Please look at the picture of the Alpha-Bits game. (Sorry, it’s a bit blurry. I have fired my camera man.) You draw from a deck of cards, containing each of the following numbers, 1, 2, 3, and 4. If you draw a 1, you move your piece, an Alpha-Bit letter, ahead one square, and so on. However, there is something funny about the game. Unsettling even.
Starting the game, it is impossible to land anywhere but on the 3rd square, the one with the bee. For . . .
If you draw a 1, then you move to square 1. The result tells you to move ahead two squares to square 3, the one with the bee.
If you draw a 2, then you move to square 2. The result tells you to move ahead one square to square 3, the one with the bee.
If you draw a 3, then you move to square 3, the one with the bee.
If you draw a 4 then you move to square 4. The result tells you to move back square to square 3, the one with the bee.
No matter what you draw, you end up on the bee.
Similarly, if your piece is on square 13, the one with the rocket, no matter what you draw, you’ll finish the game.
Working backwards from square 13, you can determine the expected numbers of turns needed to win the game.
THE EXCITING RESULTS
Square # Expected Numbers of Turns needed to win the game.
13 – 1.00
8 – 2.00
6 – 2.25
5 – 2.56
3* – 3.15
Start – 4.15
* = Mathematical excitement abounds if you’re starting on square 3. If you draw a 1, you have to go back to your bee. If in your second turn, you again draw a 1, you will once more be back at the bee square. Fear not! By using the mathematical formula for infinite sums, you can calculate how many turns you can expect to be stuck in this purgatory. (It’s 1.33 turns.) Knowing this, our calculations become simple again.
Note that is impossible to end your turn on squares: 1, 2, 4, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, and 20.
I have come to believe that the designers of this game never really played it. I am quite certain they never subjected their creation to mathematical analysis.
Please do not use this analysis for betting purposes. And if you do, do not employ a doubling cube as in backgammon.
At any rate, my years of mathematics has served me well. And you get a gold star if you read this blog all the way through.
Paul De Lancey, The Comic Chef, Ph.D., nerd