Puzzle: label the blank squares with distinct letters of the alphabet (with A...

David Eppstein - 2018-05-18 16:12:28-0700 - Updated: 2018-05-18 16:12:28-0700

Puzzle: label the blank squares with distinct letters of the alphabet (with A and Z already labeled as shown) so that each two consecutive letters are on adjacent squares. Squares can be adjacent in either of two ways: they can share an edge or they can be connected by a green tunnel. The tunnels at the sides of the board wrap around to the other side.

For the graph-theoretic explanation for the puzzle, see http://www.mathematica-journal.com/issue/v11i3/contents/superhamilton/superhamilton.pdf and for more about the underlying graph see https://en.wikipedia.org/wiki/26-fullerene_graph

The Hamiltonian Puzzle - The New York Times

Jan Moren - 2018-05-18 17:02:21-0700

Not sure I understand. Or, I'm sure I don't understand. The puzzle would be trivially unsolvable since there's no letter that is consecutive to both a and z so square 3 can't be filled.

David Eppstein - 2018-05-18 17:11:24-0700

+Jan Moren Each consecutive pair of letters must be adjacent. That doesn't mean that each adjacent pair of squares must contain consecutive letters. Each square has three neighbors, only two of which can (and for all letters other than A and Z must) contain consecutive letters. The third neighbor can be anything else.

Jan Moren - 2018-05-19 04:15:42-0700

It's probably a language issue then. I don't know the definitions of the words you use well enough to grasp the idea.

Alan Ball - 2018-05-19 06:55:36-0700

What are consequtive letters in this context because in a word consecutive letters are letters that are near each other, but that's also what adjacent means.

Alan Ball - 2018-05-19 06:56:33-0700

Do you mean consequtive alphabetically like a,b or d,e ?

David Eppstein - 2018-05-19 09:14:10-0700

Exactly. The sequence abcdefghijklmnopqrstuvwxyz should form a path through the squares.

Allen Knutson - 2018-05-20 06:25:46-0700

+Jan Moren the implication only goes one way. The B must be next to the A; say it's on the right, above the Z. Then the C would have to be on the other end of the green pipe, in position 4.

Jan Moren - 2018-05-20 16:57:35-0700

+Allen Knutson So, why do you need to follow the rule for squares 2-3, but not squares 3-6? It can't be that it only goes horizontally, or there wouldn't be vertical green lines in the diagram.

David Eppstein - 2018-05-20 17:21:31-0700

Again, you're getting the implication backwards. The rule is not for the squares 2-3, but for the letters B-C. B and C must be placed in adjacent squares. But adjacent squares (such as 3 and 6) are not required to contain consecutive letters.