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• Motivation
• The game
• The dictionary
• The program
• Results

Motivation

Online fast-paced word games, often a Boggle derivative, require studying word lists to be competitive. So, I wondered, what would be the most useful words to study? Hence I got the idea of writing a program to solve boggle boards, keeping track of the most frequent words. The results are on this page.

Elsewhere on the internet, there's an older related study for 5x5 Boggle. Also interesting is the highest scoring known boards (not restricted to Boggle dice).

If you spot any mistakes here, or missing information, send me an email.

The game

There are three versions of Boggle that I studied. There are 2 versions of 4x4 boggle and one version of 5x5. See the dice. The 4x4 games have a word minimum of 3 letters, the 5x5 minimum is 4 letters. Scoring for either size is the standard 1, 1, 2, 3, 5, 11 for 3, 4, 5, 6, 7, and 8+ letters respectively. I used the convention that QU must be used as a unit--words such as SUQ and QAT are not allowed.

The dictionary

SOWPODS was used for words of at most 15 letters. Since larger words are possible, I added ENABLE for words of 16 or more letters. However words of 16 or more letters are sufficiently rare that they have no effect on the total score results. You can download the word list.

I don't plan to run the study for TWL, as it is simply a subset of SOWPODS.

Not relevant to the rest of this page, but great for studying, I've constructed SOWPODS and TWL wordlists arranged by anagram.

The program

The key requirement of this study is to have a very efficient board solver. I've made the source available, but it's a mess. (I'm not posting the binary to avoid contributing to the problem of cheating online.)

The dictionary is put into an efficient tree structure in a two-pass process (one pass puts it into a tree, the second pass alphabetizes and reduces the memory requirement). See trie at wikipedia for the data structure. The board is solved in the obvious recursive fashion. The tree is very fast to traverse, and it instantly rejects any branches that can't lead to words. I wrote some awkward code to reduce branching (all the if's instead of a for loop), and pruned away any code not required for simply adding up scores (displaying the solutions is handled by an altogether separate subroutine). Results are tallied at the nodes of the tree in a parallel array, in such a way that memset's aren't necessary between runs.

That said, this program isn't optimal. It would be possible to further reduce branching by completely unrolling the recursive function--have a different function for each square of the board. However that code is not maintainable. Reducing the dictionary to only words that can actually occur would be a further speedup, although probably very small.

After some optimization tweaks I was able to solve 3550 4x4's per second and 1325 5x5's per second on a 1.4 Ghz P4 laptop, and 6950 4x4's and 2530 5x5's on a 2.0 Ghz Core 2 Duo laptop. The compiler is MSVC++ 6. Undoubtedly a modern compiler could do better.

Results

I ran 2,000,000,000 games of boggle for each of the three dice setups. This took a few weeks on my 2 ghz Core 2 laptop.

I've made 4 versions of the output available here.

• Program Output The first is the program output, which spits out the dictionary and the number of points each word scored. At the bottom of the file is the number of games for each score. This is probably not useful unless you want to do your own study of the data.
• Words and PPG The second file is the words sorted by points per game.
• Words by PPG The third is similar, but the numbers are omitted to make it more readable.
• Anagrams by PPG The fourth aggregates by anagram, and lists the anagrams in order of PPG. I find the fourth list the easiest to study, because I prefer to learn words by anagrams.

Program Output	Words and PPG	Words by PPG	Anagrams by PPG
New 4x4	New 4x4	New 4x4	New 4x4
Old 4x4	Old 4x4	Old 4x4	Old 4x4
5x5	5x5	5x5	5x5

Here's a summary of some of the interesting results.

Statistic	New 4x4	Old 4x4	5x5
Distinct words	220203	231198	247000
Words making half the points	3944	4441	3815
Average Score	193.88	171.78	343.49
1 percentile score	26	23	92
Median Score	165	147	448
99 percentile score	613	542	1485
Chance of 0 points	1 in 7454	1 in 6801	1 in 12.3M
Highest score	2573	2640	5347
3's per game	46.56	45.59	N/A
4's per game	49.02	45.54	112.48
5's per game	25.47	21.91	74.90
6's per game	9.495	7.652	38.77
7's per game	2.502	1.893	14.62
8+'s per game	0.5786	0.4013	5.494
Largest Words	15 letters ENTEROHEPATITIS GRANDILOQUENCES INEQUITABLENESS ITERATIVENESSES	15 letters INFRANGIBLENESS	18 letters ANTIADMINISTRATION NONCOOPERATIONISTS
10 best words	TOE TEE TEN NET TEA SET TES ATE ETA TAE	LEA TEA SEA NAE EAN EAT TAE EAS SAE ANE	TEER TEEN RETE NETE ETEN TEAR ARET TREE ERNE NEAT
Most common 8	TONETTES	SEQUELAE	ENTERATE

Best new 4x4 found, 2573 points:

OTEH
SRTS
PEAI
ALMS

Best old 4x4 found, 2640 points:

GESD
LARH
MITE
AONS

Best 5x5 found, 5347 points:

JSLAT
DPRES
TRATG
ESINA
FTEDE