Dana Ernst
January 29, 2016
620

# The mathematics of Boggle logic puzzles

Boggle is a popular word search game where players compete to find as many words as they can in a 4 by 4 grid of letters. On the other hand, a Boggle logic puzzle is the game of Boggle played in reverse. A list of words is given and you need to recreate the board. In this talk, we will discuss some of the mathematics behind Boggle logic puzzles. In particular, we will summarize some of the known results and highlight a few open problems.

In the second half of FAMUS, I plan to discuss my path from hating mathematics as a child to falling in love with mathematics to eventually earning my PhD and becoming a professor of mathematics. I hope to share a bit about what I love about mathematics, as well as the joys and struggles of teaching.

This talk was given at the Northern Arizona University Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) on Friday, January 29, 2016..

January 29, 2016

## Transcript

1. the mathematics of boggle logic puzzles
Dana C. Ernst
Northern Arizona University
January 29, 2016

2. boggle game
The Game
∙ Boogle is a word game played with lettered
dice in a 4 × 4 grid.
∙ The game was designed by Allan Turnoff and is
owned by Hasbro.
∙ Players have 3 minutes to ﬁnd as many English
language words as possible subject to:
∙ Words must be in sequences of adjacent
letters (vertical, horizontal, diagonal)
∙ Words at least 3 letters long
∙ A cube in the grid may only be used once
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3. boggle game
Scoring
∙ Each player reads off the words on their list.
∙ If a word appears on more than one list, it is
eliminated from all lists.
∙ Note that “QU” is a single tile, but counts as
two letters.
∙ Words within words are okay. For example:
∙ If “MASTER” appears, then one can also use
“MAST” and “ASTER”
∙ If “WALKING” appears, then one can also use
“WALK”
Word Length Points
3, 4 1
5 2
6 3
7 5
8+ 11
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4. boggle game
Longest Possible Words
In a standard Boggle set, the longest words that can be formed are the following 17-letter
words:
∙ “INCONSEQUENTIALLY”
∙ “SESQUICENTENNIALS”
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5. boggle game
Example Game
Let’s play a game. In 3 minutes, ﬁnd as many sequential words as possible on the
following board.
R E I B
W
E
T
S
H
R
I
T M F
A
R
Check out fuzzylogicinc.net/boggle.
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6. boggle logic puzzles
Boggle Logic Puzzles
Informally, a Boggle Logic Puzzle is a list of words that can be found in a unique n × n
board (up to rotation and reﬂection) satisfying the rules of Boggle.
Example
Fill a 3 × 3 board with the following words so that the rules of Boggle are satisﬁed.
ACT, APE, ATE, COP, END, OLD
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7. boggle logic puzzles
Example (continued)
This example is harder than it looks! Let’s make some observations. Recall the list of
words:
ACT, APE, ATE, COP, END, OLD
What letters are needed?
A, C, D, E, L, N, O, P, T
Sweet, we need exactly 9 letters to ﬁll the 9 spaces. Now, it remains to discover the
adjacency relationship. Graph theory ought to be useful here.
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8. boggle logic puzzles
Example (continued)
Let’s draw an adjacency graph for our word list:
ACT, APE, ATE, COP, END, OLD
A
T
P
O
N L
E
D
C
Our job is to now place the letters in the grid so that the adjacency is preserved.
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9. boggle logic puzzles
A Formal Approach to BLPs
∙ We assume each letter from word list appears exactly once in Boggle board.
∙ Suppose B is ﬁlled Boggle board. Let W(B) be the set of all strings of adjacent letters
(not necessarily actual words) that are at least two letters long.
∙ We say that two ﬁlled Boggle boards B and B′ are equivalent iff W(B) = W(B′).
∙ An n × n Boggle Logic Puzzle is a list of words P such that
1. There exists an n × n ﬁlled Boggle board B with P ⊆ W(B), and
2. If there exists an n × n ﬁlled Boggle board B′ with P ⊆ W(B′), then B is equivalent to
B′.
That is, P can be found in exactly one board up to equivalence.
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10. boggle logic puzzles
A Formal Approach to BLPs (continued)
∙ We can view a Boggle board as a King’s Graph.
∙ For a board B, W(B) is the set of possible paths with no repeat vertex in King’s graph.
∙ We can view a puzzle P as an adjacency graph, called G(P). Letters in words of P are
vertices and two vertices share an edge if the corresponding letters are adjacent in at
least one word in P.
∙ Solving P becomes exercise in ﬁguring out how to get G(P) as subgraph of King’s graph.
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11. boggle logic puzzles
Labeling Subgraphs
Let H be a subgraph of a graph G. Then H is a labeling subgraph of G if whenever any
subgraph H′ of G is isomorphic to H by ϕ, then the isomorphism ϕ induces an
automorphism on G.
Example
The graph on the left is a labeling subgraph of the King’s graph while the one of the right
is not.
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12. boggle logic puzzles
Theorem
A list of words P is a BLP iff G(P) is a labeling subgraph.
Comment
This is related to the Subgraph Isomorphism Problem, which is known to be
NP-complete.
Open Problem?
What are the labeling subgraphs of the n × n King’s graph? What about other graphs?
Questions
1. What are the fewest words needed to create a BLP?
2. How long does a list of words need to be in order to guarantee that you can uniquely
recreate a ﬁlled Boggle board?
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13. boggle logic puzzles
Theorems
1. Any 3 × 3 BLP with no repeated letters and using only 3-letter words must contain at
least 6 different words. (This addresses Question 1.)
2. For 3 × 3 ﬁlled Boggle boards with no letters repeated, one needs 137 different (out of
160 possible) 3-letter words to guarantee a unique solution. (This is a partial answer to
Question 2.)
3. Any 3 × 3 BLP with no repeated letters and using only 2-letter words must contain at
least 11 different words. (This addresses Question 1.)
Open Problem
To date every BLP found involving only 2-letter words uses at least 12 2-letter words. Is
there a 3 × 3 BLP with 9 distinct letters that only has 11 2-letter words.
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14. boggle logic puzzles
One More Example
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15. boggle logic puzzles
References
∙ Richard M. Green (my PhD advisor) wrote a short post about Boggle Logic Puzzles over
on Google+, which is what inspired me to give this talk.
∙ Jonathan Needleman wrote a paper titled 10 Questions about Boggle Logic Puzzles