1,2

The 15-block puzzle is often referred to (incorrectly) as Sam Loyd's 15-Puzzle (see Slocum and Sonneveld). The actual inventor was Noyes Chapman, the postmaster of Canastota, New York, who applied for a patent in March 1880.

Comment from Dan Hoey (Nov 10 2003): The set of moves from a given position depend on where the blank is. There is also a variant in which sliding a row of tiles counts as a single move. For the 8-puzzle I find:

Move....Blank...Maximum....Number of maximal-distance positions and

slide...home....distance...(position of blank in those positions)

tile....corner..31...........2 (adjacent edge)

tile....edge....31...........2 (adjacent corner)

tile....center..30.........148 (88 corner, 60 center)

row.....corner..24...........1 (center)

row.....edge....24...........2 (diagonally adjacent edge)

row.....center..24...........4 (corner)

The maximum number of moves required to solve the 2 X 3 puzzle is 21. The only (solvable) configuration that takes 21 moves to solve is (45*)/(123). - Sergio Pimentel (ferdiego(AT)suddenlink.net), Jan 29 2008. (See A151943. - N. J. A. Sloane, Aug 16 2009)

For additional comments about the history of the m X n puzzle see the link by Anton Kulchitsky. - N. J. A. Sloane, Aug 16 2009

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, see Vol. 2 for the classical 4 X 4 puzzle.

A. Br\"ungger, A. Marzetta, K. Fukuda and J. Nievergelt, The parallel search bench ZRAM and its applications, Annals of Operations Research 90 (1999), 45-63.

J. C. Culberson and J. Schaeffer, Computer Intelligence, vol. 14.3 (1998) 318-334.

Richard E. Korf, Depth-First Iterative-Deeping: An Optimal Admissible Tree Search, Artificial Intelligence, 27(1), 97-110, 1985.

Richard E. Korf and Larry A Taylor, Finding Optimal Solutions to the Twenty-Four Puzzle, Proceedings of the 11th National Conference on Artificial Intelligence, 756-761, 1993.

Richard E. Korf and Larry A Taylor, Disjoint pattern database heuristics, in "Chips Challenging Champions" by Schaeffer and Herik, pp. 13-26.

D. Ratner and M. Warmuth, Finding a shortest solution for the (N x N)-extension of the 15-puzzle is intractable, J. Symbolic Computation, 10: 111-137, 1990.

J. Slocum and D. Sonneveld, The 15 Puzzle, The Slocum Puzzle Foundation, 2006.

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 262.

Joseph C. Culberson and Jonathan Schaeffer, Searching with Pattern Databases

E. D. Demaine, Playing Games with Algorithms: Algorithmic Combinatorial Game Theory, 2001.

Filip R. W. Karlemo and Patric R. J. Ostergard, On Sliding Block Puzzles, J. Combin. Math. Combin. Comp. 34 (2000), 97-107.

Richard E. Korf, Home Page

Anton Kulchitsky, Comments on the Fifteen Puzzle

A. Reinefeld, Complete Solution of the Eight-Puzzle ..., Internat. Joint Conf. Artificial Intell., pp. 248-253, 1993.

Eric Weisstein's World of Mathematics, 15 Puzzle in MathWorld

All solvable configurations of the Eight Puzzle on a 3 X 3 matrix can be solved in 31 moves or fewer and some configurations require 31 moves, so a(3)=31.

Cf. A151943, A046164.

Sequence in context: A155097 A025524 A042841 this_sequence A096959 A112562 A024447

Adjacent sequences: A087722 A087723 A087724 this_sequence A087726 A087727 A087728

hard,nonn,bref,nice

Jud McCranie (j.mccranie(AT)comcast.net), Sep 30 2003

Edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 11 2003

a(3) is from Reinefeld, who used the method of Korf.

a(4) was found by Br\"ungger, Marzetta, Fukuda and Nievergelt (thanks to Patric Ostergard for this reference)

a(5) >= 114 from Korf and Taylor.