On a standard Boggle board game how many possible patterns are there?

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Clarification of Question by summer95-ga on 08 Oct 2004 15:54 PDT

This is a 5 x 5 board.

With 25 6-sided dice, you have 6^25 or 28,430,288,029,929,701,376
different combinations of how the dice can land.  Of course, I haven't
played boggle in a while, so I don't remember if the same letter
appears on multiple sides of a die.  If so, then the total number of
combinations would be significantly less.

If each die has 1 repeated letter, then you have 5^25 combinations, or
298,023,223,876,953,125 possible combinations of how the dice land.

If each die has 2 repeated letters, then you have 4^25 combinations,
or 33,554,432 possible combinations of how the dice land.

touf is right about the number of combinations available.  I think
regular boggle is a 4x4 board.  So, it would actually be 6^16 or
2,821,109,907,456.

However, position has to be taken into account.  The general equation
would be ((sides)^(# of dice)) x (slots!) where slots is the openings
on the board.  Specifically, the result with positioning is
(6^16)(16!) or 59,025,489,844,657,012,604,928,000.  However, this is
still not correct as a flip or rotation of the board has no bearing on
the resulting playfield.

For example, the mirror image of a given board produces the same
possible letter sequences accross adjacent squares.  This is where it
begins to get very complicated and my logic a bit fuzzy.  You have to
multiply by (1/(2^n)) for each board equivalence.  Each board
equivalence is generated by flipping (horizontal, vertical, or
diagonal-either) or rotation (90, 180, 270).  I think the answer is
(6^16)(16!)/(2^7) or 461,136,639,411,382,910,976,000 possible unique
boards.  But I may be off by some factor.  Hey, what do you want for
free?

Doh, i can't read.  He did say 5x5.  Just replace the 16s with 25s
(5^2).  It yields around 3.4452200720674866830186861158497e+42
possiblilities.  Uh, it's safe to say you will never see the same
board twice.  However, as the original poster said, duplicated letters
either on the same die or between dice very greatly reduces this
effect.  This formula only accounts for the appearence of faces on the
dice.  I would think to get a definitive answer on this subject you
would have to offer a bit more reward.  Someone might take it up jsut
for the challenge though.  I would work further on it if I had a
boggle set to consult for the duplications of letters.  I think our
work is a good starting point for further investigation.  Good luck.