How many Boggle boards exist?

Counting every board without listing one

Count a board the way the game makes one: which of the D dice lands in which tray cell (D! ways), times which face lands up on each (6^D ways), divided by the 8 ways you can turn or flip the tray without changing the game. That is exact — no estimate, no sampling (how that works) — and every digit below is exact with it.

Careful, though: that counts rolls, not pictures. Two different rolls can look identical — a different cube showing the same letter, and Big Boggle even carries two cubes printed alike. Counting only the distinct letter grids gives a smaller number, and unlike this one it has no clean formula.

Fun fact: Boggle's cubes have been re-lettered over the years — the 1976 Classic set carried four fewer T faces and two more A faces than today's, and Big Boggle Deluxe re-cut the 5×5 set entirely — but none of it moves this number by a single digit. Any D cubes at all, printed with anything at all, make exactly D!·6^D ÷ 8 boards.