A fair handmade die is a carefully measured cube of dense bone or hardwood, and testing it takes far more rolls than people expect: a 60-roll tally detects a face that comes up 5 percentage points too often only about 10% of the time, while 720 rolls scored with a chi-square statistic, flagging any result above 11.07, detects it about 80% of the time. Measure the cube, then roll it 720 times in six sets of 120 and score it. Keep opposite faces summing to 7.
Carving and marking dice, how big a test sample is needed to catch a biased die, and the arithmetic of the chi-square test.
Make The Cube Accurate
A die is fair if all six faces are equally likely, which needs a cube whose faces are equal in size and square to each other, made of material of uniform density.
- Bone. Cut the cube from the thickest, densest part of a long-bone shaft. The parent article mentions cow knuckle bones. A knucklebone (the astragalus) is an irregular lump, not a cube: it lands on four of its faces with unequal chances, and a cube cannot be carved from it without wasting most of it. Use a long-bone shaft with walls thick enough for the cube side you want; measure the wall first.
- Wood. Dense hardwood, dry and free of knots and grain tilt; see Wood Seasoning. Seal with plain wax or oil to slow moisture change, which swells wood unevenly; see Oil Finishing.
- Measure. Use a ruler or calipers and measure all three pairs of opposite faces. Differences between sides are the first source of bias. File down the long dimension until all three agree to the finest reading you can get; I give no sourced tolerance.
- Mark the pips. Standard dice put 1 opposite 6, 2 opposite 5, 3 opposite 4, so opposite faces sum to 7. Drilling pips removes material, more from the 6 face than the 1 face. To keep the balance, drill shallow, use the same pip size on every face, or fill the pips with a similar-density paste, such as charcoal mixed with a little hide glue; see Hide Glue. Do not fill them with paints that could contain lead.
How Many Rolls Does A Fairness Test Need?
The parent article says to roll 60 times, expect about 10 of each face, and adjust the die if a face is “significantly more”. This is much too small a test.
What a fair die does in 60 rolls. Each face’s count has mean 60 ÷ 6 = 10 and standard deviation √(60 × 1/6 × 5/6) = 2.89. A count of 15 or more is normal: for a fair die the chance that a given face shows 15 or more is about 6.5% (exact binomial). By simulation of 200,000 fair 60-roll runs, about 27% of fair dice show at least one face below 5 or above 15. So if “significant” means a face outside 5 to 15, a 60-roll test wrongly condemns about a quarter of fair dice and misses most real bias.
The chi-square test. After n rolls, with expected count E = n ÷ 6 per face, compute
χ² = Σ (observed − E)² ÷ E
summed over the six faces. For a fair die, χ² follows a chi-square distribution with 5 degrees of freedom. The standard 5% critical value is 11.07: if χ² is above it, a fair die would produce a result that extreme only 5% of the time. I computed 11.07 as the 95th percentile of that distribution; it matches the standard table.
How many rolls to catch a bias. Suppose one face has probability 1/6 + 0.05 = 0.2167 and the other five share the rest, 1/6 − 0.01 = 0.1567 each. The squared deviations sum to Σ(p − 1/6)² = 0.05² + 5 × 0.01² = 0.0025 + 0.0005 = 0.0030. Dividing by 1/6 gives 0.018 per roll, so the test’s non-centrality is λ = 0.018 × n.
- n = 60: λ = 1.08, detection power about 10% (from the non-central chi-square distribution, computed with scipy; a simulation agreed)
- n = 360: λ = 6.5, power about 47%
- n = 720: λ = 13.0, power about 80%
- n = 1,000: power about 93%
So a 5-percentage-point bias, 0.2167 against 0.1667, needs about 720 rolls to catch reliably. At about 5 seconds a roll that is 720 × 5 = 3,600 s, an hour. Smaller biases need more.
Worked Example: Scoring A 720-Roll Test
Expected count per face E = 720 ÷ 6 = 120. Suppose the tally is 150, 112, 116, 118, 98 and 126, which sums to 720.
- Deviations: +30, −8, −4, −2, −22, +6
- Squares: 900, 64, 16, 4, 484, 36 (sum 1,504)
- χ² = 1,504 ÷ 120 = 12.53
That is above 11.07 (the probability of such a result from a fair die is about 2.8%). The die is suspect, and the face with the largest deviation, the first at +30, is the one to inspect. A second tally of 720 rolls that also exceeds 11.07 is strong evidence of bias; one that does not is inconclusive.
With a tally of 138, 117, 119, 126, 101 and 119: deviations +18, −3, −1, +6, −19, −1; squares 324, 9, 1, 36, 361, 1 = 732; χ² = 732 ÷ 120 = 6.1, below 11.07. This gives no evidence of bias; it does not prove fairness, since a small bias can hide in 720 rolls.
What You Need
- Dense dry bone or hardwood
- Calipers or a marked ruler
- Files and an abrasive to square the faces
- A pointed drill for pips; see Drill and Auger
- A flat rolling surface and a tally sheet
- Patience: an hour for a decent test
Method: Make And Test
Step 1: Cut the cube slightly oversize.
Step 2: Square it by filing and checking with calipers or a ruler on all three axes.
Step 3: Round the corners lightly so it rolls.
Step 4: Mark the pips with 1/6, 2/5 and 3/4 opposite.
Step 5: Roll 720 times in six sets of 120, tallying each face.
Step 6: Compute χ² and compare it with 11.07.
Step 7: If flagged, repeat once before reworking; a fair die is flagged one time in twenty. If flagged twice, re-measure and file the offending dimension.
Common Mistakes
- Trusting a short test. 60 rolls cannot confirm fairness.
- Adjusting after one flagged test. One in twenty fair dice fails.
- Drilling the pips deep. The 6 face loses more material.
- Loading by accident. A knot or soft spot shifts the balance.
Safety
Bone dust, paint and small dice
Bone and wood dust irritate the lungs, so file and sand outdoors and avoid breathing it. Bone must come from animals you butchered or know to be healthy; do not use bone from a carcass of unknown history. Do not colour pips with pigments that may contain lead, mercury or arsenic; see Mineral Pigments. Dice are small enough for a young child to swallow.
What’s Next
- Learn next: Dice Games for Groups — what to play with a tested die
- Combine with: Handmade Playing Cards — the other chance-game equipment