Note: The following explanation was largely AI-generated (via Gemini), and could likely benefit from additional manual verification by humans.
Why the Induction Chart Diagonal Matches the Cotesian Numbers
The diagonal numbers on the Go First Dice induction chart match the Cotesian numbers because constructing a fair (n+1)-die set from an n-die set requires solving the exact same mathematical problem as Newton–Cotes numerical integration: finding discrete weights at equally spaced points that integrate all polynomials of degree ≤ n exactly.
1. The Fairness Condition is a Polynomial Moment Problem
In a set of permutation-fair dice (Go First Dice), when any subset of k dice is rolled, every die must have an equal probability 1 / (k + 1) of rolling the highest value:
- If a die's face values are mapped continuously onto the normalized interval [0, 1], the probability that a value x ∈ [0, 1] exceeds k other uniformly distributed dice is proportional to xk.
- For the set to be fair for all subset sizes k = 0, 1, 2, …, n, the expected value of xk over the dice distribution must match the continuous uniform distribution:
E[xk] = ∫01 xk dx = 1 / (k + 1)
(for all k = 0, 1, …, n)
2. The Induction Step Uses Equally Spaced Positions
When inducing an (n+1)-die set from an existing n-die set using m = n copies, the faces of the new die are interleaved across the existing dice blocks at n+1 equally spaced relative positions:
xi = i / n
(where i = 0, 1, …, n)
To achieve permutation fairness, the discrete number of faces (or weights wi) assigned to each position xi must satisfy:
∑i=0n wi · (i / n)k = ∫01 xk dx
(for every monomial xk with k ≤ n)
3. The Unique Solution: Newton–Cotes Quadrature Weights
By definition, the Cotesian numbers (the weights in the (n+1)-point Newton–Cotes quadrature rule) are the unique set of weights for n+1 equally spaced points that integrate all polynomials of degree ≤ n exactly:
- Because any polynomial is a linear combination of the monomials {1, x, x2, …, xn}, the moment-matching condition for Go First Dice induction and the polynomial-exactness condition for Newton–Cotes quadrature are algebraically identical.
- Because face counts on physical dice must be whole numbers, the weights must be scaled up to integers. The diagonal entries on the induction chart are the sums of these integer Cotesian weights:
| n (Players Added) | Quadrature Rule Name | Normalized Cotesian Weights | Integer Weights | Sum (Diagonal Value) |
|---|---|---|---|---|
| n = 1 | Trapezoidal Rule | 1/2, 1/2 | 1, 1 | 2 |
| n = 2 | Simpson's 1/3 Rule | 1/6, 4/6, 1/6 | 1, 4, 1 | 6 |
| n = 3 | Simpson's 3/8 Rule | 1/8, 3/8, 3/8, 1/8 | 1, 3, 3, 1 | 8 |
| n = 4 | Boole's Rule | 7/90, 32/90, 12/90, 32/90, 7/90 | 7, 32, 12, 32, 7 | 90 |
| n = 5 | 6-point Rule | 19/288, 75/288, 50/288, 50/288, 75/288, 19/288 | 19, 75, 50, 50, 75, 19 | 288 |
| n = 6 | Weddle's Rule | 41/840, 216/840, 27/840, 272/840, 27/840, 216/840, 41/840 | 41, 216, 27, 272, 27, 216, 41 | 840 |
| n = 7 | 8-point Rule | 751/17280, 3577/17280, 1323/17280, 2989/17280, 2989/17280, 1323/17280, 3577/17280, 751/17280 | 751, 3577, 1323, 2989, 2989, 1323, 3577, 751 | 17280 |
Key Takeaway
The induction chart diagonal numbers are the numerators of the Normalized Cotesian numbers because fairly ranking n+1 dice is isomorphic to integrating degree-n polynomials over n+1 equispaced nodes.
Why is n = 8 a Dash on the Chart?
An interesting consequence of this equivalence explains why the diagonal entry at n = m = 8 on the induction chart is a dash (-):
- For n = 8 (a 9-point closed Newton–Cotes rule), the quadrature weights include negative numbers (specifically, the integer weights are 989, 5888, -928, 10496, -4540, 10496, -928, 5888, 989).
- Because physical dice faces cannot be negative, no valid die can be formed from this direct unmixed induction at m = 8. (Solutions for n = 8 only start appearing at m = 9).
