Mathematical Modeling Lecture

BlackOut Game and Its Modeling– Mathematical Models in the Game

2009. 11. 17

Sang-Gu Lee, Duk-Sun KimSungkyunkwan University

sglee@skku.edu

http://www.cjent.co.kr/beautifulmind/

Blackout game(or Lights Out game) consists of a 5by 5 grid of lights; when the game starts, a set ofthese lights (random, or one of a set of storedpuzzle patterns) are switched on. Pressing one ofthe lights will toggle it and the four lights adjacentto it on and off. (Diagonal neighbours are notaffected.) The game provides a puzzle: to switch allthe lights off, preferably in as few button pressesas possible.

http://www.cjent.co.kr/beautifulmind/

Blackout game(or Lights Out game) consists of a 5by 5 grid of lights; when the game starts, a set ofthese lights (random, or one of a set of storedpuzzle patterns) are switched on. Pressing one ofthe lights will toggle it and the four lights adjacentto it on and off. (Diagonal neighbours are notaffected.) The game provides a puzzle: to switch allthe lights off, preferably in as few button pressesas possible.

Example : 3x4 case Blackout Game

Offline Simulator : Blackout Game Simulator

http://matrix.skku.ac.kr/bljava/Test.htmlhttp://matrix.skku.ac.kr/bljava/Test.html

The Tool of Black-Out Game with Java

Example : 21x15 size Black-Out

game

Example : 21x15 size Black-Out

game

Find the SolutionFind the Solution

Green Button is solution

Complete result(Press all green

button!)

Complete result(Press all green

button!)

Merin GameParker Brothers

• Developed in 1970

• Start from 3x3 size

Lights Out

XL-25Vulcan Electronics

• Developed in 1983

• For the Home Video Game

Black OutLights Out

Tiger Toy

• Developed in 1997• 5x5 size• Most popular style

Black OutMany Developers

• After the dvelopement of the LightsOut Game by the Tiger Toy Inc.

• Developed for the PDA, Cellular Phone, and PC

• 5x5 size game is popular.

Black OutKorea

CJ Enterainment introduce this game on the movie

site, “Beautiful Mind”.

Anderson, Feil1998 / 3x3 – 5x5

• Turning lights out with linear algebra / Mathematics Magazine, Vol. 71, No. 4, pp.300-303

• Introduced the solution of the 3x3sized Blackout Game.

Losada

Araujo2000 / 5x5 Solutions

• How to Turn All the Lights Out / Elem. Math. 55, p. 135 – 141

• Some solution in 5x5 cases wasshown

• Some patterns for the initial configration

sized Blackout Game.

• Tried to 5x5 cases: There is no solution for initial configuration

Losada2002 / Various Types

• ALL LIGHTS AND LIGHTS OUT –An investigation among lights and shadow / SUMA, Vol 40• Introduced in Spanish• Introduced various styles.

Korea2004 - 2006

• 2004 : Exapnd to the 19x19 size from the 3x3 size Blackout Game/ Developed the simulator for this

game.

• 2006 : Expand to the nxn szie game.Find some relations with Sigma+ Game

Recents2007 – Recents

• Expand to the mxn size game

• Survey to the general initial condition.

• Application to the teaching of the gifted students

§ Size : 3x41. Suppose the following configuration was given.

11243

010011010010

´´ ==÷÷÷

ø

ö

ççç

è

æbB 11243

010011010010

´´ ==÷÷÷

ø

ö

ççç

è

æbB

We like to make all cells to have same color by clicking some cells

We like to make all cells to have same color by clicking some cells

§ Size : 3x42. Think a (0,1)-matrix which change color of cells with adjacent edges

÷÷÷

ø

ö

ççç

è

æ

000000010011

÷÷÷

ø

ö

ççç

è

æ

001001110010

÷÷÷

ø

ö

ççç

è

æ

000000010011

÷÷÷

ø

ö

ççç

è

æ

001001110010

§ Size : 3x43. We have 12 matrices of all actions on each clicksiM

Each can be considered as a 12x1 vectorsEach can be considered as a 12x1 vectorsiM

§ Size : 3x44. Consider linear combinations of 12 matrices

BMaMaMa ii -=++++ 121211 LL (or )BJ -( has all 1-component )J Make vector from B( has all 1-component )J

[ ] bmm -=úúú

û

ù

êêê

ë

éÞ

12

1

121

a

aML

bx -=Þ A (or ) : This is linear equation problem!

bj-

Convert to the column vectors

§ Other Case : 3x51.We can get these matrices in 3x5 case.

11

Including 3 block matrices with 5x5Including 3 block matrices with 5x5

The size of A is 15(=3x5)x15.The size of A is 15(=3x5)x15.

where

§ General Case : mxn1. We can make this equation in mxn case.

Including m block matrices with nxnIncluding m block matrices with nxn

n rows

n columns

mm

Tridiagonal matrixTridiagonal matrix

Block Tridiagonal matrixBlock Tridiagonal matrix

The size of A is mn(=mxn)xmn.The size of A is mn(=mxn)xmn.

