Source: Annals of Improbable Research Nr. 4(6): 4-5 (1998)

Improbable Research: Solution for the Chicken/Egg Problem by use of Toothpaste Arithmetic

by Mark Benecke, co-editor, Annals of Improbable Research (AIR/Cambridge, USA)

Some science problems are eternal, and others nearly so. Pierre de Fermat's famous mathematical poser, for example, appeared in 1619, and lurked for 374 years before someone found a way to solve it. An even harder question has been wlth us practically forever:

"Which came first - the chicken or the egg?" Now, thanks to a simple discovery, I have cracked this heretofore maddening problem.

**Serendipity: The German Dental ltem** A German toothpaste company has made it possible to tackle the chicken-and-egg riddle in an entirely new way. Neither complex formulae nor any scientific knowledge is needed. Figure 1 shows the document which presented the key to the problem. Although I recently moved to New York City, I lived in Germany most of my life. I happened upon this item one day while shopping. It is a carton containing a tube of toothpaste.

**The Generalized Toothpaste Arithmetic Technique** Here is how to solve the chicken-and-egg problem, or any other problem.

Buy a bicarbonate toothpaste in any German *Drogeriemärkten* (drugstores). Then follow the simple arithmetic rules given on the cardboard packaging. (You can throw away the tube of toothpaste; you won't need it.)

Choose the two quantities, qualities, properties, or whatever it is you want to analyze. Write their names, one at the top left, the other at the top right. Then draw a vertical stroke-dot line down the page. After that, think of any curve that you find pretty, and plot it so that the line divides your pretty curve in two. Finally, draw two axes; there is no need to label them.

**The Solution Appears** Using the method just described, I created some sample diagrams that the reader may wish to contemplate (see Figure 2). For each of them, stare at the graph and think about the consequences of the data. Keep staring. Sooner or later, you will feel that everything in the universe is intimately related. Keep staring. Eventually, you will understand that all things are one. After you have reached this point, keep on staring, if that pleases you. That's the entire technique. An interesting thing here is that inversion of any of the mathematical relationships does not lead to any significant change in the graph's meaning (see Figure 3). This is the great advantage of toothpaste arithmetic over other, more complex mathematical procedures.

**Stunningly Simple** One happy aspect of this technique is that, unlike almost anything else in mathematics, it produces results that are obvious-stunningly so. The chicken-and-egg problem is a sterling example of this. The solution is so simple and compelling that there is no need to spell it out in words, and so I will not annoy the reader by attempting to do so here.