The exact odds, with no sampling
How you analyze boards you could never list, and the closed-form expected count for every word — no sampling.
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Three ways to know something about 7.38 × 1024 boards
Every Boggle board is a roll of real dice: each die lands in some tray cell with some face up. That makes the number of possible boards astronomically large — far too many for any computer to ever enumerate. So we got clever instead of brute-forcing:
- Count exactly with math. A closed formula (dice arrangements × face choices ÷ tray symmetries) gives the exact integer number of distinct boards — no computer enumeration needed.
- Sample fairly. We roll 1,000,000 perfectly uniform random boards per configuration (deterministically seeded, so every number is reproducible) and solve each one with a fast word-finder.
- Cross-check with exact expectations. For word frequencies, a second, independent closed-form calculation gives the exact expected count per board — sampling and math must agree, and they do. The table on which words come up most prints both side by side, and the table below is that closed form on its own.