Optimal Solution to a Transformation Problem of a Triangular Array
Reduces the coin-triangle inversion problem to finding the nearest integer point on the plane k₁ + k₂ + k₃ = n, recovering the minimum ⌊n(n+1)/6⌋ and every optimal configuration. Extends the argument to d-dimensional piles: optimal placements are balanced, tetrahedral piles have a closed form, and the disjoint-corner pattern fails from dimension four.