Urdhva-Tiryagbhyam — vertical and crosswise multiplication

For two-digit numbers the method has three steps: (1) multiply the ones digits; (2) cross-multiply — each number's tens digit times the other's ones digit, then add the two products; (3) multiply the tens digits. 13×12: 3×2=6; 1×2+3×1=5; 1×1=1 → 156. When a step gives more than 9, the excess carries to the left: in 32×41, first 2×1=2; then 3×1+2×4=11 — write 1, carry 1; finally 3×4+1=13 → 1312.

Unlike Nikhilam, this method needs no closeness to a base — it works for any numbers and extends to three and four digits. A child learns to hold several partial products in parallel, and the same insight makes school column multiplication meaningful, since the steps are essentially identical.

• Age: 10+, with the times table known cold.
• Common mistake: pairing the wrong digits in the cross-multiplication step.
• Practice tip: write down the partial results at first, then go fully mental as speed grows.