# Mathematical modeling in word games: apportionment of letter tiles.

In mathematical modeling, the topic of apportionment provides powerful tools for the design of games. In particular, letter-frequency based modeling can be used to compute appropriate letter distributions for a variety of word games. This paper will provide possible letter distributions for two different games: Scrabble and Boggle. Throughout the paper, a familiarity with the five standard methods of apportionment-Hamilton, Jefferson, Webster, Adams, and Huntington-Hill--will be assumed. For a review of these methods, please consult [1] or the following link: http://www.ctl.ua.edu/mathl03/apportionment/appmeth.htm

Computations were done using various online applets, the most important of which can be found at: http://www.cut-the-knot.org/Curriculum/SocialScience/ApportionmentApplet.shtml

Scrabble

For the purposes of this paper, letter distributions were calculated as follows: (1) regard the 26 English letters as states; (2) regard the 98 letter tiles (without blanks) as the total number of members in the house; (3) regard letters' relative frequency percentages (calculated to the thousandths) as their populations (after multiplication by 1000). A table of letter frequencies cited in [2] can be found below, followed by a full chart of the suggested letter distributions in Scrabble:
``` Frequency of letters in the English
language
Letter   Frequency
iu English
Language
E         12.702% T          9.056 A          S.167 O          7.507 I
6.966 N          6.749 S          6.327 H          6,094 R
5.9S7 D          4.253 L          4.025 C          2.782 U
2.758 M          2.406 W          2.360 F          2.228 G
2.015 Y          1.974 P          1.929 B          1.492 V
0.97S K          0.772 J          0.153 X          0.150 Q
0,095 Z          0,074
```
``` Letter distributions
suggested by the five apportionment methods
Huntington-
Letter  Scrabble  Hamilton  Jefferson  Webster  Adams     Hill
A           9         8         8         8       7         8 B
2         2         1         1       2         1 C           2
3         3         3       3         3 D           4         4
4         4       4         4 E          12        12        13
13      11        12 F           2         2         2         2       2
2 G           3         2         2         2       2         2 H
2         6         6         6       6         6 1           9
7         7         7       6         6 J           1         0
0         0       1         1 K           1         1         0
1       1         1 L           4         4         4         4       4
4 M           2         2         2         2       3         2 N
6         7         7         7       6         6 0           8
7         8         7       7         7 P           2         2
2         2       2         2 Q           1         0         0
0       1         1 R           6         6         6         6       6
6 S           4         6         6         6       6         6 T
6         9         9         9       8         8 U           4
3         3         3       3         3 V           2         1
1         1       1         1 W           2         2         2
2       2         2 X           1         0         0         0       1
1 Y           2         2         2         2       2         2 Z
1         0         0         0       1         1
```

Note that only Adams' method and the Huntington-Hill methodgive at least one tile to each letter. The latter is of particularinterest, since it is currently used by the United States House ofRepresentatives. Recall that Huntington-Hill willnever assign a state (i.e., letter) 0 representatives (i.e., tiles), sincethe geometric mean of 0 and 1 is 0. Thus, computations in the followingsection will be performed only for Huntington-Hill.

Boggle

For a brief introduction to therules of Boggle and some of the mathematics it entails, see [3]. For adiscussion of the role of problem solving strategies in Boggle andsimilar games, see [4]. Using the frequency table from the previoussection, distributions were computed as follows: (1) regard the 26English letters as states; (2) regard the 96 cube faces as the totalnumber of members in the house; (3) regard letters' relativefrequency percentages (calculated to the thousandths) as theirpopulations (after multiplication by 1000). Next, the cube faces wereassigned to the 16 cubes one by one, beginning with the most frequentletter, and cycling through the cubes [C.sub.1], ..., [c.sub.16]. Forexample, a total of 12 cube faces were assigned to the most frequentletter, E, so each of [c.sub.1] through [c.sub.12] received an E-face.The letter of the next highest frequency was T, to which a total of 8cube faces were assigned, so the next eight cubes-i.e., [c.sub.13],[c.sub.14], [c.sub.15], [c.sub.16], [c.sub.1], [c.sub.2], [c.sub.3],[c.sub.4]--each received a T-face. And so forth. The table below hasbeen alphabetized to improve readability.
``` Letter distribution suggested by Huntington-Hill

Boggle   Huntington-Hill
AAEEGN       ABCEHN ABBJOO       ACEHIP ACHOPS       ACEHNP AFFKPS
AEGHIL AOOTTW       AEHILY CIMOTV       AEHILY DEILRX       AENRUV
DELRVY       DEFIST DISTTY       DEFOST EEGHNW       DEOSTW EEINSU
DOSTWZ EHRTVW       EGILST EIOSST       EKONRU ELRTTY       JNORTU
HIMNQU       MNORTX HLNNRZ       MOQRST
```

Furtherinformation on how the creators of Boggle determined the letterdistribution (which, incidentally, has changed among various editions)was unavailable. For the reader interested in programming, a possiblefollow-up would be to generate a large number of boards using the twodistributions above, and answer questions such as: Which words appearedmost often for each of the distributions? What was the average (mean,median) total score for each of the distributions? What was the average(mean, median) length of the longest word for each of the distributions?Answers to these questions or similar ones would be of great interest tothe author, who welcomes any communication related to this paper.

Acknowledgment: The author wishes to thank Andrew Sanfratello for early discussion ofapportionment methods in the context of Scrabble tiledistribution.

References

[1]Consortium for Mathematics, & Its Applications (US]. (2003). Forall practical purposes: mathematical literacy in today's world.WH Freeman & Company.

[2] Beker, Henry; Piper, Fred(1982]. Cipher Systems: The Protection of Communications.Wiley-Interscience. p. 397.

[3] Ash, C. Boggle. Mathematicsin School, Vol. 16, No. 1 (Jan., 1987], pp. 41-43. Retrieved fromhttp://www.jstor.org/stable/30214170.

[4] Dickman, B. (2013]."Problem Solving Strategies in Boggle-like Games." WordWays, 46(1], 66-72.

Benjamin Dickman

bmd2118@columbia.edu

"Willy boy, Willy boy, where are you going?

Iwill go with you, if I may."

" I am going tothe meadows to see them mowing ; I am going to see them make thehay."

Find one of the mowers/b>