Bulletin of JSSAC(2014) Vol. 20, No. 2, pp (Received 2013/11/27 Revised 2014/3/27 Accepted 2014/5/26) It is known that some of number puzzles ca

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1 Bulletin of JSSAC(2014) Vol. 20, No. 2, pp (Received 2013/11/27 Revised 2014/3/27 Accepted 2014/5/26) It is known that some of number puzzles can be solved by using Gröbner bases. In this paper, we discuss methods for solving four kinds of number puzzles (Sudoku, Kakkuro, Hashiwokakero and Tile Paint) by using Gröbner bases. Especially, we study the following question. Of what kind of difficulty can Gröbner bases computation programs solve number puzzles? In order to give some answers of this question, we analyze data of the computation experiments. Moreover, in this paper, we introduce efficient techniques for solving number puzzles. Number puzzles have a lot of hidden mathematics rules and useful properties. We introduce these rules and properties, and describe how to translate these ones into algebraic systems of polynomial equations. In general, the Gröbner bases technique for solving number puzzles, is not considered as a good solver for number puzzles, but we show that the Gröbner bases technique can be a good solver for one of the four puzzles (Tile Paint). 1 [1, 2, 3, 4, 7, 9] 1 Risa/Asir[18] nabeshima@tokushima-u.ac.jp c 2014 Japan Society for Symbolic and Algebraic Computation

2 Risa/Asir (1) (2) [1, 3, 4] (3) [3, 7] 2 PC1: PC2: PC OS:Windows 7 Professional (64bit)CPU: Intel(R) Xeon (R) X560@ 2.67GHz 2.66GHzRAM:48GB PC OS:Windows 7 Home Premium(64bit)CPU: Intel(R) Core (TM) i CPU@ 3.40GHz 3.40GHzRAM:4GB time CPU seconds> 2h> 3h > 5h Risa/Asir version Kobe Distribution x i [3] Proposition 3 [3] Proposition 3

3 Bulletin of JSSACVol. 20, No. 2, I x 1... x n I {x i a i i = 1,..., n} a i { } 2 n n n 1 x i i 1 1 n 1 1. x i F i := n (x i j) j= n 2 x i x j G i j := F i F j x i x j [3] [1, 2, 3, 5, 6, 7, 16] [5, 6, 7] Risa/Asir dp_gr_main [1, 3, 16] Risa/Asir dp_gr_main x j x j [3]

4 ) dp_gr_main 1 PC2 Risa/Asir 1 time time time > 2h 1 17 > 2h > 2h 2 17 > 2h > 2h 3 17 > 2h > 2h > 2h 5 18 > 2h > 2h 6 25 > 2h > 2h 7 17 > 2h > 2h 8 17 > 2h > 2h 9 17 > 2h > 2h : 2.2 (1) (2) (3) 1 1)

5 Bulletin of JSSACVol. 20, No. 2, [10] Web (1) (3) (2) [3] 1: 2: Risa/Asir 2 kakuro([[5,[x1,x2]],[17,[x3,x4,x5,x6]],[6,[x7,x8]],[4,[x9,x10]],[10,[x11,x 12,x13,x14]],[3,[x15,x16]],[14,[x3,x7]],[11,[x1,x4,x8,x11]],[4,[x2,x5]],[3,[x12,x15]],[10,[x6,x9,x13,x16]],[3,[x10,x14]]]); [x5-1,-x16+1,x15-2,-x14+2,x13-4,-x12+1,-x11+3,x10-1,-x9+3,-x8+1,-x6+2,-x4+ 5,-x2+3,x1-2,x7-5,-x3+9] 2 x 1 = 2x 2 = 3x 3 = 9x 4 = 5x 5 = 1x 6 = 2x 7 = 5 x 8 = 1x 9 = 3x 10 = 1x 11 = 3x 12 = 1x 13 = 4x 14 = 2x 15 = 2x 16 = 1 [3] [3] 1 2

6 [3] 1 Wikipedia 1 2) E EPSILON DELTA 3) 2 4) PC1 time [3] Wikipedia (2007/10/25) > 5h "E " "E " > 5h > 5h > 5h > 5h > 5h 2: ) Wikipedia :50 3) 4)

