composition of solutions for the n+k queens separation problem biswas sharma jonathon byrd morehead...
TRANSCRIPT
![Page 1: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/1.jpg)
Composition of Solutions for the n+k Queens Separation Problem
Biswas SharmaJonathon Byrd
Morehead State UniversityDepartment of Mathematics, Computer Science and Physics
![Page 2: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/2.jpg)
Queen’s Movements
• Forward and backward• Left and right• Main diagonal and
cross diagonal
![Page 3: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/3.jpg)
n Queens Problem
• Can n non-attacking queens be placed on an n x n board?
• Yes, solution exists for n=1 and n ≥ 4.
![Page 4: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/4.jpg)
n Queens Problem
11 non-attacking queens on an 11 x 11 board
![Page 5: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/5.jpg)
n + k Queens Problem
• If pawns are added, they block some attacks and hence allow for more queens to be placed on an n x n board.
• Can we place n + k non-attacking queens and k pawns on an n x n chessboard?
• General solution exists when n > max{87+k, 25k}
![Page 6: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/6.jpg)
n + k Queens Problem
11 x 11 board with 12 queens and 1 pawn
![Page 7: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/7.jpg)
n + k Queens Problem
• Specific solutions for lesser n-values found for k=1, 2, 3 corresponding to n ≥ 6,7,8 respectively
• We want to lower the n-values for k-values greater than 3
k values Min board size (n)
1 6
2 7
3 8
k n > max{87+k, 25k}
4 100
5 125
6 150
![Page 8: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/8.jpg)
Composition of Solutions
Step 1: Pick and check an n Queens solution
![Page 9: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/9.jpg)
Composition of Solutions
Step 1: Pick and check an n Queens solution
![Page 10: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/10.jpg)
Composition of Solutions
Step 1: Pick and check an n Queens solution
![Page 11: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/11.jpg)
Composition of Solutions
Step 1: Pick and check an n Queens solution
![Page 12: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/12.jpg)
Composition of Solutions
Step 1: Pick and check an n Queens solution
![Page 13: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/13.jpg)
Composition of Solutions
Step 1: Pick and check an n Queens solution
![Page 14: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/14.jpg)
Composition of Solutions
Step 1: Pick and check an n Queens solution
![Page 15: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/15.jpg)
Composition of Solutions
Step 1: Pick and check an n Queens solution
![Page 16: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/16.jpg)
Composition of Solutions
Step 2: Copy it!
![Page 17: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/17.jpg)
Composition of Solutions
Step 3: Rotate it!
![Page 18: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/18.jpg)
Composition of Solutions
Step 3: Rotate it!
![Page 19: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/19.jpg)
Composition of Solutions
Step 3: Rotate it!
![Page 20: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/20.jpg)
Composition of Solutions
Step 3: Rotate it!
![Page 21: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/21.jpg)
Composition of Solutions
Step 3: Rotate it!
![Page 22: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/22.jpg)
Composition of Solutions
Step 3: Rotate it!
![Page 23: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/23.jpg)
Composition of Solutions
Step 3: Rotate it!
![Page 24: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/24.jpg)
Composition of Solutions
Step 3: Rotate it!
![Page 25: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/25.jpg)
Composition of Solutions
Step 3: Rotate it!
![Page 26: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/26.jpg)
Step 4: Overlap it!
This is how we compose a (2n-1) board using an n board…
… and so all the composed boards are odd-sized.
![Page 27: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/27.jpg)
Step 5: Place a pawn
![Page 28: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/28.jpg)
Step 5: Place a pawn
![Page 29: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/29.jpg)
Step 6: Check diagonals
![Page 30: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/30.jpg)
Step 6: Check diagonals
![Page 31: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/31.jpg)
Step 6: Check diagonals
![Page 32: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/32.jpg)
Step 6: Check diagonals
![Page 33: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/33.jpg)
Step 6: Check diagonals
![Page 34: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/34.jpg)
Step 6: Check diagonals
![Page 35: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/35.jpg)
Step 6: Check diagonals
![Page 36: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/36.jpg)
Step 6: Check diagonals
![Page 37: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/37.jpg)
Step 7: Move Queens
![Page 38: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/38.jpg)
Step 7: Move Queens
![Page 39: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/39.jpg)
Step 7: Move Queens
![Page 40: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/40.jpg)
Step 8: Check Diagonals
![Page 41: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/41.jpg)
Step 8: Check Diagonals
![Page 42: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/42.jpg)
Final Solution!
