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Claude Shannon's Long Division

January 15, 2018

2016 was the centennial of the birth of Claude Shannon (1916-2001), the founder of information theory. I wrote an article about Shannon in that year (Claude Shannon, April 28, 2016). For Christmas, 2017, I received from my son the well written biography of Shannon by Jimmy Soni and Rob Goodman.[1] The book is intended for general audiences, so it contains a lot of interesting, but ancillary, information that doesn't relate directly to Shannon, which is my only complaint. My son knew exactly what I wanted for Christmas, since he's always careful to ask me beforehand. For Christmas, 2016, he gave me a Raspberry Pi 3, a single board computer that runs the Linux operating system.

Claude E. Shannon

Claude Shannon demonstrating a maze-solving mouse, called Theseus, in 1950.[2]

Electronic computing devices were still primitive in 1950, so the necessary computation and memory operations for Theseus were done using electromechanical relays.

The mouse was named Theseus in reference to Θησευς, who escaped the Minotaur's Labyrinth.

(Still image from a YouTube video by BinarycoreMedia, April 30, 2013, modified for artistic effect.)[2]


Shannon, like many other young scientists and engineers, tinkered with technology during his childhood. He built mechanical and electrical devices, including a radio transmitter and receiver. Since he enjoyed both electronics and mathematics, Shannon graduated from the University of Michigan in 1936 with degrees in both electrical engineering and math. As a graduate student at MIT, he wrote a Master's thesis entitled, A Symbolic Analysis of Relay and Switching Circuits. This thesis was on the use of Boolean algebra for efficient switching of telephone circuits.

Figure 30 from Claude Shannon's 1940 Master's thesis

Figure 30 from Claude Shannon's 1940 Master's thesis. (Via MIT Library.)[3]


After earning his Ph.D., Shannon became a National Research Fellow at the Institute for Advanced Study (Princeton, New Jersey). He then joined Bell Labs to work on military projects, including cryptography, during World War II. In his cryptographic work, Shannon proved that properly constructed one-time pads were unbreakable.

At Bell Labs, Shannon published his most famous paper, "A Mathematical Theory of Communication," in 1948 in two parts in the Bell System Technical Journal.[4-5] This paper was the foundation of information theory. In it, Shannon defined the concept of information entropy.

Shannon joined MIT's faculty and its Research Laboratory of Electronics in 1956. He was a faculty member at MIT until 1978. Shannon developed Alzheimer's disease, and he died on February 24, 2001. Among his many honors, Shannon received the 1966 National Medal of Science, presented by President Lyndon B. Johnson. Through it all, he was modest about his accomplishments, and he worried that they were being "oversold."[6]

Early in their biography of Shannon, on page 19, Soni and Goodman write about Shannon's first publication, accomplished at age 17.[1] This was the solution to a problem in long division posed a few months earlier in The American Mathematical Monthly by R.M. Sutton of Haverford College, Pennsylvania.[7-8] This wasn't a standard long division; rather, it was a long division encoded by a cryptographic substitution of alphabet letters for numbers, as shown below.

Claude Shannon's_AMM_problem

As can be surmised, all the numbers from zero through nine have been substituted by some letter from the set, J,L,M,N,R,S,T,U,X, and Y. Not only was the solution difficult, but so was Sutton's finding a simple long division that contained all the digits. In this age of spreadsheets and calculators, it's rare when a person does a long division by hand. While long division by hand seems inelegant, a longer step from elegance was my decision to solve this problem by a brute force computer program.

A brute force attack, also called a proof by exhaustion and a proof by cases, involves checking each possible case. One famous example of a brute force attack is the computer-assisted proof of the Four-Color Conjecture that you need only four colors to color a map such that no adjacent regions have the same color. This conjecture was not easy to prove analytically, but Kenneth Appel and Wolfgang Haken used a computer to analyze every conceivable map to prove the conjecture true in 1976.[9-10] They analyzed all 1,936 cases of the problem using available computers of that time, an IBM 360-75, an IBM 370-158, and an IBM 370-168.[10]

The Sutton long division has ten variables, each of which has ten possible values. This long division requires the four equations shown below to be simultaneously true. This leads to 1010 (ten billion) cases involving four calculations each. The C language program that I wrote is a trivial assemblage of ten nested loops (source code, here, shannon.c).

Necessary conditions for long division problem

This is not the type of program you should ever admit writing, and you can wear out your curly-brackets keys. However, all these nested loops executed on my Ubuntu 16.04.3 LTS 64-bit four core Intel i3-4160 3.60GHz desktop computer with 8 gigabytes of RAM in just 17.171509 seconds, obtaining the first (and only) solution in just 2.786426 seconds. The solution is J = 1, L = 6, M = 0, N = 3, R = 5, S = 8, T = 4, U = 9, X = 7, and Y = 2.

Claude Shannon's_AMM_solution

My son, who's an actual computer scientist, produced an improved version of the program, with source code here, shannon2.c. His program obtains the solution in 0.605299 seconds, nearly five times faster, on my desktop computer.

References:

  1. Jimmy Soni and Rob Goodman , "A Mind at Play: How Claude Shannon Invented the Information Age," Simon & Schuster, July 18, 2017, ISBN-13: 978-1476766683, 384 pp. (via Amazon).
  2. Claude Shannon demonstrates machine learning, YouTube Video by Sean Palmer, May 17, 2014. In this video, Shannon describes himself as a mathematician.
  3. Claude Elwood Shannon, "A symbolic analysis of relay and switching circuits," Master's Thesis, MIT Library, 1940 (PDF File).
  4. Claude E. Shannon, "A Mathematical Theory of Communication", Bell System Technical Journal, vol. 27, no. 3 (July, 1948), pp. 379-423.
  5. Claude E. Shannon, "A Mathematical Theory of Communication", Bell System Technical Journal, vol. 27, no. 4 (October, 1948), pp. 623-656.
  6. G. Pascal Zachary, "Celebrating Claude Shannon," IEEE Spectrum April, 2016, p. 8.
  7. R. M. Sutton, Problem E58, The American Mathematical Monthly, vol. 40, no. 8 (October, 1933), p. 491f., DOI: 10.2307/2302079.
  8. C. E. Shannon, Problem E58 Solution, The American Mathematical Monthly, vol. 41, no. 3 (March, 1934), p. 191f., DOI: 10.2307/2302271.
  9. K. Appel and W. Haken, "Every planar map is four colorable. Part I: Discharging,"Illinois Journal of Mathematics, vol. 21, no. 3 (1977), pp. 429-490. A PDF file (4817 KB) can be found here.
  10. K. Appel, W. Haken, and J. Koch, "Every Planar Map is Four Colorable. II. Reducibility", Illinois Journal of Mathematics, vol. 21 no. 3 (1977), pp. 491-567. A PDF file (3811 KB) can be found here.

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