**Counting** Count the number of paths from S (start) to G (goal) that stays along the white roads. Paths cannot use the same road twice but may visit the same intersection twice. As an example, the pink path shows one valid path.

**Expected difficulty** Insane • Answer • Comment/E-mail if you want a solution to be published

Puzzle 80: Hardcore Mode

Counting

Erm. You may use a calculator or a program. (If you managed to program the solution, then you deserve the answer. I’m not responsible if you miscalculate when you’re multiplying large numbers by hand.)

This is a rejected puzzle of a set born out of a stupid idea. Why did I even think of this? The good news is I have a free Brilliant.org problem idea, and I can practice my programming skills.

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Quite a diverting little puzzle, but not very hard.

I think the answer is x + 6x^2 + 2x^3 + 20x^4 + 6x^5, where x= 1225 (=35^2).

Cheers,

Andrew.

The thing is that in a competition setting, you’ll be all shaky to check whether you have obtained all the paths. (Not to mention 35^10 is a pain to compute.) That’s why the puzzle is rated insane. In a more casual setting I’d call it hard.

Spoilers ahead!

Let a_i be the number of paths from A to B in this graph (hopefully, the spacing will come out right in the actual comment as well as in the editor)

`o-A-o`

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o-o-o

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o-B-o

of length 2i which do not repeat arcs, for i = 1,2,3,4,5. Then the answer is… hidden by chao 😉

Does this count?

Ahh, my previous comment has become unreadable without the LaTeX. I thought that it worked in the comments, too Please delete it (and this comment as well), the first commenter has done the job already.

I’ve edited your comment. I guess you can use code tags in place of LaTeX?