![Page 43: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/43.jpg)
Composition of Solutions
• Dealing with only k = 1• Always yields composed
boards of odd sizes
n Solution Composed Size (2n -1 )
7 13
8 15
9 17
10 19
![Page 44: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/44.jpg)
Some boards are ‘weird’
• E.g. boards of the family 6z, i.e., n = 6,12,18… boards that are known to build boards of sizes (2n-1) = 11,23,35…
![Page 45: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/45.jpg)
Some boards are ‘weird’
n = 12 board with no queen
![Page 46: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/46.jpg)
Some boards are ‘weird’
n = 12 board with 11 non-attacking queens
![Page 47: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/47.jpg)
Some boards are ‘weird’
n = 12 board with 11 originally non-attacking queens and one arbitrary queen
in an attacking position
![Page 48: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/48.jpg)
Some boards are ‘weird’
n = 23 board built from n = 12 boardThis board has 24 non-attacking queens and 1 pawn
![Page 49: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/49.jpg)
Future Work
• Better patterns for k = 1• Composition of even-sized boards• Analyzing k > 1 boards
![Page 50: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/50.jpg)
Thank you
• Drs. Doug Chatham, Robin Blankenship, Duane Skaggs
• Morehead State University Undergraduate Research Fellowship
![Page 51: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/51.jpg)
ReferencesBodlaender, Hans. Contest: the 9 Queens Problem. Chessvariants.org. N.p. 3
Jan. 2004. Web. 12 Mar 2012. <http://www.chessvariants.org/problems.dir/9queens.html>.
Chatham, R. D. “Reflections on the N + K Queens Problem.” College Mathematics Journal. 40.3 (2009): 204-211.
Chatham, R.D., Fricke, G. H., Skaggs, R. D. “The Queens Separation Problem.” Utilitas Mathematica. 69 (2006): 129-141.
Chatham, R. D., Doyle, M., Fricke, G. H., Reitmann, J., Skaggs, R. D., Wolff, M. “Independence and Domination Separation on Chessboard Graphs.” Journal of Combinatorial Mathematics and Combinatorial Computing. 68 (2009): 3-17.
![Page 52: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/52.jpg)
Questions?
Thank you all
![Page 53: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/53.jpg)
![Page 54: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/54.jpg)
A ‘differently weird’ board
2+6z board (n=14)
![Page 55: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/55.jpg)
All-nighters (may) yield solutions
![Page 56: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/56.jpg)
All-nighters (may) yield solutions
![Page 57: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/57.jpg)
Example that doesn’t work
Step 1: Pick and check an n Queens solution
![Page 58: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/58.jpg)
Example that doesn’t work
Step 1: Pick and check an n Queens solution
![Page 59: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/59.jpg)
Example that doesn’t work
Step 1: Pick and check an n Queens solution
Problem!
![Page 60: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/60.jpg)
Example that doesn’t work
Step 1: Pick and check an n Queens solution
![Page 61: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/61.jpg)
Example that doesn’t work
Step 1: Pick and check an n Queens solution
![Page 62: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/62.jpg)
Example that doesn’t work
Step 1: Pick and check an n Queens solution
![Page 63: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/63.jpg)
Example that doesn’t work
Step 1: Pick and check an n Queens solution
![Page 64: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/64.jpg)
Example that doesn’t work
Step 1: Pick and check an n Queens solution
![Page 65: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/65.jpg)
Composition of Solutions
Step 2: Copy it!
![Page 66: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/66.jpg)
Composition of Solutions
Step 3: Rotate it!
![Page 67: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/67.jpg)
Composition of Solutions
Step 3: Rotate it!
![Page 68: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/68.jpg)
Step 4: Overlap it!
![Page 69: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/69.jpg)
Step 5: Place a pawn
![Page 70: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/70.jpg)
Step 5: Place a pawn
![Page 71: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/71.jpg)
Step 6: Check diagonals
![Page 72: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/72.jpg)
Step 7: Move Queens
![Page 73: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/73.jpg)
Step 7: Move Queens
![Page 74: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/74.jpg)
Step 8: Check Diagonals
![Page 75: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/75.jpg)
Step 8: Check Diagonals
![Page 76: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/76.jpg)
Review: Check Diagonals
![Page 77: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/77.jpg)
Review: Check Diagonals
![Page 78: Composition of Solutions for the n+k Queens Separation Problem Biswas Sharma Jonathon Byrd Morehead State University Department of Mathematics, Computer](https://reader037.vdocuments.mx/reader037/viewer/2022110304/551b9267550346a6148b604c/html5/thumbnails/78.jpg)
Review: Check Diagonals
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Review: Check Diagonals