From time to time, a journalist may face a soul-shattering dilemma. A scoop so shocking it cannot be withheld, yet so terrible it cannot be told.
And what goes for journalists, goes double for stick-figure cartooning math teachers. Thus, as one who loves truth even at its ugliest, I choose to divulge a fact sure to rattle your faith in humanity itself:
The game of dreidel is built on a lie.
Dreidel, of course, is a beloved Chanukah game. (Happy Chanukah, everybody!) First, each player places a chocolate coin in the center. Then, you take turns spinning a four-sided top (the dreidel), obeying the commands that appear on its ides:
The top functions like a die, with an equal chance of landing on each side—at least, in theory.
The reality is far more sinister.
Fearless and groundbreaking research by Robert and Eva Nemiroff reveals that on the typical dreidel, not all sides are equally likely.
I quote here from their startling abstract:
all three dreidels tested—a cheap plastic dreidel, an old wooden dreidel, and a dreidel that came embossed with a picture of Santa Claus—were not fair… it is conjectured that hundreds of pounds of chocolate have been distributed during Chanukah under false pretenses.
It I worth asking: Why?
No, not “why does a Jewish toy come embossed with a picture of Santa Claus,” although this too is a vexing matter. I mean: Why is the dreidel unfair?
Is it shoddy craftsmanship?
A manufacturer’s deviousness?
The likeliest answer: none of these. It seems that, across the board, spinning is a poor randomization process. A classic study by three Stanford researchers called Dynamical Bias in the Coin Toss found that spinning coins on a table was less effective for randomization than flipping them through the air.
One can imagine why. The long duration of a spin, from rapid beginning to wobbly end, allows time enough to amplify a tiny difference in weight distribution. The heavier side falls down. The lighter side lands up. Invisible deviations in density become visible disparities in chocolate allocation.
What’s the solution?
One drastic measure: change randomizers. Use a tetrahedral die, or two coins (with HH, HT, TH, and TT as the four outcomes). But this would remove the dreidel from dreidel. Unacceptable. When a patient comes with chest pains, you don’t yank out her heart.
Instead, I have a different solution: each turn, you spin the dreidel three times, and interpret the outcome according to this table:
Each row follows the same pattern. It consists of four permutations: one without nun, one without shin, one without he, and one without gimmel. Because order does not affect the probability of a permutation, each row is therefore equally likely.
Via this system, the underlying probabilities of the dreidel itself are rendered irrelevant. Even a grossly asymmetric dreidel can be used to play a fair and balanced game.
Now, is this hyper-complicated? Yes.
Liable to confuse and alienate children? Almost certainly.
Totally unnecessary, given that nobody cares whether the four sides of the dreidel come up with equal likelihood? Perhaps.
But mathematics has never been about “understandable” or “desired.” It has always been about insinuating itself, over all manner of protests, into nostalgic memories and cherished holiday sentiments. And I refuse to let that tradition die.