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Triangular Arrays

November 5, 2010

The Earth may be round, but the world is rectangular. That's the shape of our buildings and nearly everything else we make. Numerical arrays are likewise rectangular, whether business charts, data plots and the array of pixels on your computer screen. Triangular arrays are much less common. The most famous triangular array is probably Pascal's triangle, which is an easy way to determine the coefficients of a binomial equation. These are the coefficients of the terms you get when you expand the equation
(x + y)n

A physical manifestation of Pascal's triangle is Galton's box, devised by Francis Galton to demonstrate the normal distribution by dropping beads through an array of pegs. An interesting example of this device is seen in this YouTube video.

I encountered another triangular array during a trip back from visiting the exhibits at the May, 1999, Conference on Lasers and Electro Optics in Baltimore, Maryland. Since this location was an easy day trip from our laboratory, most of my department traveled in the same van to attend. This trip is remembered more for the team camaraderie than anything we saw at the exhibits. On the way home, we stopped at a Cracker Barrel Restaurant and were introduced to a peg solitaire game.[1,2] Its layout is shown in the figure, and the object is to remove one peg initially, then jump pegs, as in checkers, until only one peg remains. It's actually quite difficult to do manually, so I did what any computer person would do. I wrote a solver program, which is something many others have done, and there are simulations available on the Internet.[2] I wrote my solver in Visual Basic, since I wanted to include a graphical interface, and the object code runs in my version of Wine 1.0.1 on Linux. It might still run on Microsoft Windows systems, but I haven't used them since Windows 2000, so I don't know. A zip file of the solver is available here.

Peg Solataire Game

Peg Solitaire Game

Triangular arrays of objects such as pennies are easy to make, so they've been used as a teaching tool.[3] As the figure shows, the task of inverting a triangular array of pennies is an exercise in optimization. It's a fun way to introduce children to some aspects of mathematics.[3]

Two triangle inversions

A little strategy goes a long way. Two triangle inversions from Ref. 3.[3]

An article by Jörg Pretz on the arXiv preprint server demonstrates the use of a triangular array of lights as a binary clock display.[4,5] Jörg Pretz's version, built from bicolor LEDs, has green lights for AM, and red for PM. As he points out in his article, a clock like this is possible since the number of minutes in twelve hours (720) is six factorial,
6! = 6 x 5 x 4 x 3 x 2 x 1

so we get our rows of 1, 2, 3, 4 and 5 elements with multipliers between the rows of 1/3, 1/4, 1/5 and 1/6, as shown in the figure. The time displayed in this example is (1 x 6 hours) + (2 x 2 hours) + (2 x 30 minutes = 1 hour) + (1 x 6 minutes) + (4 x 1 minute) = 11:10.

Triangular Clock

Triangular clock showing 11:10. See Refs. 4, 5.


  1. Dan O’Brien, Peg Board Puzzle Solution Page
  2. Cracker Barrel Triangle Puzzle
  3. Arthur J. Baroody and Jesse L. M. Wilkins, "Inverting a Triangular Array: Involving Students in Mathematical Inquiry," Mathematics Teaching in the Middle School, vol. 9, no. 6 (February, 2004), pp. 306-313.
  4. Jörg Pretz, "A different way to read off the time - A new idea for a binary clock," arXiv, June 7, 2010.
  5. Jörg Pretz, Binary Clock Web Site.                                    

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Linked Keywords: Sphere; rectangle; doughnut; arrays; pixels; triangular arrays; Pascal's triangle; binomial equation; Galton's box; Francis Galton; Conference on Lasers and Electro Optics; CLEO; Baltimore, Maryland; Cracker Barrel Restaurant; peg solitaire; Wine 1.0.1; Linux; Microsoft Windows; Windows 2000; arXiv preprint server; factorial.