Slice Algorithms for Counting in U-Dimensional Space

This paper presents a computationally efficient algorithm for solving the following well-known die problem: Consider a “crazy die” to be a die with \( n \) faces where each face has some “cost”. Costs need not be sequential. The problem is to determine the exact probability that the sum of costs from \( U \) throws of this die is \( \geq T \), \( T \in \mathbb{R} \). Our approach uses “slice” volume computation in \( U \)-dimensional space. Detailed algorithms, complexity analysis, and comparison with traditional generating functions approach are presented.