Project Euler/312

Problem Statement

A Sierpiński graph of order-1 ($ S_1 $) is an equilateral triangle.

$ S_{n + 1} $ is obtained from $ S_n $ by positioning three copies of $ S_n $ so that every pair of copies has one common corner.

Let $ C(n) $ be the number of cycles that pass exactly once through all the vertices of $S_n$.

For example, $ C(3) = 8 $ because eight such cycles can be drawn on $ S_3 $, as shown below:

It can also be verified that:

$ C(1) = C(2) = 1 $

$ C(5) = 71328803586048 $

$ C(10\,000) \bmod 10^8 = 37652224 $

$ C(10\,000) \bmod 13^8 = 617720485 $

Find $ C(C(C(10\,000))) \bmod 13^8 $

Notes

This problem is asking about Hamiltonian cycles on Sierpinski graphs.

The problem combines elements of graph theory, combinatorics, and fractal geometry.

Key concepts:

• Graph theory fundamentals
  • Definition of a graph
  • Cycles and paths
  • Hamiltonian cycles (precise definition is important - cycle visiting every vertex precisely once)
  • Graph properties (number of vertices and edges as function of recursion level)

• Sierpinsky Graphs
  • Recursive construction ($ S_{n+1} $ in terms of $ S_n $)
  • Vertex and edge growth
  • Vertex degrees

• Hamiltonian Cycle Theory

• Recursion and mathematical induction
  • Exploiting recursive structure is central principle for Sierpinski graphs
  • Inductive proofs (arguments based on the recursion level n, base case of 0 or 1)
  • Recursive counting: how to construct Hamiltonian cycles in $ S_{n+1} $ from those in $ S_n $, recurrence relation would be a formula for H(n+1) in terms of H(n) (and potentially other related counts), where H(n) is number of hamiltonian cycles in S(n)

• Combinatorial Counting
  • Recurrence relation may require using standard techniques
  • Symmetry: Sierpinsky graphs possess significant symmetry (rotational/reflectional), so exploiting this symmetry can potentially greatly reduce counting arguments. This looks like counting "real different" cycles, and multiplying by a symmetry factor.
  • Casework within recursion: when constructing a cycle in $ S_{n+1} $ from $ S_n $, need to analyze how cycle traverses connections between the $ S_n $ copies, which might involve careful casework depending on which corner vertices are entry/exit points

Possible Strategy

C(C(C(10_000))) mod 13^8 indicates you don't compute full values, but instead exploit mathematical properties:

• Periodicity:
  • check if the sequence C(n) (mod M) becomes periodic at some point.
  • If a period P can be found, then C(X) (mod M) = C(X (mod P)) (mod M)
  • Chinese Remainder Theorem if M is composite, or properties relate to powers of primes (such as 13^8)

• Matrix exponentiation:
  • If recurrence C(n) is linear homogeneous recurrence, such as C(n) = a*c(n-1) + b*C(n-2), then it can be solved with matrix exponentiation
  • Technique works extremely well with modular arithmetic
  • Allows computing C(N) (mod M) even for very large N in O(log N) matrix multiplications

• Fixed points or specific properties:
  • Specific property related to C, and modulus 13^8, potentially?
  • C(n) (mod M) might converge to a fixed point for large n,
  • C(n) (mod M) might simplify greatly if n is a value modulo M,