Scrabble algebra

Word Ways, Nov, 2007 by David Morice

(In Scrabble, the letters of the alphabet are assigned values from 1 to 10, which are printed on the wooden tiles. The values may double or triple depending on where they are played on the board to score points. The Scrabble Values are: (AEILNORSTU) = 1, (DG) = 2, (BCMP) = 3, (FHVWY) = 4, (K) = 5, (JX) = 8, (QZ) = 10. Note: In this article "number" can refer to either the number itself or its number name.)

In the Aug 1991 Kickshaws, I noted that TWELVE is the only self-referential number in Scrabble. That is, the Scrabble values of the six letters add up to the number itself: 1 4 1 1 4 1 = 12. In the Nov 1991 Kickshaws, JeffGrant suggested spelling the numbers on a Scrabble board, which has double and triple letter and word squares, and using that method he found that the following are self-referential. In reply to an email, Jeff writes:

   ... by including the blank tiles some others can be
   [self-referential] too--ONE (2 blanks), TWO (blank for W), THREE
   (blanks for T and H), FIVE (blanks for F and I), SEVEN (blank for
   S), and EIGHT (blank for E). According to the Aug 1991 Kickshaws,
   by placing the letters on the board with its double- and
   triple-letter and -word scores, all numbers between ONE and
   TWENTYFIVE can be self-referential, except TEN and NINETEEN (NINE
   can be by using one blank and playing across a triple word score
   3x3 = 9--per Nov 1991 Kickshaws).

   I never actually studied which numbers could be made
   self-referential on the board. My quest was to find the highest
   such number which was EIGHTHUNDREDTEN played for a
   triple-triple-triple word score along one side of the board (See
   Nov 1991 Kickshaws). Obviously this would be most unlikely (and
   disallowed!) in a real game. Many more self-referential numbers
   could theoretically occur in a Scrabble game, but it would involve
   playing them adjacent to other words and taking the score for those
   words too. Including other words in the play is probably going too
   far. Jeremy Morse suggested THIRTY as the largest self-referential
   number that could reasonably occur in an actual game.

In Scrabble algebra, the letter values can be added, subtracted, multiplied, and divided. Using the letters that spell out a number, Scrabble equations can be formed in this fashion: (1) put the letters of the number in a row, (2) beneath each letter put its Scrabble value, (3) place , -, x, and / signs, as well as any parentheses needed, to form an equation whose results equal the number itself, (4) (optional) replace the Scrabble values with the letters to show the equation (but this is optional, since the solution consisting of math operations appears in the third step). Here is THIRTY:

(1) T H I R T Y

(2) 1 4 1 1 1 4

(3) (1 4 1)((1 x 1) 4) = 30

(4) (T H I)((R x T) Y) = 30

There are other ways to arrange math operations so that the Scrabble values equal THIRTY. Here are two:

THIRTY = ((1 x 4) 1)(1 1 4)

THIRTY = (-(1 4)/-1)(-1 (-1)) 4

Regular algebraic equations (a b-c=1) differ from Scrabble equations (O N-E=1) in a basic way. (1) In an algebraic equation, letters and math operations are given, and numeric values of the letters have to be found. (2) In a Scrabble equation, letters and their numeric values are given, and math operations that make the equation work have to be found.

SIXTEEN of the numbers from ONE to TWENTY form Scrabble equations. The four that don't are FIVE, NINE, TEN, NINETEEN. As mentioned above, TWELVE is the only number that has a solution using just addition, but it also forms equations using other math operations; one alternative solution is-(1-4)(1-1 (4x1)). No other number has a solution that uses a single math operation. Furthermore, no number name has just one solution: Every number name has either several solutions or no solution at all. The ONE-to-TWENTY list shows only one solution for each number that has a solution.

ONE = 1 1-1

TWO = 1x2x1

THREE = (-1-4-1)/(-1-1)

FOUR = 4/1/1/1

FIVE = none

SIX = -1-1 8

SEVEN = (-1-1))-4)-(-1/-1)

EIGHT = (1x1x2x4)/1

NINE = none

TEN = none

ELEVEN = (1 1 1)(4)-(1x1)

TWELVE = 1 4 1 1 4 1

THIRTEEN = (1x4)(1 1 1) (1x1x1)

FOURTEEN = (4 1 1 1)(1/1/1 1)

FIFTEEN = (4x1)(4x1)-(1x1x1)

SIXTEEN = (1 l)x8x(-1)(-1)x1/1

SEVENTEEN = (1 1 4 1 1)(1 1) (1x1)

EIGHTEEN = (1 1 2 4)(1 1) 1 1

NINETEEN = none

TWENTY = (1 4)(1x1x1x4)

All Scrabble equations can be represented by "opposite" Scrabble equations. That is, take an equation, change all the minuses to plusses and all the plusses to minuses. Then put the equation in parentheses and place a minus sign before the first parenthesis. This is a trivial way. It could be continued indefinitely by repeating the process with each new equation. Here is how it could begin with TWELVE, converting the positives to negatives.

TWELVE = 1 4 1 1 4 1 TWELVE = -(-1 -4 -1 -1 -4 -1)

Which is the highest number that forms a Scrabble equation? Which number forms the fewest equations? Which forms the most? Does any number form one and only one equation? ZERO is self-referential in a different way: It can form 0 Scrabble equations. Are there any other numbers that can form a total number of Scrabble equations equal to the number itself?