Devlin's Angle

July 1998

Buttered Toast and Other Patterns

Last month's column took a look at the book The Bible Code, Michael Drosnin's runaway bestseller that attached mystical and possibly religious significance to what is surely the familiar behavior of sufficiently random sequences. Specifically, by taking a suitably long text (in Drosnin's case, the book of Genesis), choosing a starting letter, and then skipping through the text to pick up every N'th successive letter for a chosen integer N, various recognizable words will be spelled out. As I pointed out in my article -- and as many others have pointed out elsewhere -- there are so many choices of starting point and skip-length N available that it is inevitable that, for some choices of starting letter and skip-length, the process will lead to recognizable words, and indeed to phrases that can be regarded as having significance.

The fact, is, human beings are remarkably good at seeing patterns. Indeed, it is arguable that the ability to see patterns is our key evolutionary trick, the ability that has ensured our survival as a species in the Darwinian battle for survival. Armed with modern computers, our pattern finding ability is even greater. Hence the Bible Code.

But spotting a pattern is one thing; deciding whether or not that pattern has any significance is quite another. That was where Drosnin went astray. The patterns he saw had no significance. Sometimes, however, we perceive a pattern that is not simply a random effect. Take, for instance, the oft-cited observation that a slice of buttered toast knocked from the breakfast table will usually land buttered side down. Most of us believe we have noticed this annoying tendency of buttered toast. But what is going on?

One explanation would be that the toast lands butter-up half the time and butter-down the other half, in an entirely random fashion. According to this explanation, our perception of a pattern of butter-down behavior is a result of our paying more attention to those occasions when the toast drops butter-side down. After all, the butter-side down landing causes us to clean up the carpet, and hence makes the event more significant -- and more memorable -- than when it lands dry-side down. For a dry-side down landing, we either blow on the dry-side to remove any imagined dust and then eat the toast or else throw it in the bin, depending on the status of the floor -- no big deal either way.

An alternative explanation would be that there is a physical law that says toast tends to fall buttered side down, and the pattern we notice is for real.

Surprisingly, the second explanation is correct. Not that there is a law of nature specifically dedicated to buttered toast. Rather, it's a matter of the height of the typical breakfast table, coupled with the fact that when the toast is on the table, it is assuredly butter-side-up. As the toast slides across the edge of the table, with part of the toast hanging off the table, gravity causes the toast to start to rotate. Thus, when the toast falls, it is rotating. For a typical breakfast table, there is just enough fall time for the toast to complete one half a revolution before it hits the deck. The result: the toast lands butter-side-down.

This explanation can be easily checked experimentally without ruining your carpet. Just place a book on your desk, title-side-up, and push it off the edge. It is likely to land title-side-down.

Another case where our pattern-spotting ability also turns out to be reliable is our sense that, when we stand in line at the supermarket checkout, one of the lines next to us moves ahead faster than the line we have chosen. The explanation is straightforward. Assuming that the delays that slow up lines affect every checkout line randomly, and that we join a line having two neighboring lines, the probability that our line moves fastest of the three is 1/3, while the probability that one of the other two adjacent lines moves faster is 2/3, twice as high. The odds are two-to-one that you will watch an adjacent line move ahead of you!

Of course, this explanation does not work if you join one of the lines at the edge of the checkout area. In that case, there is only a 50% chance that the neighboring line will move faster than yours. Much better odds. Moreover, there is the added advantage that you are probably more likely to find yourself standing in line with another mathematician who understands elementary probability theory, and the two of you can pass the time bemoaning the slow progress of your line.

One final example where our perception of a pattern turns out have a rational explanation is the annoying tendency that when we want to use a map, the main location we are interested in lies on or near to a crease in the map, where it is difficult to make out all the details (especially in an older map), or else near the edge, where we need a second map to complete the picture. In this case, some simple geometry provides the explanation.

Given a numerical value for being "near a crease or edge", say, within a distance from a crease or edge of one-tenth the width of the map, then for most maps -- certainly for the kinds of maps designed to fit into your automobile glove compartment -- the ratio of the total map area that lies "near a crease or an edge" to the overall map area will be greater than one-half. For example, for a square map with just a single crease down the middle, the ratio works out to be 0.52. For a more generous definition of "near a crease or an edge", the ratio will in fact be much greater than one-half. Frustrated map readers might like to perform a few calculations, with different maps and different values of "nearness". In this way, you will be able to reassure yourself that the map-makers are not out to get you. Rather, it's a simple matter of relative areas.

- Keith Devlin


Devlin's Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary's College of California, in Moraga, California, and a Senior Researcher at Stanford University. He was a lead consultant of the Life By The Numbers television series, and has written the companion book by the same title, published in April by John Wiley and Sons.