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**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 [[induction_chart|Go First Dice induction chart]] match the **[[http://oeis.org/A100642/table|Cotesian numbers]]** because constructing a fair (//n//+1)-die set from an //n//-die set requires solving the exact same mathematical problem as [[https://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas|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 [[fairness#permutation-fair|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 **//x//<sup>//k//</sup>**.
  * For the set to be fair for all subset sizes //k// = 0, 1, 2, ..., //n//, the expected value of //x//<sup>//k//</sup> over the dice distribution must match the continuous uniform distribution:

> //E//[//x//<sup>//k//</sup>] = ∫<sub>0</sub><sup>1</sup> //x//<sup>//k//</sup> //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:

> //x//<sub>//i//</sub> = //i// / //n//
> (where //i// = 0, 1, ..., //n//)

To achieve permutation fairness, the discrete number of faces (or weights //w//<sub>//i//</sub>) assigned to each position //x//<sub>//i//</sub> must satisfy:

> ∑<sub>//i//=0</sub><sup>//n//</sup> //w//<sub>//i//</sub> · (//i// / //n//)<sup>//k//</sup> = ∫<sub>0</sub><sup>1</sup> //x//<sup>//k//</sup> //dx//
> (for every monomial //x//<sup>//k//</sup> with //k// ≤ //n//)

===3. The Unique Solution: Newton–Cotes Quadrature Weights===
By definition, the [[http://oeis.org/A100642/table|Cotesian numbers]] (the weights in the (//n//+1)-point [[https://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas|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//, //x//<sup>2</sup>, ..., //x//<sup>//n//</sup>}, 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  |  **[[m2n2|6]]**  |
|  **//n// = 3**  |  Simpson's 3/8 Rule  |  1/8, 3/8, 3/8, 1/8  |  1, 3, 3, 1  |  **[[m3n3|8]]**  |
|  **//n// = 4**  |  Boole's Rule  |  7/90, 32/90, 12/90, 32/90, 7/90  |  7, 32, 12, 32, 7  |  **[[m4n4|90]]**  |
|  **//n// = 5**  |  6-point Rule  |  19/288, 75/288, 50/288, 50/288, 75/288, 19/288  |  19, 75, 50, 50, 75, 19  |  **[[m5n5|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  |  **[[m6n6|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  |  **[[m7n7|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|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).
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