Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Letters, Colors, and Words: Constructing the Ideal Building Blocks Set

Ricardo Salazar, Shahrzad Jamshidi

TL;DR

The study defines a six-cube, six-color building-block puzzle intended to maximize mono and rainbow words drawn from a short English word dataset. It formulates a high-dimensional combinatorial optimization problem and evaluates multiple heuristics—random search, simulated annealing, three tree-search strategies (constrained greedy, best-first, greedy), reinforcement learning, and a genetic algorithm—under strict alphabet-coverage constraints. The results show that, within the constrained setup, a greedy tree-search with the base permutation achieves the best single performance (~2386 words), while a stochastic GA can reach higher counts (up to 2846 words) when constraints are relaxed; RL fails to converge effectively. These findings illustrate the feasibility of heuristic search for combinatorial language-building puzzles and offer insights for designing educational puzzle toys and scalable versions with more cubes or colors.

Abstract

Define a building blocks set to be a collection of n cubes (each with six sides) where each side is assigned one letter and one color from a palette of m colors. We propose a novel problem of assigning letters and colors to each face so as to maximize the number of words one can spell from a chosen dataset that are either mono words, all letters have the same color, or rainbow words, all letters have unique colors. We explore this problem considering a chosen set of English words, up to six letters long, from a typical vocabulary of a US American 14 year old and explore the problem when n=6 and m=6, with the added restriction that each color appears exactly once on the cube. The problem is intractable, as the size of the solution space makes a brute force approach computationally infeasible. Therefore we aim to solve this problem using random search, simulated annealing, two distinct tree search approaches (greedy and best-first), and a genetic algorithm. To address this, we explore a range of optimization techniques: random search, simulated annealing, two distinct tree search methods (greedy and best-first), and a genetic algorithm. Additionally, we attempted to implement a reinforcement learning approach; however, the model failed to converge to viable solutions within the problem's constraints. Among these methods, the genetic algorithm delivered the best performance, achieving a total of 2846 mono and rainbow words.

Letters, Colors, and Words: Constructing the Ideal Building Blocks Set

TL;DR

The study defines a six-cube, six-color building-block puzzle intended to maximize mono and rainbow words drawn from a short English word dataset. It formulates a high-dimensional combinatorial optimization problem and evaluates multiple heuristics—random search, simulated annealing, three tree-search strategies (constrained greedy, best-first, greedy), reinforcement learning, and a genetic algorithm—under strict alphabet-coverage constraints. The results show that, within the constrained setup, a greedy tree-search with the base permutation achieves the best single performance (~2386 words), while a stochastic GA can reach higher counts (up to 2846 words) when constraints are relaxed; RL fails to converge effectively. These findings illustrate the feasibility of heuristic search for combinatorial language-building puzzles and offer insights for designing educational puzzle toys and scalable versions with more cubes or colors.

Abstract

Define a building blocks set to be a collection of n cubes (each with six sides) where each side is assigned one letter and one color from a palette of m colors. We propose a novel problem of assigning letters and colors to each face so as to maximize the number of words one can spell from a chosen dataset that are either mono words, all letters have the same color, or rainbow words, all letters have unique colors. We explore this problem considering a chosen set of English words, up to six letters long, from a typical vocabulary of a US American 14 year old and explore the problem when n=6 and m=6, with the added restriction that each color appears exactly once on the cube. The problem is intractable, as the size of the solution space makes a brute force approach computationally infeasible. Therefore we aim to solve this problem using random search, simulated annealing, two distinct tree search approaches (greedy and best-first), and a genetic algorithm. To address this, we explore a range of optimization techniques: random search, simulated annealing, two distinct tree search methods (greedy and best-first), and a genetic algorithm. Additionally, we attempted to implement a reinforcement learning approach; however, the model failed to converge to viable solutions within the problem's constraints. Among these methods, the genetic algorithm delivered the best performance, achieving a total of 2846 mono and rainbow words.

Paper Structure

This paper contains 29 sections, 2 theorems, 9 equations, 6 figures, 24 tables.

Table of Contents

  1. Introduction
  2. Set up
  3. Data set
  4. One-dimensional representation of a cube set
  5. Letter repetition
  6. Search space
  7. Search algorithms
  8. Random Search
  9. Simulated Annealing
  10. N-ary tree search algorithms
  11. Constrained greedy search
  12. Best-first search
  13. Greedy search
  14. Reinforcement Learning
  15. Environment
  16. ...and 14 more sections

Key Result

Lemma A.1

\newlabellem:legal_moves0 Suppose two entries in a 6-cube building block set need to be swapped. This can be done in three of fewer 'legal moves'.

Figures (6)

  • Figure 1: Cube colouring template.
  • Figure 1: Representation of a cube in one dimension.
  • Figure 1: Example of swaps within a cube.
  • Figure 2: Example of Mono and Rainbow words.
  • Figure 3: Base cube set.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Lemma A.1
  • Proof 1
  • Theorem A.2