Fibonacci Pinball

Fibonacci Pinball

The picture shows a type of pinball machine that you can build yourself. You will need 10 finishing nails, 5 small cups, a wooden board and a pinball (marble). Nail the nails part way into the board in the triangular pattern shown, with one nail in the top row, two in the second, three in the third and so on, and with enough space for the pinball to fit between the nails.

To operate the machine, tilt the board at a slight angle and release the pinball so that it hits the top nail dead center. If the machine is not tilted the pinball will be deflected either left or right with equal probability by the first nail. It will then continue falling and hit one of the nails in the second row and be deflected either left or right around that nail with equal probability.

The result is that the pinball follows a random path, deflecting off one pin in each of the four rows of pins, and ending up in one of the cups at the bottom. The various possible paths are shown by the gray lines and one particular path is shown by the red line.

How many random paths are there through your pinball machine, and what are they?

The answer is 16. The brief explanation is:

The first row has one pin. The number of possible paths through the first row = 2.

The second row has two pins. Since what happens in the second row is completely independent of what happened in the first row, the number of possible paths the pinball could travel from the top through the second row = 4 (2 x 2).

The third row has three pins. Since what happens in the third row is completely independent of what happened in the second row, the number of possible paths completed from the top through the third row = 8 (2 x 2 x 2).

The fourth row has four pins so the number of possible paths from the top through the fourth row = 16 (2 x 2 x 2 x 2).

If you drop 16 pinballs into the top of your machine and repeat that event one million times, what is the average number of pinballs per event that will fall into each cup at the bottom? The answer, from left to right, shown in our pinball machine image below is 1-4-6-4-1. The image to the right is known as Pascal's Triangle. Pascal's Triangle is very useful for analysing the pinball machine. Pascal's Triangle also pops up in a variety of other seemingly unrelated areas. First we mention that the triangle continues on forever and we have only shown the first five rows. Can you see the pattern and guess what the next row of numbers is?

If we superimpose Pascal's Triangle on top of the pinball machine then we see the connection between the two: Each number of Pascal's Triangle represents the number of distinct paths that a pinball can take to arrive at that point in the pinball machine. Without knowing any more at this point it is still fair to say that Pascal's Triangle is a logically ordered description of the outcome of a series of completely random events.

Although Pascal did not discover the sequence of numbers that bears his name, the origin is believed to be hundreds of years earlier in China, he did popularize the sequence in the 17th century from his research, of all things, on improving his betting odds at the gaming tables. If Blaise Pascal were around today he would probably be running some hundred billion dollar derivative heddge fund that kept the Fed Chairman up at nights.

Pascal's Triangle is an oddity. The construction of the triangle is simple. The numbers on each new row are derived by adding the numbers immediately above and to the right and left. We use letters to make words, words to make sentences, and sentences to tell stories that inform us. Numbers are really no different. Clusters of numbers are scale. Sequences of numbers are a process. For us numbers are abstract symbols but to the Pythagoreans numbers had an actual form and a shape. The dots on the right side of the page are the number 34 – a triangle and a square. Sometimes it is useful to think of numbers, including stock and commodity prices, as shapes and forms. Shapes occupy space. They have scale. And they reside in time.

Here's an image of Pascal's Triangle filled out to ten rows.

It looks interesting. But so what? That would be the normal and expected reaction from a generation with hundreds of years of learning that numbers are only abstract symbols used as a convenience to measure something else that is tangible and real. But wait. Didn't we say that Pascal's Triangle is a logically ordered description of the outcome of a series of completely random events? Could there not also be a hidden order within the description itself?

It gets curiouser and curiouser but finally we arrive. Maybe even back to the beginning. When you add the diagonal rows of Pascal's Triangle. Left to right and right to left. You derive the sequence of Fibonacci ratios.

The deeper you get into the heart of Pascal's Triangle the closer you get to the Divine Proportion. What does it all mean? Who knows for sure. Perhaps it is enough to leave with the thought that the detritus of large numbers of simple binary decisions, like the left or right of the pinball, or the buy or sell in the pits, leave footprints in space and time that may be impossible to recognize while happening but become clear enough down the road if you know where to look.
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The pinball machine discussion and illustrations are from the math department at the British Columbia Institute of Technology.