*Gaius Julius Caesar (100 BC - 44 BC) (left) and Ulysses S. Grant (1822-1885) (right). (Left image, a bust of Caesar from the 1902 book, History of the World, edited by H. F. Helmolt, from Wikimedia Commons, courtesy of The General Libraries, the University of Texas at Austin. Right image, a portrait of Grant taken sometime between 1855 and 1865, from the Library of Congress, American Memory Collection, Digital ID, cwpb 06971, via Wikimedia Commons.)*

Grant's memoirs were in a long tradition of military memoirs that date back to at least the time of Gaius Julius Caesar (100 BC - 44 BC). Caesar wrote his account of the nine-year Gallic Wars (58 BC - 50 BC) as a third-person narrative in the well known book,

*Building a triangle from a broken strand of dry spaghetti, or from a broken stick. (Created using Inkscape)*

The probability that a stick broken into three random lengths can build a triangle is only 25%.[6-7] An analytical solution of this is given in ref. 6, and a Monte Carlo calculation to verify this is quite simple; so simple, in fact, that even I was able to create a very short computer program to do this (source code here). The histogram of successful triangle builds for 10,000 trials of 10,000 broken sticks appears below.

*Histogram of 10,000 iterations of 10,000 trials for success in building a triangle from a stick broken into three pieces. (Click for larger image)*

The saga of the twice broken stick continues, as evidenced by a recent paper on arXiv by Steven R. Finch, a mathematician at the Massachusetts Institute of Technology. In this paper, he calculates the median area for objects built from randomly broken sticks, not only for triangles, but for quadrilaterals as well. In the case of triangles, the median area is quite small. For a stick of unit length, the median area is just 0.031458...[8] The above truncated decimal for the calculated median area belies the extreme precision of the calculation. Finch gives the actual value as 0.0314584607846627648007001...[8] I write this as an example of why you shouldn't be too satisfied with the results of your computer simulations. As an example, I wrote a Monte Carlo simulation of this problem (source code here) that gives a good value on my desktop computer in a few minutes. My program is not that elegant, so more iterations could be done in a shorter time, but the precision of the computer results are a long way from an actual calculation (see figure).

*Histograms of mean and median area computations for random triangles. We can be fairly certain that the median area falls between 0.0313 and 0.0317, but this precision is far behind the value of 0.0314584607846627648007001... of an actual calculation. (Created using Gnumeric. Click for larger image).*

- I Will Send a Barrel of This Wonderful Whiskey to Every General in the Army, from Garson O’Tool's Quote Investigator, February 18, 2013.
- G. Julius Caesar, "Commentarii de Bello Gallico," Latin and English texts on Tufts University Project Perseus.
- Feynman's Interest in Spaghetti, from Scott Roberts' heelspurs.com.
- RWD Nickalls, "The Dynamics Of Linear Spaghetti Structures," June 14, 2006 (PDF File).
- Basile Audoly and Sébastien Neukirch, "Fragmentation of Rods by Cascading Cracks: Why Spaghetti Does Not Break in Half," Phys. Rev. Lett., vol. 95, no. 9 (25 August 25, 2005), Document 95.095505 (4 pages).
- Eugen J. Ionascu and Gabriel Prajitura, "Things to do with a broken stick," arXiv, April 20, 2013.
- MIT PRIMES/Art of Problem Solving, CROWDMATH 2017: The Broken Stick Problem.
- Steven R. Finch, "Median Area for Broken Sticks," arXiv, April 25, 2018.