Working on the railroad

by Brian Hayes

Published 10 February 2007

The March-April issue of American Scientist is now available on the Web; paper copies should be on their way soon. My column is about hump yards and turnouts and wyes—in other words, about algorithms for railroad workers. “Computing with locomotives and box cars takes a one-track mind.” There’s a small puzzle near the end of the column. You’re welcome to post comments, complaints and solutions here.

In the new issue I also recommend a “Macroscope” article on Avogadro’s number by Ronald M. Fox and Theodore P. Hill of Georgia Tech. For those who have forgotten their chemistry, Avogadro’s number is the number of molecules in a mole of a substance (an amount in grams numerically equal to the molecular weight). Specifically, N_A is defined as the number of carbon atoms in 12 grams of carbon-12, and its value is roughly 6.02 × 10²³. Fox and Hill suggest turning the definition upside-down: Instead of trying to count the atoms in a gram, define the gram as a certain number of atoms. They have a specific number to recommend: 602,214,141,070,409,084,099,072. I invite you to deduce what’s so special about this particular number and why they favor it over other candidates in the same range.

Responses from readers:

• A comment from Barry Cipra, 12 February 2007 at 12:54 pm

  I’d like to suggest 602,214,205,251,903,400,125,971 as an alternative Avogadro’s number. It’s the only value in the Fox-Hill range that is the cube of a *prime* number. (My thanks to Paul Zorn for doing the exact arithmetic.)

  The prime, by the way, is 84,446,891. The Fox-Hill cube root is 84,446,888 = 8 x 17 x 620,933.

• A comment from Barry Cipra, 12 February 2007 at 9:43 pm

  A triangle pyramid, aka cannonball arrangement, with 1 ball sitting on 3 sitting on 6 sitting on 10, etc., for a total of k layers, has a total of k(k+1)(k+2)/6 balls. This gives the very nice value

  602,214,150,661,210,865,396,970

  by setting k=153,450,179. Moreover, this value for k is, once again, a prime number!

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Tags for this article: mathematics, problems and puzzles, science.

Publication history

First publication: 10 February 2007

Converted to Eleventy framework: 22 April 2025