Evaluation of GPT-5 on an Advanced Extension of Kashihara's Problem

This work presents the solution of Problem 1 of Question #30 from Comments and Topics on Smarandache Notions and Problems, published in 1996 by Kenichiro Kashihara, corresponding to Problem 16 of Only Problems, Not Solutions!. It is a recreational mathematics problem that remained open until 2010, when an extended version was published in Appendix 2 of Divisibilità per 3 degli elementi di particolari sequenze numeriche, Rudi Mathematici Bookshelf. The question concerns OEIS sequence A001292 (the so-called "circular sequence") and the probability that a generic element ends with a given digit c∈{0,1,2,...,9}. In 2010, an enhanced version of the problem was studied, providing the general formula of this probability as a function of the length of the last complete block and of c, and using the exact value to bound the interval in which the probability lies in the inter-block (partial block) case, as reported in note 14 on page 17 of the aforementioned work. On August 9, 2025, GPT-5 independently solved the same enhanced version of Kashihara's question, proposing a more compact formula and extending it exactly to the incomplete block case as well.