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 and the recently introduced 6-by-6 Super Big Boggle 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
AACIOT	AAEEGN
ABILTY	ABBJOO
ABJMOQu	ACHOPS
ACDEMP	AFFKPS
ACELRS	AOOTTW
ADENVZ	CIMOTU
AHMORS	DEILRX
BIFORX	DELRVY
DENOSW	DISTTY
DKNOTU	EEGHNW
EEFHIY	EEINSU
EGKLUY	EHRTVW
EGINTV	EIOSST
EHINPS	ELRTTY
ELPSTU	HIMNUQu
GILRUW	HLNNRZ

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 set	Average  # of words	Average  Boggle score	Average length of longest word
New Boggle	~104	~150	6.8
Classic Boggle	~93	~128	6.6

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 set	Average  # of words	Average  Boggle score	Average length of longest word
New Boggle w/o one corner cube	~92	~131	6.6

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 set	Average  # of words	Average  Big Boggle score	Average length of longest word
Big Boggle	~190	~395	8.3

(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.)

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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.

Letter distributions in Bananagrams and other games

What determines how many of each letter there are in various word games? The inventor of Scrabble designed the frequency of his tiles around the frequency of letters used on the front page of the New York Times. (Somebody must have sat and done a lot of counting back in 1938.) The distribution for Snatch is said to be based on the frequency distribution of letters in the English language as calculated by World War II cryptographers.

This photograph visually displays the letter distribution for Bananagrams. (Click for a larger version.)
The following table gives the letter distributions for Bananagrams, Scrabble (without the blanks), and Snatch:

Game	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y	Z
Bananag.	13	3	3	6	18	3	4	3	12	2	2	5	3	8	11	3	2	9	6	9	6	3	3	2	3	2
Scrabble	9	2	2	4	12	2	3	2	9	1	1	4	2	6	8	2	1	6	4	6	4	2	2	1	2	1
Snatch	5	2	4	4	12	4	2	5	5	1	1	5	4	5	6	3	1	5	5	7	4	2	3	1	3	1

By comparing the Bananagrams and Scrabble sets, we learn that the Bananagrams set of tiles is a superset of the Scrabble set of tiles (except for the blanks). In fact, for each letter of the alphabet, the Bananagrams set has at least one more tile than the Scrabble set. Since there are 100 Scrabble tiles and 144 Bananagrams tiles, this additional 26 tiles explains a lot of the discrepancy. It also gives a hint as to how the inventors of Bananagrams may have derived their distribution.


The Snatch set, on the other hand, is NOT a subset of the Bananagrams set. If you want to try to play Snatch with a Bananagrams set of tiles (and if you are kind of a stickler), you can get close to the right distribution by throwing out these letters in these numbers: (A,7), (E,5), (I,6), (O,4), (R,3). (I am reducing each difference by one to allow for a slightly larger Bananasnatch set (121 tiles instead of 100).) You can remember this with the mnemonic lAttIcEwORk and noting that you start by throwing out 7 tiles lettered A, decreasing the number by one for each of the vowels, and finally setting aside 3 tiles with the letter R on them.

UPDATE:
I found the letter distribution for Spanish Bananagrams here and the French Bananagrams letter distribution here.

Here is the revised table, in vertical format (to accommodate the extra letters):

	B a n a n a g r a m s	S p a n i s h B G	F r e n c h B G	S c r a b b l e	S n a t c h
A	13	17	13	9	5
B	3	3	3	2	2
C	3	5	4	2	4
D	6	7	6	4	4
E	18	17	18	12	12
F	3	2	3	2	4
G	4	3	2	3	2
H	3	3	2	2	5
I	12	8	12	9	5
J	2	2	2	1	1
K	2	1	1	1	1
L	5	5	6	4	5
M	3	3	4	2	4
N	8	7	9	6	5
O	11	12	12	8	6
P	3	3	4	2	3
Q	2	2	2	1	1
R	9	7	10	6	5
S	6	8	6	4	5
T	9	6	10	6	7
U	6	7	6	4	4
V	3	2	4	2	2
W	3	1	1	2	3
X	2	1	2	1	1
Y	3	2	1	2	3
Z	2	2	1	1	1
CH		2
LL		2
Ñ		2
RR		2

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Suggested reading:

• The Boggle cube redesign and its effect on the difficulty of Boggle
• The longest word that you can make in Bananagrams