7 Bulletin of JSSACVol. 20, No. 2, [16] [17] 1 [20] [5, 6, 7, 16] 2 [4] 2 [8, 9] 3 5) (1) (2) (3) 2 (4) (5) 5) 1990

8 [3] [3] 3 3 x 1 3 x x 2 x 4 x x 3 3: 4: 3 5: 3.1 [3] 1. (2) 2. (3) x 1 x(x 1)(x 2) = 0 3. (4) 4. (5) (3) x i 012 x i (x i 1)(x i 2) = 0 i = 1,..., 6 (4) x 1 + x 5 2 = 0x 1 + x 6 3 = 0x 2 + x 6 2 = 0x 2 + x 4 3 = 0x 3 + x 4 3 = 0 x 3 + x 5 1 = 0 (5) x 2 x 5 0 x 2 x 5 = x 1 = 2x 2 = 1x 3 = 1x 4 = 2x 5 = 0

9 Bulletin of JSSACVol. 20, No. 2, x 6 = 1 (2)(3)(4)(5) (1) (1) (1) 6 7 6: 7: (1) [3] (1) (1) :

10 (1) (1) x x(x 1) = 0 3: (1) 8 (2) (3) 2 (4) (5) (1) i = 1, 2, 3, 4 j = 1, 2, 3k = 1, 2 2 Risa/Asir 6 2 1[11] [11] easy easy 3

11 Bulletin of JSSACVol. 20, No. 2, x 2 x 1 x 3 8 () x 2 x 1 x 7 3 x 2 x 1 x 3 6 x 4 x x i 2 = 0 (x i 2)(x i 1) = 0 x j 2 = 0 () x 2 x 1 x x 1 x 2 x 4 1 x (x j 2)(x j 1) = 0 x k 2 = 0 (x k 2)(x k 1) = 0 4: I PC easy 9 16

12 () x x x 1 x 2 x 4 x (1 4 ) 2 x 2 = 0 x 1 = 0 x i (x i 1) = 0 5: II easymedium (1) (2) 1 (3) 6) 9 web [10] [12, 13, 14, 15] 10 6) 53

13 Bulletin of JSSACVol. 20, No. 2, size time time > 3h > 3h > 3h > 3h > 3h > 3h > 3h > 3h > 3h > 3h > 3h 85 > 3h > 3h > 3h 89 1 > 3h > 3h > 3h > 3h > 3h > 3h > 3h > 3h > 3h 77 > 3h > 3h 101 > 3h > 3h 105 > 3h > 3h > 3h > 3h 84 > 3h > 3h 74 1 > 3h > 3h > 3h 132 > 3h > 3h 129 > 3h > 3h 134 > 3h 6:

14 : 10: 9 11: x 1 x(x 1) = x i (x i 1) = 0 i = 1,..., 11 1 x 1 + x 5 + x 6 + x 9 = 12x 2 + x 7 + x 10 = 4x 3 + x 4 + x 7 + x 11 = 12x 4 + 2x 8 = 2 x 1 + x 2 + x 3 + x 4 = 1x 5 + x 2 + 2x 4 = 2x 6 + 2x 7 + x 8 = 3 x 9 + x 10 + x 11 + x 8 = 2 Risa/Asir [113] gr([x1+x5+x6+x9-1,2*x2+x7+x10-4,x3+x4+x7+x11-1,2*x4+2*x8-2,x1+x2+x3+ x4-1, x5+x2+2*x4-2,x6+2*x7+x8-3,x9+x10+x11+x8-2,x11^2-x11,x10^2-x10,x9^2-x 9,x8^2-x8,x7^2-x7,x6^2-x6,x5^2-x5,x4^2-x4,x3^2-x3,x2^2-x2,x1^2-x1],[x1,x2, x3,x4,x5,x6,x7,x8,x9,x10,x11],2); [x11,x10-1,-x9,-x8+1,-x7+1,-x6,-x5+1,x4,-x3,-x2+1,x1] x 1 = x 3 = x 4 = x 6 = x 9 = x 11 = 0 x 2 = x 5 = x 7 = x 8 = x 10 = 1

