The mathematics of Boggle logic puzzles

77d59004fef10003e155461c4c47e037?s=47 Dana Ernst
January 29, 2016

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..


Dana Ernst

January 29, 2016


  1. the mathematics of boggle logic puzzles Friday Afternoon Mathematics Undergraduate

    Seminar 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 1
  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 2
  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” ∙ “QUADRICENTENNIALS” ∙ “SESQUICENTENNIALS” 3
  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 4
  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 5
  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. 6
  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. 7
  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. 8
  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. 9
  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. 10
  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? 11
  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. 12
  14. boggle logic puzzles One More Example 13

  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 that is available for free on the arXiv. ∙ The Wikipedia article on Boggle has some cool information. ∙ Image source: 14