# Connecting the Die Hard Problem and Pythagorean Triples

## Josh Canter

### Abstract

A connection between the Pythagorean Triples and the cup problem (given two unmarked cups find a measure between them) was made during a Mathematics course. Although a solution was found for most of the Pythagorean Triples, the research was broadened in hopes of finding a solution for all Pythagorean Triples. To get a solution, one can use the a linear combination produced by the Euclidean Algorithm. This result was a solution for the greatest common divisor, found by the Euclidean Algorithm. Using that same linear combination one can get solutions for all the multiples of the greatest common divisor less than the largest cup. During all this research several methods for solving with the same two cups became apparent. We furthered the exploration seeking to find the optimal solution, the one with the lease number of steps. We used a theorem about Linear Diophantine Equations and optimize a value n to find to a new solution, the most optimal. All of this research was able to relate back to Pythagorean Triples, solving the original problem.