by Mason Wright
Who doesn’t love a horse ‘n’ goggle?
Whether campers are hoping for a spot in the MOCA kitchen or an extra dollop of brownie glop, the system of selection known as the horse ‘n’ goggle has become a Camp Nebagamon constant. In fact, look it up on Wikipedia. First, you’ll find a description: “To use the system, all participants stand in a circle. An arbitrary member of the group is selected by the leader as a starting point. All participants simultaneously throw between zero and five fingers. The leader counts the total number of fingers thrown, then counts that many people around the circle. The selected person is the winner.”
But scan the footnotes, and you’ll discover this: “Camp Nebagamon, Camp Horseshoe, and North Star Camp, all located in Wisconsin, have references to this game on their websites.”
In Keeping the Fires Burning, Nardie and Sally Stein describe the horse ‘n’ goggle as “a way of ‘throwing fingers’ to decide who gets chosen for an extra dessert—or perhaps an extra job. It is usually (unless the counselor conducting it is a skilled mathematician) a game of chance.”
Anyone who has visited Nebagamon during recent summers may have witnessed a horse ‘n’ goggle renaissance. These days, the campers shout—ein, zwei, drei, horse ‘n’ goggle!—with gusto. Loud. Really loud. And the counselor horse ‘n’ goggle tournament, started in 1986, has evolved into the king of counselor competitions, with the two finalists often arriving for the championship throw like heavyweight prizefighters entering the ring. It is all pomp and circumstance.
And randomness, right?
Well, actually, it turns out that the horse ‘n’ goggle is not necessarily the fair and responsible way of allocating resources that the MOCA staff would have you believe.
I own a master’s degree in computer science from the University of Michigan, and I am currently a PhD candidate in computer science and engineering in Ann Arbor. I’ve studied artificial intelligence, machine learning, and computational complexity. I’ve worked in data analysis for Microsoft. I have co-authored studies with names like “An Investigation of the Effect of Competition on the Way Students Engage in Game-Based Deliberate Practice.” So numbers and predictors are my game.
Inspired by four summers as a Nebagamon camper (1997-2000) and two as a counselor (2006 and 2007), I decided to delve into the mysteries of the horse ‘n’ goggle. And here’s what I discovered: There is a way—a small way, and not a sure thing by any means, but a way—to beat the system, sort of. The secret: Position yourself halfway around the circle.
With more than six players, a standard horse ‘n’ goggle game is actually biased against Player One, the first person to be counted. The person directly across from Player One—halfway around the circle—has better odds of being picked, and this only becomes more probable if there are lots of players.
How can this be? In a game with six players, each player controls his own destiny—in the sense that whatever number the others throw, there is some number (0 to 5) that he can throw that will prove to make him the winner. With more than six players, however, there is the possibility that no matter what he throws, the count may not land on him. He is the mercy of the odds.
On a popular MOCA night with lots of hands in the horse ‘n’ goggle, the “law of large numbers” says the average throw will be almost exactly 2.5. That’s because (0 + 1 + 2 + 3 + 4 + 5) / 6 = 2.5. This means that the count will often pass around the circle 2.5 times and finish across from Player One.
In fact, a computer simulation finds that with 50 players in the horse ‘n’ goggle, Player One is over four times less likely to win than the player halfway around the circle. And with even more players—imagine, say, a camp-wide horse ‘n’ goggle—the odds are even better that the count will finish halfway around the circle.
It is the same concept as flipping a coin. Flip in ten times, and it might land on heads eight times. Flip it 1,000 times, and it’ll be close to 50-50.
So now you know the dark secret of the horse ‘n’ goggle. If you find yourself in a particularly big circle, position yourself about halfway around it—directly across from where they start counting. Unless, of course, someone decides to randomize the spot where the counting begins. Because there’s one thing of which we can be certain: In a horse ‘n’ goggle, nothing is certain.