In many Greek households, one of the highlights of New Year's Day is the cutting of St. Basil's cake.
Made from flour, eggs, butter, sugar, orange flavoring, and other ingredients, this special round cake also contains a surprise—a foil-wrapped coin. In some recipes, the coin is added to the batter before baking; in others, it's slipped under the cake when the cake is placed on a serving platter.
In Greek tradition, the cutting of St. Basil's cake reveals what the new year has in store for the family, and the person who gets the slice containing the hidden coin is considered to be the luckiest one of all.
Sometimes, however, as the cake is being cut into sectors, the knife actually hits the hidden coin. Christina Savvidou of the University of Cyprus in Nicosia wondered what the probability of such an occurrence is and how it depends on the size (or number) of slices. She reported her findings in the February 2005 Mathematics Magazine. Savvidou was a member of the Cyprus team in the 2001 International Mathematical Olympiad, earning a bronze medal.
Savvidou assumed that the coin is parallel to the base of the cake. Mathematically, she worked out the probability that a disk (representing a coin) is contained entirely within one of the sectors into which a circle in the plane has been equally divided. She ended up with a formula giving the probability that a radial cut hits a coin.
Stan Wagon of Macalester College, however, wasn't satisfied with Savvidou's analysis. She had assumed that the coin was equally likely to be near the center as near the edge. "But there is much more area near the edge," Wagon notes. Indeed, given a disk of radius R, the area of the inner disk of radius R/2 is a quarter that of the full disk of radius R. So, the coin is much more likely to be closer to the cake's outer edge than near its center.
Savvidou had also assumed that the coin would be horizontal. In practice, however, a coin slipped into batter could end up in any orientation.
In revisiting the problem, Wagon and his colleague Dan Flath came up with a new set of results that provides a more realistic take on the probability of finding a hidden coin in a St. Basil's cake—an effort that required extensive computer modeling, some algebraic manipulation, and a bit of baking. Their findings will appear in an upcoming issue of the UMAP Journal (http://www.comap.com/undergraduate/products/).
Flath and Wagon first derive a formula for finding a marble in a cake. Savvidou's original assumption of a horizontal coin is really the same as mixing a sphere, such as a marble, into the batter. So, if a cake has a radius of 16 centimeters and the marble a radius of 1 centimeter, then, for 2 cuts, the probability of a hit is 0.16, and for 4 cuts, it's 0.31. You can also figure out how wide a cake has to be if you want a certain probability of finding a marble of a given size.
Suppose, however, that a coin can be in any orientation. In this case, because a coin's horizontal projection is an ellipse, working out what happens for arbitrary coin orientations involves dealing with skewed ellipses. Flath and Wagon ended up formulating an algorithm for computing the probability that a cut strikes an ellipse.
Even the new model may not fully represent what might happen to a coin randomly embedded in a cake. "If the cake has finite height," Flath and Wagon point out, "then a coin near the top or bottom is more likely to be horizontal than vertical, since a vertical coin would not be entirely within the cake." Similarly, a coin near the cake's outer edge is more likely to have a vertical orientation than a horizontal one.
There's no simple way to account for these effects. So, Flath and Wagon note, "we simply live with the fact that our assumption of purely random orientations might not conform exactly to physical reality, but is of little importance when the coin is small relative to the cake."
Flath and Wagon also looked at an interesting special case. They computed the probability of finding a needle hidden in a cake—or more generally of hitting a needle if n, randomly oriented needles happen to be embedded in the dough.
"Without baking a lot of St. Basil's cakes, one has very little intuition about the probabilities in question," Flath and Wagon conclude, "but the modeling done here, though imperfect in some fine detail, gives us good approximations to the actual probabilities."