Find the shortest string of a given length that meets specific "beautiful" criteria, prioritizing the lexicographically smallest one if multiple solutions exist. The beauty criteria involves avoiding palindromic substrings of length two or three.

Determine if an equation where words represent numbers can be solved such that each letter maps to a unique digit. You need to check if a valid assignment of digits to letters exists to make the equation true.

You are given an array of numbers. Minimize the largest number in the array by repeatedly replacing pairs of numbers with their product.