A Generalization of 3d Queens Adam Pendry Dr. Robinson May 2 nd 2012

8 Queens Rules of the 8 Queens Puzzle, 8 Queens 8x8 Chess Board 8 Directions of movement No conflicts allowed

Four Distinct Solutions for 8 Queens

Generalizing Into N Queens Rules of the N Queens Puzzle N Queens NxN Chess Board 8 Directions of Movment No Conflicts allowed N: Soln ,20073,312

Generalize Into 3d NxN Queens NxNxN Chess Board 26 Directions of Movement No Conflicts allowed

Generating a Solutions Bank Using Java to Create Solutions Using Java to Quantify Solutions queenSolutions.get(0) = {1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0, 0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0}

Creating a Template Start with a solution from the bank Create a corresponding template Isolate a queen Determine what layer Add appropriate numbers to indexes

Sample Code for Template

Trimming The Solution Bank

Finding 3d Solutions

Fit Method

Combine Method

Sample Output Base layer is now 0 Entering Conditional Statement fits on top of 0 Entering Conditional Statement fits on top of 975 and 0 Entering Conditional Statement 7 97 fits on top of 2357 and 975 and 0 Entering Conditional Statement fits on top of 97 and 2357 and 975 and 0 Entering Conditional Statement fits on top of 1370 and 97 and 2357 and 975 and 0 Entering Conditional Statement fits on top of 2365 and 1370 and 97 and 2357 and 975 and 0 Entering Conditional Statement fits on top of 325 and 2365 and 1370 and 97 and 2357 and 975 and 0 Entering Conditional Statement fits on top of 1780 and 325 and 2365 and 1370 and 97 and 2357 and 975 and 0 Entering Conditional Statement fits on top of 2592 and 1780 and 325 and 2365 and 1370 and 97 and 2357 and 975 and 0 Entering Conditional Statement fits on top of 578 and 2592 and 1780 and 325 and 2365 and 1370 and 97 and 2357 and 975 and 0 This is solution number 1 0, 975, 2357, 97, 1370, 2365, 325, 1780, 2592, 578, 2108,

Results Size # Of Solutions Found # Of Solutions Previously Claimed

Proof Of Concept