Any permutation can be represented as a product of independent cycles. The minimum number k needed to return cards to their start order is equal to Least Common Multiply of these cycles. So you need to factorize k = p₁^a₁p₂^a₂... and check whether sum p₁^a₁ + p₂^a₂ + ... is greater than 10⁵. If it is greater, then there is no solution, else answer is a permutation with lengths of cycles equal to p₁^a₁, p₂^a₂, ...

Let's make a graph, words in dictionary are the vertices, and if two words differs only in one letter, then there is an edge between them. To construct such a graph, put all words in map or trie and then for each word try to change letter at each position to all other possible 25 letters; if such a word exists in dictionary, then make an edge between this words. Now you need to run BFS from first word to last word.

slides 19-21