A blog for fans of Bananagrams, word games, puzzles, and amazing things

Thursday, October 17, 2013

The Boggle cube redesign and its effect on the difficulty of Boggle

I wanted to buy a copy of Boggle. This seemingly simple mission was complicated by the facts that a) there are different kinds of Boggle out there and b) I like to make things complicated.

If you exclude variations such as Big Boggle [affiliate link] and the recently introduced 6-by-6 Super Big Boggle [affiliate link] and just limit yourself to the original 4-by-4 Boggle configuration, there are three principle versions of Boggle:

1) Boggle Reinvention (now sold as just "Boggle") - While the new sealed case design and the integrated timer mean that you don't need to worry about losing any of the pieces, there are reports that it is possible for two dice to become jammed together in such a way that it is essentially impossible to separate them without opening up the case and destroying the game. In my opinion, Boggle should not be a game that can break.

2) Plain old Boggle, made from about 1976 to 1986 (which I will call "classic Boggle").

3) The version of Boggle sold from 1987 to ~2008 - essentially the same as classic Boggle except that the letter distribution on the cubes was completely redesigned. I'll call this "New Boggle".

Below, you can see a side-by-side comparison of the classic and new sets of Boggle dice.

Classic
Boggle Dice 
New
Boggle Dice
AACIOTAAEEGN
ABILTYABBJOO
ABJMOQuACHOPS
ACDEMPAFFKPS
ACELRSAOOTTW
ADENVZCIMOTU
AHMORSDEILRX
BIFORXDELRVY
DENOSWDISTTY
DKNOTUEEGHNW
EEFHIYEEINSU
EGKLUYEHRTVW
EGINTVEIOSST
EHINPSELRTTY
ELPSTUHIMNUQu
GILRUWHLNNRZ
To help visualize the differences between these distributions, I sorted the classic letter distribution by number of letters (shown on the left below) and used that order to sort the new letter distribution (shown on the right).

<= Classic New =====>
EEEEEEEEEE EEEEEEEEEEE
  AAAAAAAA AAAAAA
   IIIIIII IIIIII
    OOOOOO OOOOOOO
     LLLLL LLLL
     NNNNN NNNNNN
     SSSSS SSSSSS
     TTTTT TTTTTTTTT
      DDDD DDD
      RRRR RRRRR
      UUUU UUU
       BBB BB
       CCC CC
       GGG GG
       HHH HHHHH
       MMM MM
       PPP PP
       YYY YYY
        FF FF
        KK K
        VV VV
        WW WWW
         J J
         Q Q
         X X
         Z Z
In some ways (such as increasing the number of Ts and Hs), the new distribution is closer to the letter frequency in English words, but that motive alone would not explain why the number of As was decreased and the number of Os was increased. It has been suggested that this change was designed to reduce the frequency of harder letters (like K and G) and make finding words easier.

One other interesting property of the new set of dice is that since it concentrates certain letters all on the same die, it is never possible to make words that combine F and K (like FAKE, FORK, SKIFF,...) or words that combine B and J (JOB, JAB, BANJO,...). It is also not possible to make words with three Ps (like PINEAPPLE) or two Ks (like SKOOKUM, which is a slang term in the Pacific Northwest, derived from the Chinook language, and having multiple meanings: as an adjective it refers to something that is massive or powerful or reliable or simply really cool; as a noun, it can refer to an evil spirit or demon or a monster somewhat like Bigfoot or Sasquatch; it is pronounced /SKOO kum/).

So does this change in letter distribution have an effect on the game? To find out, I ran some simulated Boggle games, generating random boards with each set of dice and using a Boggle solver (written by GitHub user cespare) to determine the number of words in each board, the resulting Boggle score, and the longest word in each grid.

[The default word list used by the Boggle solver only contains words that are 15 letters long or shorter. While it's highly unlikely to find a random 4-by-4 Boggle board containing a 16-letter word, I decided to augment the word list to include 16-letter words, as well as the 17-letter words that could be made with the Qu cube (like QUATTUORDECILLION [which means 1045 in the U.S. and 1084 in Britain] and SESQUIPEDALIANISM).]

Dice setAverage 
# of words
Average 
Boggle
score
Average
length of
longest word
New Boggle~104~1506.8
Classic Boggle~93~1286.6

The results show that there are about 12% more words to be found in a New Boggle board. These results are from simulating 10,000 boards for each set of dice, so the numbers in the table may be off by a few percent.

To try to give a little more insight into the difference between these versions of the game, I ran simulations for a New Boggle board in which a randomly chosen cube from a corner of the board was removed. The corresponding results,

Dice setAverage 
# of words
Average 
Boggle
score
Average
length of
longest word
New Boggle
w/o one
corner cube
~92~1316.6

are really close to the classic Boggle results, suggesting that if you want to make your New Boggle game about as hard as a classic Boggle game, you can just remove a die from the corner of the board before hunkering down to find words.

Out of curiosity, I also ran simulations for 5-by-5 Big Boggle, using the requirement that words be at least four letters long (unlike the three-letter limit in regular Boggle) and using the Big Boggle scoring system, which yielded these results:

Dice setAverage 
# of words
Average 
Big
Boggle
score
Average
length of
longest word
Big Boggle~190~3958.3

(For comparison with 4-by-4 Boggle, if you include the three-letter words that are in Big Boggle boards, the average number of words increases to about 260.)


Of course, these calculations only confirmed what I already knew: the older version of the game is harder and is the one for me. I bought a copy of classic Boggle from eBay. The dice are made of wood rather than plastic. The timer has sand in it and doesn't make some noise to tell me when time is up. Succinctly, I think it is skookum.