Bulletin of the Transilvania University of Braşov Vol 8(57), No Series III: Mathematics, Informatics, Physics,

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1 Bulletin f the Transilvania University f Braşv Vl 8(57), N Series III: Mathematics, Infrmatics, Physics, ALGORITHM FOR SOLVING A PUZZLE PROBLEM Adrian DEACONU 1 Abstract We present an algrithm fr slving a puzzle which cnsists in a set f p pieces that have t cver an m x n rectangle Mathematics Subject Classificatin: 91A24. Key wrds: puzzle game, backtracking. 1 Intrductin In this paper, we present an algrithm fr slving the fllwing puzzle prblem: - A rectangle f m x n squares it is given, where m and n are psitive integer numbers. - The rectangle is divided int p disjint pieces, each piece is a cmbinatin f s i squares (i = 1, 2,..., p) s that: p s i = m n (1) i=1 - The pieces are scrambled. - Slving the prblem cnsists in placing the pieces n the m x n rectangle s that the pieces d nt verlap and all the squares are cvered. There are applicatins based n this prblem that can be dwnladed frm Ggle Play (Andrid), Windws Marketplace and App Market (ios). Smetimes it is very difficult t slve the prblem in the advanced levels and the implementatin f the algrithm we prpse can be a very useful tl t thse wh are nt able t pass these levels. Lets cnsider an example with m = 6, n = 5 and p = 5: 1 Faculty f Mathematics and Infrmatics, Transilvania University f Braşv, Rmania, a.deacnu@unitbv.r

2 126 Adrian Deacnu The slutin is the fllwing: Preliminaries The input f the prblem can be cnsidered as fllws: - A 0-initialized m x n matrix dented R (the given empty rectangle) and - A vectr f 0-1 matrices P i (i = 1, 2,..., p), the representatin f the given p pieces (1 where square exists and 0 where square des nt exist). On each margin (the first and the last line, the first and the last clumn) there must be at least ne 1.

3 Algrithm fr slving a puzzle prblem 127 We dente by h i and l i the dimensins f the matrix P i. Fr instance, in the abve example P 1 is: We dente: We have: N ( P i) = h i l i j=1 k=1 P i j,k (2) p N ( P i) = m n (3) i=1 since the pieces cver the rectangle and they d nt verlap. This is a checksum test fr the crrectness f the input. If the checksum test is nt passed it means that the input is incrrect, but if it is passed it des nt necessarily mean that the input is crrect. The utput is the pairs f crdinates (a i, b i ) inside R fr each piece P i, fr each i = 1, 2,..., p, and: and 1 a i m (4) 1 b i n (5) s that the pieces cver the rectangle and they d nt verlap. Fr instance, in the abve example we have: a 1 = 4, b 1 = 2 a 2 = 1, b 2 = 3 a 3 = 2, b 3 = 3 a 4 = 2, b 4 = 1 a 5 = 1, b 5 = 1 3 The algrithm We prceed with psitining the pieces inside R nly if they pass the checksum test (see 3). Well apply a backtracking strategy t slve ur prblem. We try t psitin sequentially the pieces n the rectangle. We try every pssible psitining f each piece till we place all the pieces. The k-th piece is

4 128 Adrian Deacnu psitined at the crdinates i and j inside R if it fits inside R and it des nt verlap with the previus k 1 placed pieces. The number f psitining f the k-th piece decreases with increasing f k because the number f empty spaces decreases and because the shape f the k-th piece matches mre difficult with the previus placed nes. It is mre efficient t psitin the pieces P with bigger N(P ). S, we can start the algrithm by srting descending by N( ) in rder t btain a mre time efficient implementatin. The pseud-cde f the recursive psitining is as fllws: bl SlveRecursively(k) if k = p+1 then return true; fr i=1 t m d fr j=1 t n d if PieceCanBePsitinedAt(k, i, j) then a[k] = i; b[k] = j; if SlveRecursively(k+1) then return true; endfr; endfr; return false; The cmplete algrithm is the fllwing: Srt (P[i])i=1,2,...,n descending by N( ); Sl = false; if ChecksumTest(p, P) then if SlveRecursively(1) then Print(p, a, b); Sl = true; endif endif if nt Sl then Print( N slutin ); 4 Cnclusin and sme pssible extensins We presented an algrithm t slve a puzzle which cnsists in a set f p pieces that have t cver an m x n rectangle. The pieces cannt be rtated.

5 Algrithm fr slving a puzzle prblem 129 A pssible extensin f the prblem can be the situatin when the pieces can be rtated. The implementatin f this variant is mre time cnsuming. The input can be btained by image prcessing using OpenCV by reading the pieces directly frm a picture. This is als a pssible extensin f the applicatin that makes it easier and interactive t use. References [1] D.J. Hff, P.J. Olver, Autmatic Slutin f Jigsaw Puzzles, http : //math.umn.edu/ lver/vi /puzzles.pdf [2] P. Nrvig, Slving Every Sudku Puzzle, http : //nrvig.cm/sudku.html [3] Shlmn, D., David, O. and Netanyahu, N. S., A Genetic algrithm-based slver fr very large Jigsaw puzzles, http : // undatin.rg/penaccess/cntent cvpr 2013/papers/Shlmn A Genetic Algrithm Based 2013 CV P R paper.pdf [4] Wng, J., The Fifteen Puzzle - The Algrithm, http : //jamiewng.cm/2011/10/16/f if teen puzzle algrithm.

6 130 Adrian Deacnu