim Propp

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Re: [math-fun] A prime factor factory
by Dan Asimov 16 Jun '18

16 Jun '18

If true, that would be a colossally cool theorem! Dan W. Edwin Clark wrote: ----- If my program is correct, starting with the n-th prime (n<=10^5) the procedure always ends with {3, 5, 7, 11, 13, 19, 23, 47} which is sequence A020575: (In particular the number of primes obtained will always be finite.) -----

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[math-fun] ft/meter: all threes and ones
by Bill Gosper 16 Jun '18

16 Jun '18

In:= FromContinuedFraction@{3, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1} Quantity@ "ft"/Quantity@"m" Out= 1 --rwg Isn't it weird that to get ft/meter, you divide length(meter) / length(foot) ?

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Re: [math-fun] Schensted's two-dimensional abacus
by Bill Gosper 14 Jun '18

14 Jun '18

Or https://www.youtube.com/watch?v=q7swR8MATA0 --rwg On 2018-06-14 03:35, James Propp wrote: > I know “Ea Ea” is Babylonian, but it brings to my mind the > pseudo-Polynesian “Ea Ea” at the end of the Kellogg’s Puffa Puffa Rice > commercial (does anyone else remember that?). > > Jim Propp > > On Wednesday, June 13, 2018, Allan Wechsler <acwacw(a)gmail.com> wrote: > >> Oh, sorry for not reading this thread more carefully. Somebody else already >> said this. >> >> On Wed, Jun 13, 2018 at 10:58 PM, Allan Wechsler <acwacw(a)gmail.com> wrote: >> >> > Yes, and goes by the name Ea Ea. >> > >> > On Wed, Jun 13, 2018 at 10:50 PM, Hans Havermann <gladhobo(a)bell.net> >> > wrote: >> > >> >> > The Wikipedia page for Craige gives no death date, but uses the past >> >> tense in describing his residency in Portland, Maine. >> >> >> >> I had stated that Craige's mother's obit mentioned two grandchildren so >> I >> >> had hopefully assumed that they might be Craige's kids, he being her >> only >> >> child. I'm fairly certain now that Craige fathered no children and that >> his >> >> mother's grandchildren should more properly have been described as her >> >> step-grandchildren. >> >> >> >> In the search to uncover all this, I unexpectedly discovered that Craige >> >> Schensted is still alive! He's living in Portland, Maine. >> >>

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