Creating Dense Boggle® Boards

Description of the Boggle® board layout problem

o---o---o---o---o | A | T | G | C | o-|-\---/---\---o | L | R | J | E | o---o---o---o---o | J | R | F | G | o---o---o---o---o | M | H | E | S | o---o---o---o---o Figure 1: Sample 4x4 Boggle® board containing the word "large".

Summary of methods used

Start with N random 4x4 board configuration (or 5x5). Perturb each of the N random configurations in one of the ways described below. To better determine perturbation success, the program keeps track of the percentage of times that a particular perturbation type given a particular parameter succeeds in improving the board configuration. Choice of the perturbation to be performed on a board is picked randomly among the possible perturbations and parameters using the weighting based on past performance. Change k randomly selected letters on the board. Swap k randomly selected adjacent pairs of letters. Swap k randomly selected pairs of letters. Rotate the board k1 positions vertically and k2 positions horizontally. Concatenate the original list of N configurations with the new list of N perturbed configurations, sort by board score (computed by finding total Boggle® score possible for the board, which is not necessarily the same as the maximum number of words), and keep the N best configurations. Go to step 2 and repeat.

Results and relevant files

S E R S P A T G L I N E S E R S which contains 124 3-letter words ( 124 points) 281 4-letter words ( 281 points) 363 5-letter words ( 726 points) 304 6-letter words ( 912 points) 213 7-letter words (1065 points) 97 8-letter words (1067 points) 28 9-letter words ( 308 points) 4 10-letter words ( 44 points) 4527 total points, 1414 total words. Figure 2: Best found 4x4 Boggle board configuration.

R S C L S D E I A E G N T R P I A E S O L M I D C which contains 195 3-letter words ( 195 points) 442 4-letter words ( 442 points) 661 5-letter words (1322 points) 668 6-letter words (2004 points) 533 7-letter words (2665 points) 342 8-letter words (3762 points) 189 9-letter words (2079 points) 71 10-letter words ( 781 points) 18 11-letter words ( 198 points) 1 12-letter words ( 11 points) 13459 total points, 3120 total words Figure 3: Best found 4x4 Boggle board configuration.

D S R O D G T E M E N S R A S I T O D G N T R P R E I A E S T S C L P D which contains 232 3-letter words (232 points) 610 4-letter words (610 points) 907 5-letter words (1814 points) 1087 6-letter words (3261 points) 980 7-letter words (4900 points) 749 8-letter words (8239 points) 429 9-letter words (4719 points) 184 10-letter words (2024 points) 47 11-letter words (517 points) 13 12-letter words (143 points) 5 13-letter words (55 points) 26514 total points, 5243 total words Figure 4: Best found 6x6 Boggle board configuration.

Comments to Chuong Do (chuong.do@stanford.edu)