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

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

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 $S_{n+1},/math> from <math>S_n$ , 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

1. Precisely define Sierpinsky graph being dealt with

2. Understand its recursive construction

3. Use mathematical induction or recursive arguemnt to prove existence of Hamiltonian cycles

4. Leverage recursive structure and symmetry to develop recurrence relation for number of distinct Hamiltonian cycles

5. Solve recurrence relation to find formula for count as function of n