Talk:15 puzzle

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 Field:  Discrete mathematics

For inclusion: puzzle history at this page


I have a strong background in mathematics but did not understand the proof. Can someone please clarify the language? Asteron 17:59, 29 August 2006 (UTC)

Yes, that piece seems to be unclear, but English is not my native language, and I perhaps can't clarify this in the best way. I suggest you to follow the link on top of this Talk Page and read the last but one paragraph there. I think, it's clear enough. Cmapm 18:33, 29 August 2006 (UTC)


The page states that there IS a hueristic to solve it, but doesn't give that solution. We definitely need this... Fieari 18:08, 29 August 2006 (UTC)

There is an actual solution. I can't describe it in detail, but it involves solving the top two rows first, then the bottom row. If done correctly, the third row solves itself. 23:19, 5 August 2007 (UTC)

1) We need to distinguish a 'human-like' solution from a more optimal (in terms of minimal number of moves) solution. a) A human like solution would be based on considering one tile at a time (and on a higher level, one row at a time) b) A computer solution could consider more or even all tiles at a time

2) About the human algorithm

I think we need to be more specific on 'done correctly' as described above. It's actually the trickiest part of the puzzle. Actually, I made an algorithm that does row 1 first, then row 2 and then row 3 and 4 together by rolling up row 3 into the left lower 2x2 square and row 4 in the right lower 2x2 square. Since the last row only has 3 tiles, it can always be reordered in the right order in the lower 2x2 square together with the hole. An illustrative video is at: It analyzes the current state in a long if else statement with many cases and calculates just one step ahead. So it does not search a tree, which would be another option. I could make one that only shows the algorithm. also describes it well

3) About the computer algorithm There is a video on youtube illustrating this algorithm somewhere. Looks entirely different. —Preceding unsigned comment added by Mrcastorp (talkcontribs) 22:03, 4 March 2010 (UTC)


The parity does not change, regardless of the number of rows the empty block is moved. Every time the empty block changes rows, it creates an odd number of permutation inversions.

NP hard?[edit]

Is there a formal proof that the problema of finding the shortest solution is a NP hard problem? How is the reduction of this problem? —The preceding unsigned comment was added by (talk) 03:33, 1 January 2007 (UTC).

Requested move[edit]

The following discussion is an archived debate of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the debate was PAGE MOVED per unopposed request. -GTBacchus(talk) 04:15, 29 January 2007 (UTC)


Add "* Support" or "* Oppose" or other opinion in the appropriate section followed by a brief explanation, then sign your opinion with ~~~~


Add any additional comments

The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.


I do understand the parity function and that it divides all possible permutations into two classes: The odd and even permutations. However, I don't understand how to prove that given any two permutations from the same class, there is always a series of valid moves that leads from the one to the other. IMHO, this proof is a necessary addition to the solvability argument. Just because we define a parity function that knows only two different outcomes (odd, even) and then prove that valid moves stay within a class, it doesn't mean that maybe there really are four different classes and any valid move stays inside those, too. That's purely hypothetically of course, I know that the solvability argument given in the argument is true for any n-puzzle, but for me it's not proven yet. jlh (talk) 23:34, 12 December 2008 (UTC)

Rephrasing your question, since obviously all solvable permutations will be even, and all odd permutations will not be solvable, can there be any even permutations that are not solvable? Actually, even if there are, does it matter? The whole point was that (at least) half the permutations are not solvable... -- Jokes Free4Me (talk) 18:19, 3 September 2009 (UTC)

A second question about this solvability issue could be if the proof (that there are at least N!/2 unsolvable permutations) apply for odd N's too, like it applies for even values of N. Can anyone help with this? -- Jokes Free4Me (talk) 18:19, 3 September 2009 (UTC)

Odd permutations are unsolvable for all sizes of the puzzle. Even permutations are always solvable. The article on Mathworld gives references to the original published proofs (Johnson and Storey), which are available on JSTOR but you will need an account or cash to see them in full. SpinningSpark 21:47, 4 September 2009 (UTC)
The "always" part ("Even permutations are always solvable") is not yet justified in the current form of the article. I'll add info about Archer.
And about that JSTOR proof... it's "© 1879", isn't it in the PD already? -- Jokes Free4Me (talk) 12:33, 10 September 2009 (UTC)

The proof given in this section is different from the proof given in section Alternate proof. This section says:

The invariant is the parity of the permutation of all 16 squares plus the parity of the taxicab distance (number of rows plus number of columns) of the empty square from the lower right corner.

