Does the Japanese Multiplication Method Really Work?

Short answer: yes, the “Japanese” line method of multiplication always gives the right answer. It is not a trick: counting the crossing points is exactly the same calculation as the standard method, just drawn instead of written.

Contents7 sections

How the method works

Example: $12\times 13$.

1. Draw 12 as slanted lines going one way: 1 line (the tens), a gap, then 2 lines (the ones).
2. Draw 13 as lines crossing the first set: 1 line (tens), a gap, then 3 lines (ones).
3. The crossings form groups. Count the dots in each group, from left to right:
   • tens of 12 × tens of 13: $1\times 1=1$ crossing → hundreds
   • the two middle groups together: $1\times 3+2\times 1=5$ crossings → tens
   • ones × ones: $2\times 3=6$ crossings → ones
4. Read the digits: 1, 5, 6. So $12\times 13=156$.

Why it works

A group of $a$ lines crossing a group of $b$ lines makes $a\times b$ crossing points, so the drawing is just a way to compute products of digits. Writing the two numbers as $10a+b$ and $10c+d$:

$$\begin{aligned}(10a+b)(10c+d)&=100\,ac\\&\quad+10\,(ad+bc)\\&\quad+bd.\end{aligned}$$

The left group counts $ac$ (hundreds), the two middle groups count $ad+bc$ (tens), and the right group counts $bd$ (ones). This is the same expansion behind ordinary long multiplication and the distributive property.

When a group has 10 or more dots

Then you carry, exactly as in written multiplication. Example: $23\times 24$.

• Hundreds: $2\times 2=4$
• Tens: $2\times 4+3\times 2=14$
• Ones: $3\times 4=12$

Work from the right. Ones: 12, write 2 and carry 1. Tens: $14+1=15$, write 5 and carry 1. Hundreds: $4+1=5$. Result: $552$.

Where it gets awkward

• Large digits. $89\times 97$ needs groups of 8, 9, 9 and 7 lines, and counting 72 or 81 crossings is slower than knowing your times tables.
• Zeros. A digit 0 has no lines, so you must leave a visible empty place (some people draw a dashed line) to keep the place values aligned.
• Bigger numbers. Three-digit numbers work too, with three groups each and five diagonal columns, but the drawing gets crowded.

The method is great for seeing why multiplication by place value works. For speed, the written method or mental math is better.

Try these

Puzzle 1

Use the line method for $21\times 13$.

Show Answer

Hundreds $2\times 1=2$; tens $2\times 3+1\times 1=7$; ones $1\times 3=3$. Result 273.

Puzzle 2

Use the line method for $31\times 22$.

Show Answer

Hundreds $3\times 2=6$; tens $3\times 2+1\times 2=8$; ones $1\times 2=2$. Result 682.

Puzzle 3

For $123\times 21$, which groups of crossings make up the tens digit?

Show Answer

The tens column collects the crossings whose place values multiply to 10: (tens of 123) × (ones of 21) and (ones of 123) × (tens of 21), that is $2\times 1+3\times 2=8$. The full columns are 2, 5, 8, 3, so $123\times 21=\mathbf{2583}$.

Related

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• How many squares are on a chessboard?

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