Friday 11 February 2011

Friday Puzzles #91

And so, dearest reader, we will once more traverse the logical minefield that is numberlink. I’m 95% sure that the solution I have in mind for this week’s puzzle is rigid to the point of being unique – which is interesting because at first glance it doesn’t seem like it should be the case.

I’m hoping very much that this ambiguity is what actually makes this a rather hard puzzle. Assuming that I haven’t bodged it, this is definitely not an easy, and probably rises above medium too – although I’m not a great judge of numberlink difficulty. And as always with numberlink, I think the most valuable feedback would be directed towards any potential multiple solutions anyway, so I positively encourage you to try and break this rather than worry too much about my grading…

And to think I was going to risk yet another comment free week by doing a kakuro! Maybe next week eh. Enjoy! 
    #117 Numberlink – rated medium
All puzzles © Tom Collyer 2009-11


  1. 1. Unique – assuming all spaces are used. Not sure otherwise.

    2. Actually, rather easy for anyone assuming uniqueness – and if you have done any number of these after taking a course in these on the Mellow Melon site. I think I had one decision to actually make to crack the whole thing on the bottom middle after all of the initial obvious lines were started.

    3. I enjoy every Numberlink i can get my hands on, so please understand that I appreciate the effort. I just think that it is mis-rated for anyone with experience with them.



  2. I’m more confident on uniqueness now, I have a heuristic “proof” which convinces me but which I’ll refrain from posting for the spoiler value.

    I think following the MellowMelon school of numberlink ensures that most 10×10 puzzles become “easy”. (incidentally if Palmer is reading then it’d be great to have his view). Nevertheless I’d argue that to the uninitiated there is no easy start-off, and the quickest way to the solution is via some intuitively informed trial-and-error rather than tweaking, as opposed to anything more concrete. I guess difficulty of numberlink when taking into account meta-solving is a discussion I’m definitely not ready to conclude any time soon..

  3. To be honest, I’ve never understood how nikoli can call any 10 by 10 Numberlink hard. Some of them are deceptive enough to be medium, but not more than that. I’d put this between easy and medium because that corner in the bottom right gives so much information, even though the links are a bit stranger than usual and there’s no pseudo-straight ones.

    I was able to prove uniqueness on this one without any other assumptions; here’s how. Start by examining which numbers the 3 can wind around. After brute forcing that a bit, you find out that its path must pass above the 4,1,5 in the center. There’s just too little space to squeeze through on the left for anything else to work. After that, this means the 5 has to go around the 3 through the bottom, so you can have the 5 in the corner go down the left edge.

    After drawing in those paths, the 2s become a problem. Remember the 3 has to pass above the 4,1,5. Suppose the 2’s path does not do the same thing. Then the 3 is forced to wind around the 2 and leave no space on the bottom, sealing off whichever of the 4,1,5 numbers the 2 passed under. Therefore, the 2 and 3 both must pass above the 4,1,5. There’s just enough space for both, so you can draw in both of their paths going along the left and above the 4,1,5.

    That 2-path going along the top edge has to come down to the 2 in the bottom right. The only way for the 3-path to get to the top right is to snake under the 2. So you can draw in a path under the 2 that you know is part of the 3. This is now enough for simple applications of the “different links can’t intersect” rule to completely finish the puzzle from here.

    I’ve actually been thinking about writing a Part II of that guide, which focuses on how as a constructor to prove uniqueness on a puzzle in a manner similar to what I described above. Maybe it will encourage more people to post them, and post them with confidence. But I think I still have more to learn before I start writing it, so I don’t know when it will happen.

    Also, I’m glad to be hearing that the solving primer is so helpful though (here and elsewhere)... I was really worried I was trying to teach something that really couldn’t be put into words. I’m glad I didn’t succumb to that fear.


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