Freaks City

Notation Let \(dp_i\) denote the expected number of attempts to correctly play the prefix \(1 \to i\), with \(dp_1 = n\). The target melody is given by the array \(target_i\). Analysis When playing a melody, the current partially correct sequence falls into one of two cases: The current suffix matches some prefix of the melody. No suffix of ...

This article addresses the problem of finding the coefficient of the $x^n y^m$ term in the expansion of the polynomial $(ax + by)^k$. The solution involves applying the binoimal theorem and calculating combinations modulo 10007. Binomial Theorem Application The binomial theorem states that $(x+y)^k = \sum_{i=0}^{k} \binom{k}{i} x^{k-i} y^i$. In ...

Problem Description Given that today is Saturday, determine which day of the week it will be after 20^22 days. Note that days are represented as 1 (Monday) through 7 (Sunday). Approach This problem can be efficiently solved using modular arithmetic to avoid computing the extremely large value of 20^22 directly. Since the week cycles every 7 day ...