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

TL;DR

This paper introduces a complex combinatorial problem of assigning letters and colors to cube faces to maximize spellable words, exploring various optimization algorithms and finding the genetic algorithm most effective.

Contribution

It formulates a novel problem of optimizing letter and color assignments on cubes to maximize specific word formations, and evaluates multiple heuristic and evolutionary algorithms for solving it.

Findings

Genetic algorithm outperformed other methods in maximizing words

The problem is computationally intractable for brute force solutions

2846 words were achieved using the best algorithm

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.