Ponder This

May 2025 - Solution

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May 2025 Challenge

May 2025 Solution:

The numerical solutions are

p = 8999
q(1) = 3898

Finding the correct test results is a standard task for constraint solvers, which can also be performed manually. The data was choses such that there is only one valid solution. Given this solution, the first 8 questions give us a linear equations system which has a solution in integers over the rational numbers, meaning the same solution is also valid over any prime p. So we first solve this system.
The next equation tells us that multiplying all the results give us a number N such that the prime p we are looking for divides N-1. So we factor N-1 (easy for numbers of this size), and for every prime factor we check whether the last equation holds. Only p=8999 passes this test.
In order to find q(1) one simply find the correct test results and then uses some interpolation method (e.g. Lagrange interpolation) over F_{8999}.