15 Bulletin of JSSACVol. 20, No. 2, Step 1 Step 2 Step 1. Step k 1 n n k = s 1 s k 0 12 k = 0n = k n k = s s = < = Step 1 Step 3 Step 1. Step 2. Step

16 : web [12] p time PC1 () time 1 11 (4 4) [19] No.00 (6 6) [19] No.01(10 10) [19] No.04 (10 10) [12] p.49, 1 (10 10) [13] p.39, 1 (10 10) 42 > 5h 7 [14] p.41, 1 (10 10) 38 > 5h 8 [15] p.71, 1 (10 10) [15] p.71, 2 (10 10) 39 > 5h 10 [19] No.02 (10 10) 51 > 5h 11 [19] No.03 (10 15) 64 > 5h 12 [13] p.39, 2 (25 25) 199 > 5h () time 1 11 (4 4) [19] No.00 (6 6) [19] No.01(10 10) [19] No.04 (10 10) [12] p.49, 1 (10 10) [13] p.39, 1 (10 10) [14] p.41, 1 (10 10) [15] p.71, 1 (10 10) [15] p.71, 2 (10 10) [19] No.02 (10 10) [19] No.03 (10 15) [13] p.39, 2 (25 25) 144 > 5h 7: 8:

17 Bulletin of JSSACVol. 20, No. 2, [12, 13, 14, 15] l m 1 n 1 m n m l : 1 Step 1 Step 3 Step 1. Step 2. Step PC1 time time 0

18 () time 12 [13] p.39, 2 (25 25) 199 > 5h 13 [12] p.49, 2 (25 25) 129 > 5h 14 [14] p.41, 2 (25 25) 137 > 5h 15 [15] p.71, 3 (25 25) 173 > 5h 9: () time 12 [13] p.39, 2 (25 25) [12] p.49, 2 (25 25) [14] p.41, 2 (25 25) [15] p.71, 3 (25 25) : () time 1 11 (4 4) [19] No.00 (6 6) [19] No.01(10 10) [19] No.04 (10 10) [12] p.49, 1 (10 10) [13] p.39, 1 (10 10) [14] p.41, 1 (10 10) [15] p.71, 1 (10 10) [15] p.71, 2 (10 10) [19] No.02 (10 10) [19] No.03 (10 15) :

19 Bulletin of JSSACVol. 20, No. 2, Risa/Asir 5 Risa/Asir dp_gr_main 2 [1] Elizabeth Arnold, Stephen Lucas and Laura Taalman, Gröbner basis representations of Sudoku, The College Mathematics Journal, Vol. 41(2), pp , 2010 [2] R. M. Falcón and J. Martín-Morales, Gröbner bases and the number of Latin squares related to autotopisms of order 7, Journal of Symbolic Computation, Vol.43(11-12), pp , 2007 [3] Jesús Gago-Vargas, Isabel Hartillo-Hermoso, Jorge Martín-Morales and José María Ucha- Enríquez, Sudokus and Gröbner bases: not only a divertimento, Proc. Computer Algera in Scientific Computing, LNCS, Vol. 4194, pp , Springer, 2006 [4] Alex Griffith and Adam Parker, A Gröbner basis approach to number puzzles, Proc. Midstates Conference for Undergraduate Research in Computer Science and Mathematics, pp.57 64, Oberlin College, 2009

20 [5] Shutaro Inoue, Efficient singleton set constraint solving by Boolean Gröbner bases, Communication of JSSAC, Vol. 1, pp , 2012 [6] 1785 pp [7] 1666 pp [8] JST CREST ()2011 [9] ( )& 2012 [10] [11] [12] Vol [13] Vol [14] Vol [15] Vol [16] 2012 [17] Vol.18-2 pp [18] Masayuki Noro and Taku Takeshima, Risa/Asir- A Computer Algebra System, Proc. ISSAC 1992, pp , ACM-Press, [19] () [20] Yosuke Sato, Shutaro Inoue, Akira Suzuki, Katsusuke Nabeshima and Ko Sakai, Boolean Gröbner bases, Journal of Symbolic Computation, Vol.46(5), pp , 2011

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