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Problem of the Week
Problem B and Solution
Letters with Friends

Problem

In the game "Scribble", players place tiles containing individual letters on a board to form words. The board is divided up into squares, and each letter in a word is placed in adjacent squares in a row (reading from left to right) or a column (reading from the top down). Each letter has a point value, and the score for a word is found by adding up the point values for each of its letters. The point values for each letter are shown in the following table.

Letter Number of Points
A, E, I, L, N, O, R, S, T \(1\)
D, G, U, Y \(2\)
B, C, M, P, W \(3\)
F, H \(5\)
J, K, V \(8\)
Q, X, Z \(10\)

For example, the word POUR would have a score of \(3+1+2+1=7\).

Players alternate turns making words on the board. After the first turn, each new word must build off of one of the letters already on the board. The score for any new word includes the point value for the letter that was already on the board.

Vinny and Theo played Scribble. Their game board is shown after they each had three turns.

A game board with six words. A complete description of the board can be found at the end of the problem.

  1. Which word on the game board has the highest score?

  2. Which word on the game board has the lowest score?

  3. If Vinny made the first word, what is the maximum total score that he could have for his three words? In order to achieve this score, which word(s) could he have started with?

Extension: Suppose Vinny’s next word has a score of \(10\). Which word could this be? See how many different possible words you can find. You can assume that the game board extends past the size of what is shown above.

Solution

  1. The word KICK has the highest score. Its score is \(8+1+3+8=20\).

  2. The word ORANGE has the lowest score. Its score is \(1+1+1+1+2+1=7\).

  3. The three words with the highest score are KICK, QUESTION, and BIZARRE. Their scores are \(20\), \(18\), and \(18\), respectively. If Vinny made all three of these words, then his total score would be \(20+18+18=56\).

    It is possible for Vinny to place all three of these words. He could have started with QUESTION or BIZARRE, as shown in the tables below.

    Turn Word
    Vinny’s first turn QUESTION
    Theo’s first turn CHEF
    Vinny’s second turn BIZARRE
    Theo’s second turn ARCADE
    Vinny’s third turn KICK
    Theo’s third turn ORANGE
    Turn Word
    Vinny’s first turn BIZARRE
    Theo’s first turn ARCADE
    Vinny’s second turn QUESTION
    Theo’s second turn CHEF
    Vinny’s third turn KICK
    Theo’s third turn ORANGE

    However, if Vinny started with KICK, then Theo would have made CHEF next, and then Vinny would have made QUESTION. The only possible next word is BIZARRE, so Theo must have made BIZARRE. Thus, if Vinny started with KICK, then he could not have achieved the maximum total score of \(56\). Therefore, Vinny could have started with QUESTION or BIZARRE.

Solution to Extension:

There are many possible words. It helps by looking at the game board to find the letters that Vinny could build off of. Here are some possible words he could make that have a score of \(10\).