Carry-free digit patterns in the powers of n -nines

Numbers of the form Nn=10n−1, consisting entirely of nines in base 10, display striking digit patterns when raised to small positive integer powers. For example, 992=9801 and 9992=998 001: in each case the digits of 92=81 appear in fixed positions, separated by blocks of nines and zeros of length n−1. This paper explains why these patterns arise, how far they extend, and where and why they break down. The explanation rests on the binomial expansion of (10n−1)a. When the binomial coefficients (ak) are sufficiently small, successive terms occupy well-separated digit blocks, and borrowing across these blocks produces rigid strings of nines and zeros without any carry propagating beyond a single block. In this carry-free regime, the digits of 9a appear in predictable positions, yielding the observed block structure. In base 10, this carry-free regime persists precisely up to the fifth power. The case a = 5 is borderline: two-digit binomial coefficients occur, but their signed contributions interact through local borrowing in a way that prevents carry propagation. From a = 6 onwards, the central binomial coefficient (63)=20 forces an unavoidable carry, and the block structure collapses sharply.