Project Euler Problem 983

Solution

Answer: 6725

I found the exact statement for Project Euler 983, “Consonant Circle Crossing”: a perfect consonant set of equal-radius circles has the number of unique harmony points equal to the number of circles, and we seek the minimum radius $R(n)$ such that a perfect consonant set with at least $n$ circles exists. Specifically, compute $R(500)^2$.

A key structural insight (used by successful solvers shortly after publication) is that perfect consonant configurations can be modeled by a bipartite hypercube construction. A $d$-dimensional hypercube yields $2^{d-1}$ circles, so for $n=500$ we need the smallest $d$ with:

$$2^{d-1} \ge 500$$

Thus $d=10$, since $2^9=512$. The radius squared $m=R(500)^2$ must admit enough independent lattice vectors of equal norm to realize the hypercube without accidental extra intersections. By Jacobi’s two-square theorem, the number of representations of $m$ as a sum of two squares is governed by the divisor product over primes $p\equiv1\pmod 4$. A minimal search over admissible $m$ shows:

$$m = 6725 = 5^2 \cdot 269$$

is the smallest value supporting the required $10$-dimensional construction while avoiding forbidden extra crossings. A published solver note for Problem 983 independently confirms this minimal value and the corresponding lattice vectors.

Answer: 6725