Every Second Turn and Uniqueness

[Maxelkat]

Following on from some comments on the 'Puzzles to Inspire Youngsters' thread, I was interested to know whether just drawing a loop and marking every second turn would be likely to give rise to a puzzle with a unique solution. Dr Steve said he used this method to create his examples, and considered it was just luck that they turned out to be unique.
To give myself a good chance of a multiple solution I raised it to a 12x12 grid.
What I found what that it quite a difficult example, and whilst I strived to keep control of the logic, in order to determine whether it had a unique solution (I think it has), I may have made unwarranted leaps.

On this particular example I'd be grateful for someone to see if it does indeed have a unique answer.

A more general question is will you nearly always get a puzzle with a unique answer, and what properties would the loop have to have for there to be multiple solutions (symmetry perhaps?).

Elliott

[drsteve]

Sorry Elliot, but I think there's two solutions to it. I could be wrong though. Second opinion?

Oh, and it's Steve, not Dr Steve Makes me sound like a Californian plastic surgeon. Maybe I should rethink my username - it sounds really odd with the space take out and a couple of capital letters.

[AndreyBogdanov]

Sorry Elliot, but I think there's two solutions to it. I could be wrong though. Second opinion?

You are right, Steve. Both solutions are in the picture.

[nickdeller]

I agree with Steve. Two solutions, differing according to how you link up the path segments in the yellow sections.

EST-Maxelkat.jpg (97.68 KiB) Viewed 14256 times

Looking at it again, the ambiguous sections seem to have some interesting properties - rotational symmetry, no circles, and interchangable. I wonder if any of that is a clear-cut flag of anything? Obviously, as a brute force fix, you could make the solution unique by including any one path segment from within those blocks as a starter.

[Maxelkat]

Thanks Steve, Andrey and Nick, I might have known I'd missed something. It's not particularly obvious how you would tweak it into uniqueness either, unfortunately.
I still think just drawing a loop and then checking for uniqueness is a pretty good way of creating them, although it has the disadvantage that you have to be able to solve them too!

Elliott

[Maxelkat]

Okay, so, not one to be deterred, I had another crack at achieving uniqueness from a more-or-less randomly drawn loop. I wondered whether lack of uniqueness was a property of the loop itself, or the parity of nodes I had chosen.
So I took the very same loop that had failed to yield a unique solution previously, and selected the alternate set of nodes.
I believe that this new puzzle

12x12 every second turn 2.xls

(13.49 KiB) Downloaded 556 times

now does have a unique solution (but I thought that before and was wrong).
If any one fancies it I'd be grateful of a second opinion on its uniqueness.
Elliott

[nickdeller]

I believe that this new puzzle now does have a unique solution (but I thought that before and was wrong).
If any one fancies it I'd be grateful of a second opinion on its uniqueness.
Elliott

Unique and solvable. Interesting!

[ronaldx]

gosh that's cool! do both loops have a unique solution if you use the alternative corners? is that a property of the loops or just random luck?

[PuzzleScot]

Unique. Agreed.

[nickdeller]

gosh that's cool! do both loops have a unique solution if you use the alternative corners? is that a property of the loops or just random luck?

Even more interestingly, no they don't! The second one solves uniquely up to the position below. From there though, the path can't turn in the orange cell, but the symmetry of the wider yellow section means that there are two solutions dependent on whether we use the orange cell for a horizontal or a vertical path segment.

EST-Maxelkat2.jpg (51 KiB) Viewed 14225 times

[Maxelkat]

I've tried this a few times now, with various different sizes, and I almost always hit upon a unique solution, so I'd say it's definitely a good way of making them, and fun too.

Elliott