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Gaby numbers her 21 students with the primes between 11 and 97. She now asks them to sit around a circle making sure that any two of them sitting next to each other have either their tens or units digit equal.

In how many essentially different ways can they so sit?

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    $\begingroup$ Do you have some surprising or clever way of counting Hamiltonian cycles in mind, or is this just a brute force calculation exercise on some random graph? If the former, I'll be happy to retract the VTC on your say-so. $\endgroup$ – Bass Jan 20 at 15:36
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The graph has 21 nodes, 62 edges, and

2,924,976

Hamiltonian cycles.

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