§ General Case : mxn2. Unique solution of mxn size Black-Out Game

bx -=A : consistentÞ¹ 0)det(Abjx -=A(or )

: consistentÞ¹ 0)det(Abjx -=A(or )

3. For nonsingular matrices A, we can determine the existence of the unique solution of the general Black-Out game.

[Pattern 1][Pattern 1]

[Pattern 2][Pattern 2]

•Determinant : 00

•Determinant is not zero1

•Determinant : 0 [Patttern 2]2

Pattern Analysis of Determinants: S.-G. Lee and D.-S. Kim, Optimal solution

of the $m \times n$ size blackout game and its tiling, J. Korea Soc. Math. Ed. Ser. E:

Communications of Mathematical Education, Vol. 21 (2007) , No. 4, pp. 597--612.

n rows

mm

n columns Tridiagonal matrixTridiagonal matrix

Block Tridiagonalmatrix

Block Tridiagonalmatrix

How we can solve this linear system of equations?How we can get the determinant of this block tridiagonal matrix?

Apply to the Blackout Game Cases: L. G. Molinari, Determinant Apply to the Blackout Game Cases: L. G. Molinari, Determinant

of block tridiagonal matrices, Linear Algebra and its Applications, 429

(2008), 2221-2226

A Generating Matrix of the Fibonacci Sequence

We can assign an interpretation of the Riordan matrix to find the generating sequences.

A-sequences(in Riordan array) from two generating functions

: General polynomial to get of the T11.

[Pattern 1][Pattern 1]

Why it shows the pattern 1 in the determinant table of the tridiagonal matrix?

•Determinant : 00

•Determinant is not zero1

•Determinant : 0 [Patttern 2]2

Why it shows the pattern 1 in the determinant table of the tridiagonal matrix?

[Pattern 1][Pattern 1]

The result about the determinants of K

Repeat this numbers: A010892 in http://www.research.att.com/njas/sequences/

– M. Anderson and T. Feil, Turning lights out with linear algebra, Mathematics Magazine, 71 (1998) No. 4, 300-303– P. V. Araujo, How to Turn All the Lights Out, Elem. Math. 55 (2000), 135-141– P. Barry, A catalan transform and related transformations on integer sequences, Journal of Integer Sequence, 8 (2005), Article

05.4.5.– G. Birkhoff and S. McLane, Algebra, 3rd ed. Chelsea. 1999– N. D. Cahill, D. A. Narayan, Fibonacci and Lucas Numbers as Tridiagonal Matrix determinants, Fibonacci Quart. 42 (2004), No.

3, 216-221– T. Delgado, 'Beyond Tetris' - Lights Out, GameSetWatch, January 29, 2007.– S. Hansell, Building a Better Cat, New York Times, December 5, 2002.– S.-T. Jin, A characterization of the Riordan Bell subgroup by C-sequences, Korean Journal of Mathematics, 17 (2009), No. 2,

147-154.– S.-G. Lee and D.-S. Kim, Optimal solution of the mxn size blackout game and its tiling, J. Korea Soc. Math. Ed. Ser. E: – S.-G. Lee and D.-S. Kim, Optimal solution of the mxn size blackout game and its tiling, J. Korea Soc. Math. Ed. Ser. E:

Communications of Mathematical Education, 21 (2007) , No. 4, pp. 597-612.– S.-G. Lee, D.-S. Kim, C.-W. Ryu and Y.-M. Song, A history of the mathematical modeling on the blackout game, The Korean

Journal for History of Mathematics, 22 (2009), No. 1, 53-74.– S.-G. Lee, J.-B. Park, J.-M. Yang and I.-P. Kim, Linear algebra algorithm for the optimal solution in the Blackout game, Journal

of Korean Soc. Math. Ed. Ser. A : The Mathematical Education, 43 (2004), No. 1, 87-96– S.-G. Lee, H.-G. Seol and S.-I. Han, A Research on a Model of BL-PBL Self -Directed Linear Algebra Lecture at College, Journal

of Korean Soc. Math. Ed. Ser. E: Comm. of Mathematical Education, 19 (2005), No. 4, 769-785.– S.-G. Lee and J.-M. Yang, Linear Algebraic approach on real sigma-game, Journal of Applied Mathematics and Computing, 21

(2006), No. 1-2, 295-305– R. Losada, All lights and lights out - An investigation among lights and shadow, SUMA 40 (2002)– L. G. Molinari, Determinant of block tridiagonal matrices, Linear Algebra and its Applications 429 (2008), 2221-2226– D. Merlini, R. Sprugnolia and M. C. Verria, Combinatorial sums and implicit Riordan arrays,Discrete Mathematics, 309 (2007),

Issue 2, 475-486– S.-G. Lee, The blackout game simulator in JAVA Applet, http://matrix.skku.ac.kr/bljava/Test.html– N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/