The History and Mathematics of Six-Sided Dice
Dice are among the oldest gaming implements known to civilization. Excavations at Shahr-e Sukhteh in southeastern Iran unearthed six-sided dice dating back to nearly 3,000 BCE. Ancient peoples initially used sheep and goat anklebones (talus or astragalus) before carving cubes from ivory, wood, marble, and bronze.
The Triangular Probability Distribution of 2D6
When a single six-sided die is rolled, each outcome from 1 to 6 has an equal probability of 1/6 (approx. 16.67%). This is a uniform distribution. However, when two dice are rolled and summed—the standard mechanic in games like Catan, Monopoly, and Craps—the distribution becomes triangular:
- Sum of 2: 1/36 (2.78%) — Snake Eyes [1+1]
- Sum of 3: 2/36 (5.56%) — [1+2, 2+1]
- Sum of 4: 3/36 (8.33%) — [1+3, 2+2, 3+1]
- Sum of 5: 4/36 (11.11%) — [1+4, 2+3, 3+2, 4+1]
- Sum of 6: 5/36 (13.89%) — [1+5, 2+4, 3+3, 4+2, 5+1]
- Sum of 7: 6/36 (16.67%) — Most common outcome [1+6, 2+5, 3+4, 4+3, 5+2, 6+1]
- Sum of 8: 5/36 (13.89%)
- Sum of 9: 4/36 (11.11%)
- Sum of 10: 3/36 (8.33%)
- Sum of 11: 2/36 (5.56%)
- Sum of 12: 1/36 (2.78%) — Boxcars [6+6]
Why Physical Plastic Dice Are Often Biased
Many mass-produced board game dice are tumbled in abrasive ceramic slurries to round their edges. This industrial tumbling process frequently removes uneven amounts of material from different corners, shifting the die’s balance. Furthermore, traditionally drilled pips remove mass from the "6" face, making that side slightly lighter and slightly increasing the odds of the opposite "1" face resting on the table.
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