Act 1 · Building Intuition
4. Reasoning: Diverse Pathfinding
Working a problem out instead of looking it up has a name: reasoning. And here is the surprise: it is never one fixed procedure. Page 1 gave you 17 × 24 and one way to build it. Take a bigger one, 350 × 638, and three cultures worked it three different ways, each a real method with its own strengths and weaknesses. Pick one below, walk it to the end, then try another and compare what they cost.
1. Pick a road and walk itone problem · three methods · authored walkthroughs
350
× 638
Press Next (or Play) to walk the road.
The limit of this road: General at any size, but slow: every digit pair gets touched, and every carry is a chance for a digit to slip.
2. Three roads, three tollswhat each method is good at, and what it costs
Standard long multiplication
the school algorithm: one partial product per digit, then add.
limit: General at any size, but slow: every digit pair gets touched, and every carry is a chance for a digit to slip.
Japanese line multiplication
a visual method often taught in East Asia: draw each digit as a bundle of lines, count the crossings.
limit: No times tables needed, but big digits mean counting dozens of crossings (5 × 8 alone is 40), and a zero needs a ghost line.
Indian Vedic (Urdhva-Tiryagbhyam)
the 'vertically and crosswise' sutra: one pass, one result digit per column.
limit: The fastest road, but the crosswise pattern must be memorized for each size, and the carries stack up in your head.
There are many different ways to do multiplication. We showcase only three here.