Magazine article Word Ways

Mathematical Modeling in Word Games: Apportionment of Letter Tiles

Magazine article Word Ways

Mathematical Modeling in Word Games: Apportionment of Letter Tiles

Article excerpt

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 method give at least one tile to each letter. The latter is of particular interest, since it is currently used by the United States House of Representatives. …

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