But this is not correct. A row move will change the parity of the taxicab distance. But a row move will not change the parity of the permutation. So the parity used in the invariant is changed. --JingguoYao (talk) 07:15, 25 February 2016 (UTC)

Remember that the permutation here includes the empty square, so yes, a row move does change the parity of the permutation (it swaps the empty square with a non-empty square). Perhaps it should be clarified what "the permutation" refers to. Sniffnoy (talk) 15:45, 25 February 2016 (UTC)
It does already say "parity of the permutation of all 16 squares", so obviously that includes the empty square. To explain it any further would be tantamount to explaining that fifteen plus one equals sixteen. SpinningSpark 16:06, 25 February 2016 (UTC)
Thanks. I am clear on this. The invariant is not wrong. --JingguoYao (talk) 01:04, 26 February 2016 (UTC)

Link to 15 puzzle download[edit]

The author of game15 has repeatedly inserted a link to his site where a version of the puzzle can be downloaded. This has become very persistent. Just wanted to check whether there was anyone who disagrees with me continuing to delete this as spam. There is already a link to a Java online version which is much better and does not require a download. The site was also originally not in English, although it has now been (badly) translated and it does not render correctly in Firefox. SpinningSpark 17:49, 15 February 2009 (UTC)

Please keep removing it. The spam easily fails WP:EL. JackSchmidt (talk) 17:57, 15 February 2009 (UTC)

Deletion of category fads[edit]

This article has now been removed from category:fads for the second time. The category went through a deletion discussion which decided to remove all entries from the category but to allow back well-referenced ones. A terrible decision in my view, either the category should have been cleaned of the ones that don't belong or the decision made that we don't want it and delete outright. If ever there was a case of throwing out the baby with the bathwater....but whatever, getting back to this article, I cannot conceive how any fad could be more well documented than the 1880 craze for this one. If any article belongs, it is this one. I can think of several other puzzles that a good case could be made for, but I restored this one in particular because the article details the claim already and has good quality references. Perhaps the deleter was thinking that "craze" does not equal "fad", in which case I refer you to the opening words of the category page "A fad, also known as a craze..." SpinningSpark 22:51, 15 October 2009 (UTC)

Bobby Fischer[edit]

A story I think recounted in Profile of a Prodigy has the chess player entering a 15 tournament that happened to be being held in his hotel and he won it.--Jrm2007 (talk) 14:46, 5 March 2010 (UTC)

Sam Loyd[edit]

User:DreamGuy's deletion of the image was unconstructive and pointless, but there could be more about the unsolvable 15-14 variant, which kind of launched the 15 puzzle into a certain renewed notoriety for a few years in the 1890s(? not sure about the exact dates). AnonMoos (talk) 06:46, 17 March 2011 (UTC)

Obviously, I agree with that as I originally reverted DreamGuy. The diagram makes clear what the configuration of the whole puzzle is for the unsolvable state. It is, of course, possible to swap the 14 and 15 tiles on a solvable puzzle if the state of the rest of the puzzle is freely choosable. As for writing more on the renewal of the craze, I suggest getting a copy of Jerry Slocum's book which is the main source for this. User:Spinningspark 07:48, 17 March 2011 (UTC)

Having the exact same image as at the top of the article but with numbers flipped serves no encyclopedic purpose whatsoever. It makes the article look amateurish, and it suggests that our readers are too stupid to understand simple English. DreamGuy (talk) 13:11, 17 March 2011 (UTC)

Unfortunately for you, the "minor revision" is the difference between the original solvable puzzle vs. the revised unsolvable puzzle, and also the difference between the original ca. 1880 craze (which burned out fairly quickly) vs. the later ca. 1890s partial revival of interest in it... AnonMoos (talk) 16:05, 17 March 2011 (UTC)


The book The 15 Puzzle, Jerry Slocum & Dic Sonneveld, ISBN 1-890980-15-3 is cited four times. It is published by the Slocum Puzzle Foundation, which was founded by one of the authors. It seems to be effectively self-published — is this a reliable source? Deltahedron (talk) 18:15, 3 December 2012 (UTC)

The source NeverEndingBooks appears to be a blog [1]. Why is this a reliable source? Deltahedron (talk) 19:49, 3 December 2012 (UTC)

The definition of odd and even permutations is missing[edit]

The article does not explain what an odd or even permutation is. It also does not reference a definition. But it uses the term more than once. --Ceving (talk) 14:08, 13 February 2013 (UTC)

The first occurence of the word "permutation" in the article is in the phrase "parity of the permutations", wikilinked to Parity of a permutation. Not all readers may connect "odd/even" to "parity", but all the same I think this is sufficient.-- (talk) 14:25, 13 February 2013 (UTC)

Not necessarily square.[edit]

There are variants with non-equal dimensions, the proper name for this puzzle is "mn-1 puzzle", though it's not widely used. But mn-1 puzzles can be treated like n^2-1 puzzles with an extra row (i.e. concentrating on the row(s) first) -- 10:44, 8 October 2013‎