标题： 字母、颜色和词语：构建理想的积木集合 显示英文标题

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

摘要： 定义一个建筑块集合为一组n个立方体（每个有六个面），每个面分配一个字母和一个颜色，颜色来自m种颜色的调色板。 我们提出一个新问题，即为每个面分配字母和颜色，以最大化可以从选定数据集中拼写的单词数量，这些单词要么是单色单词，所有字母颜色相同，要么是彩虹单词，所有字母颜色各不相同。 我们考虑一个选定的英语单词集合，长度不超过六个字母，来自一个美国14岁青少年的典型词汇量，并探讨当n=6且m=6时的问题，同时增加每个颜色在立方体上恰好出现一次的限制。 这个问题难以解决，因为解空间的大小使得穷举方法在计算上不可行。 因此，我们旨在使用随机搜索、模拟退火、两种不同的树搜索方法（贪心和最佳优先）以及遗传算法来解决这个问题。 为了解决这个问题，我们探索了一系列优化技术：随机搜索、模拟退火、两种不同的树搜索方法（贪心和最佳优先）以及遗传算法。 此外，我们尝试实现一种强化学习方法；然而，在问题的约束下，该模型未能收敛到可行的解决方案。 在这些方法中，遗传算法表现最佳，总共实现了2846个单色和彩虹单词。 显示英文摘要

摘要： 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.