Powers of Eleven

The early powers of 11 spell out Pascal's triangle. 11² = 121, 11³ = 1,331, and 11⁴ = 14,641 — read off the digits and you get the rows 1-2-1, 1-3-3-1, and 1-4-6-4-1. The match is no accident: raising 10 + 1 to a power is exactly what those binomial coefficients count.

The clean pattern breaks at 11⁵ = 161,051. The next row should read 1-5-10-10-5-1, but the two-digit 10s force carries, scrambling the tidy digits while the deeper structure stays intact.

Shortcuts like this compound over time. A cashier who multiplies by 11 in their head spots rounding errors instantly. Mental math is not about speed for its own sake — it is about building number sense that catches mistakes before they become costly.