# How Many Beans Make Soup?

by Michael Reidy Tunbridge Wells, Kent, United Kingdom

America’s taste for bean soup appears to be unrelenting, and the World Wide Web offers more than a quarter of a million references to the subject. Multiple-bean soups are particularly in vogue. A methodical check on a leading search engine produced the following results which I record here for future historians of early twenty-first century food. Unexpectedly, this research also thrown up food for thought for mathematicians.

The methodology for researching multiple-bean soup was thus: The phrase “2 bean soup” was entered into the search engine, and the result recorded. Next, the phase “two bean soup” was entered. The search term producing the largest number was recorded as the most accurate number. This method was repeated until the number of beans in soup failed to produce relevant returns, thus, “Page 34, beans are the flavor of the month for soup...” was not considered a valid return for ‘34 bean soup.’

The chart (see Figure 2) plots the number of pages returned for each number of varieties of bean in soups for bean quantities ranging from 2 to 23. No soups were found using in excess of 23 varieties of bean. Figure 2. A graph of the data. This depicts the number of World Wide Web pages the author found that pertain to each number of varieties of bean in soups for bean quantities ranging from 2 to 23.

*Figure 2. A graph of the data. This depicts the number of World Wide Web pages the author found that pertain to each number of varieties of bean in soups for bean quantities ranging from 2 to 23.*

Taking the pulse of bean soup is less straight forward than originally supposed. I had reckoned to encounter a normal bell curve with a peak around 16 beans, as the diversity of recipes for bean soup would at first sight seem to be a random event.

However, the presence of six distinct peaks at 3, 5 , 9, 13, 15 and 19 beans is nothing less than startling. (One must note that these numbers do not constitute a Fibonacci sequence, despite their approximate similarity to one.). Four of the six peak numbers of beans are prime, and the remaining two numbers are the square of the first number and the product of the first two. Multiple bean soup looks to be anything but a random phenomenon.

*Figure 1. Three kinds of beans (in cans). Photo: S. Drew.*

#### Pushing the Packet

Further investigation of these relationships failed to locate a 25-bean soup (25 is 5²) or 45 bean soup (45 is 5 x 9). Similarly, I have not been able to determine why prime numbers 7, 11 and 17 fail to make popular soup. 11- and 17-bean soups have turned in a particularly alarming performance, with only 21 and 12 references, respectively. (One must note the coincidental numeric palindrome formed by the digits 2 and 1.)

Not being a cook, I cannot explain why the three major peaks occur around the numbers 3, 5 and 15. The mathematical relationship between the three numbers is startling (3 x 5 =15). Food historians may know of rules of proportion governing cuisine wherein years of experimentation have yielded rules for the use of ingredients in fixed mathematical proportions. The distance between successive peaks (major and minor: 3, 5, 9, 13, 15 19) occurs in a regular pattern as well: 2, 4, 4, 2, 4. (2, of course, is the second prime number, and 4 is its square). It is interesting to note that continuing the pattern (19 + 4) takes us to the prime number 23, which is the largest number of beans found in soup during our investigation.

#### Some Possible Explanations

There are many possible explanation for these and other patterns that are evident in this referential-bean counting exercise. Several have to do with oddity and evenness. The visual appeal of objects odd or prime in number is well known, but given the nature of soup, the number of beans used in its preparation is not readily noticeable — either consciously or unconsciously — the soup eater, or even to someone who simply observes or handles the finished soup. Seasonal influences are not applicable, because most recipes use dried, rather than fresh, beans.

Moving a step back in the process, the sale of beans is apt to be related to the marketing of the beans, in that prime and odd numbers are displayed at wholesale and retail outlets. Alternatively, there is the so-called Beethoven phenomenon. Beethoven’s odd numbered symphonies are universally acknowledged to be superior to his even numbered ones. The appeal of odd numbers may stem from the fact that humans are, for the most part symmetrical, so there may be a special appeal for things with odd numbers, which would be considered exotic. Another possibility arises at the chemical level, where the interaction of odd or prime numbers of ingredients may produce more desirable flavors than do even or non-prime numbers. (A quick glance through a book of cocktail recipes suggests there is something to this, but it requires considerable further investigation.)

#### Summary

The socio-cultural origins of bean soup may be found to play an important part in unravelling the mathematical aspects of recipes in general and of multi-bean soup in particular. (A doctoral thesis titled ‘Multi-bean soups in multi-cultural societies’ is certainly in the offing.)

#### Conclusions

While not significantly nearer knowing why the composition of multi-bean soups tends to cluster around certain numbers of beans, it has been shown that the number of beans in soups is not random and seems to have a purposiveness. That the clusters are around prime numbers (and their multiples) may be a statistical aberration arising from the fact that there is a disproportionately large number of prime numbers between 1 and 23 (there are 10 of them). However, the peaks are pronounced enough around the three major peaks (3, 5, 15) suggest that something else has a major influence over bean soup recipes. Further research is clearly called for.

#### TABLE 1

**The Raw Data: Collected using www.google.com on 1 February 2002** *Number of Beans in Soup | Number of References Returned by Search Engine* 2 109 3 261 4 35 5 336 6 108 7 100 8 29 9 128 10 118 11 21 12 49 13 98 14 26 15 355 16 166 17 12 18 44 19 128 20 3 21 3 23 1

(As a bonus, here is a recipe from Flickr user Bill McKee.)

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The article above is republished with permission from the March-April 2003 issue of the *Annals of Improbable Research*. You can download or purchase back issues of the magazine, or subscribe to receive future issues. Or get a subscription for someone as a gift!

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