Ivars Peterson's MathTrek
March 4, 2002
In the popular SCRABBLE® Brand Crossword Game, players create words from letters selected at random from a stockpile of 100 tiles. The tiles are laid down on a board 15 squares high by 15 squares wide to form an interlocking, crossword arrangement.
Each letter of the alphabet has a particular value and the number of tiles bearing each letter depends on the language. The standard (American English) edition of the game has the following letter distribution: A: 9, B: 2, C: 2, D: 4, E: 12, F: 2, G: 3, H: 2, I: 9, J: 1, K: 1, L: 4, M: 2, N: 6, O: 8, P: 2, Q: 1, R: 6, S: 4, T: 6, U: 4, V: 2, W: 2, X: 1, Y: 2, Z: 1, blank: 2. In all, 42 of the tiles feature vowels.
Players start with seven tiles, then draw additional tiles to replace any that are played on the board. An unlucky player can occasionally draw a seven-letter hand that contains no vowels, but the chances of doing so are relatively slim given the letter distribution.
When you find yourself drawing seven-letter combinations such as NQWZCGJ more often than you would expect, however, it pays to take a closer look at what's going on. That's what happened when retired geologist Charles J. Robinove of Monument, Colo., tried out an electronic travel edition of the game.
While playing Hasbro's Scrabble Express, Robinove got the impression that he was drawing too few vowels and too many low-frequency (high-scoring) consonants. He decided to investigate the situation by comparing the performance of the handheld device with that of a computer edition of the game. Robinove describes his findings in the current issue of Chance.
For each of the two versions of the game, Robinove drew 100 initial hands, totaling 700 letters, then looked at the vowel frequency. The computer version of Scrabble provided an average of 2.84 vowels per hand, quite close to the expected number, 2.94 vowels per hand. Scrabble Express provided only 1.36 on average.
Moreover, out of 100 hands, the Scrabble computer game produced only two hands with no vowels. In contrast, the handheld Scrabble Express unit produced 23 hands with no vowels.
"There are too few vowels and too many rare consonants such as Q, W, and V in the Scrabble Express game," Robinove concluded. "The huge difference in letter-frequency deviation reinforces the conclusion from the analysis of vowel frequency that the pick of letters in a hand does not represent a statistically random letter selection and is therefore unfair to the player of the game."
Indeed, Robinove's data suggest that Scrabble Express assumes the following letter distribution: A: 6, B: 3, C: 4, D: 4, E: 4, F: 3, G: 5, H: 3, I: 4, J: 4, K: 4, L: 3, M: 4, N: 5, O: 3, P: 4, Q: 2, R: 4, S: 4, T: 4, U: 3, V: 5, W: 4, X: 2, Y: 6, Z: 2, blank: 5. Rounding to the nearest integer means that the total number of tiles isn't precisely 100.
The results of Robinove's investigation leave several mysteries. Was the problem caused by a programming error or the use of a faulty random number generator? Was it precipitated by the implementation of the wrong letter distribution? Was the game changed in some way to fit within the limitations of the chosen electronic format, leading to anomalies in the letter frequency?
How pervasive was the problem? Was it an isolated case, affecting only Robinove's particular unit or perhaps one manufacturing run of the device? If not, was the problem ever identified and fixed in later production runs of the handheld game?
Robinove contacted the game's manufacturer, but received no reply. "I still have no idea why the game is biased," he says.
I welcome comments from any readers (and Scrabble players) who can provide additional information about this curious anomaly.
Copyright 2002 by Ivars Peterson
Robinove, C.J. 2002. Letter-frequency bias in an electronic Scrabble game. Chance 15(No. 1):30-31.
SCRABBLE® is a registered trademark. See the "Official Worldwide Scrabble Home Page" at http://www.scrabble.com/.
The National Scrabble Association has a Web site at http://www.scrabble-assoc.com/index.html.
Links to additional information about Scrabble and other crossword game resources can be found at http://home.teleport.com/~stevena/scrabble/faq.html. The Web site of the Elliott Avedon Museum and Archive of Games at the University of Waterloo, Ontario, features Scrabble at http://www.ahs.uwaterloo.ca/~museum/vexhibit/Whitehill/scrabble/.
Comments are welcome. Please send messages to Ivars Peterson at email@example.com.
A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the MAA book Mathematical Treks: From Surreal Numbers to Magic Circles.