Line Multiplication: The Stick Method, Japanese, Chinese or Neither?

Line multiplication draws each digit of two numbers as a bundle of parallel sticks, crosses the two sets of bundles, counts the crossing points in each diagonal zone and then carries, so 21 x 13 gives zones of 2, 7 and 3 and the answer 273. It works because each zone count is the sum of digit products for one place value. No historical source supports the "Japanese" or "Chinese" name.

That is the short answer. The rest of this post does what the viral videos skip. It works the fixed problem set from our mathematics traditions series with full line diagrams: 98 x 97, 47 x 53, 386 x 524 and 85². It says plainly why the method cannot take √7,744, divide 7,429 by 37 or turn 1/19 into a decimal. It proves the method with algebra, counts every line and every crossing so you can judge "fast" for yourself, and traces the names "Japanese multiplication", "Chinese stick multiplication" and "Mayan multiplication" as far back as any source will go.

Every crossing count, zone total, carry and answer below was computed with Python first; the script is in the walkthrough near the end.

Key takeaways

• The method in one line. Draw each digit as that many parallel lines, cross the bundles of the two numbers, count crossing points zone by zone (one zone per place value), then carry from right to left.
• Why it works. Two bundles of a and b lines that are not parallel meet in exactly a x b points. A zone gathers every pair of digits whose place values multiply to the same power of ten. So each zone count is a column of the ordinary long multiplication, before carrying. That is the whole proof, written out with algebra below.
• It is a picture of a known algorithm. The zone totals are exactly the diagonal sums of the lattice (gelosia) method and exactly the "vertically and crosswise" sums of Urdhva-Tiryagbhyam. In algebra, they are the coefficients of the product of two polynomials evaluated at x = 10.
• "Japanese", "Chinese" and "Mayan" are folklore names. We found no historical source for any of them. The earliest dated use we could find is a teacher's blog from 17 January 2011 that already calls it "Japanese Multiplication". Japan's own national curriculum teaches the 9 x 9 table, column multiplication and the soroban, not counting crossings.
• Fast only for small digits. 21 x 13 needs 12 crossings. 98 x 97 needs 272, and 9,999 x 9,999 would need 1,296. A single 8 x 9 pair is 72 dots to count, where a times table gives one fact.
• Multiplication only. There is no line method for square roots, division or decimals of fractions. Lines can check a guessed root or quotient, nothing more.

The method, step by step, on 21 x 13

The video version of the method usually starts with 21 x 13 or 21 x 23 because the digits are small. Here is the full procedure on 21 x 13 = 273, written so it works for any pair of whole numbers.

1. Draw the first number. For 21, draw 2 parallel lines slanting one way for the tens digit, leave a gap, then 1 parallel line for the units digit. The highest place value goes first.
2. Draw the second number across it. For 13, draw 1 line slanting the other way for the tens digit, a gap, then 3 lines for the units digit, so that every line of 13 crosses every line of 21.
3. Mark the zones. The crossings fall into groups. The group where the tens of 21 meet the tens of 13 is the hundreds zone. The two groups where a tens bundle meets a units bundle together make the tens zone. The group where the two units bundles meet is the units zone.
4. Count the crossings in each zone. Hundreds: 2 x 1 = 2 points. Tens: 2 x 3 + 1 x 1 = 6 + 1 = 7 points. Units: 1 x 3 = 3 points.
5. Carry from right to left. Units 3, nothing to carry. Tens 7, nothing to carry. Hundreds 2. Read the digits: 273.

Here is the classic picture for 21 x 13, drawn in text. The \ lines are 21 (one units line on the left, two tens lines to its right) and the / lines are 13 (one tens line, then three units lines). Each X is a crossing point. In this drawing the diamond of crossings points down, so the three zones are stacked as horizontal bands rather than side by side.

     \     \ \             /     / /
      \     \ \           /     / /
       \     \ \         /     / / /
        \     \ \       /     / / /
         \     \ \     /     / / /
          \     \ \   /     / / /
           \     \ \ /     / / /
            \     \ X     / / /      <- hundreds band: 2 crossings
             \     X \   / / /       <- hundreds band
              \   / \ \ / / /
               \ /   \ X / /         <- tens band: 7 crossings
                X     X X /          <- tens band
               / \   / X X           <- tens band
              /   \ / / X \          <- tens band
             /     X / / \ \         <- units band: 3 crossings
            /     / X /   \ \        <- units band
           /     / / X     \ \       <- units band
          /     / / / \     \ \
         /     / / /   \     \ \
        /     / / /     \     \ \
       /     / / /       \     \ \
      /     / / /         \     \ \
     /     / / /           \     \ \
    /     / / /             \     \

There are 12 crossings in all: 2 in the hundreds band, 7 in the tens band and 3 in the units band. Twelve is also 3 x 4, the sum of the digits of 21 (2 + 1 = 3) times the sum of the digits of 13 (1 + 3 = 4). That is not a coincidence, and it gives the method a free error check that we use throughout.

How to read the grid diagrams in this post

Slanted lines are hard to draw in text, so for every problem below we turn the classic picture by 45 degrees. It is the same drawing, rotated.

• Rows are the lines of the first number. The top bundle is its highest place value, separated from the next bundle by a blank row.
• Columns are the lines of the second number. The left bundle is its highest place value, separated from the next by a wider gap.
• Each crossing point is printed as the power of ten of its zone: 2 for hundreds (10²), 1 for tens, 0 for units. You count the 2s, the 1s and the 0s.
• Zones run along the diagonals from top right to bottom left, exactly as the bands do in the classic picture. Top left is the highest zone, bottom right is the units zone.
• A zero digit is drawn as one dotted phantom line (.... across, : down). Its crossings are printed as o and are never counted. We explain why below.

Here is 21 x 13 again as a grid. Count the 2s (2 of them), the 1s (7) and the 0s (3).

    |     |  |  |
----2-----1--1--1----
----2-----1--1--1----
    |     |  |  |
----1-----0--0--0----
    |     |  |  |

Japanese, Chinese, Mayan or neither: where the names come from

The method travels online under at least six names: Japanese multiplication, Chinese multiplication, Chinese stick multiplication, Mayan multiplication, Vedic or Indian multiplication, and the neutral line or stick multiplication. A name that points at a country implies a history. So we went looking for that history.

What we looked for

We searched for any historical source that describes multiplying by drawing bundles of lines and counting their crossing points: a mathematical text, a printed arithmetic book, a school curriculum, a museum object or a scholarly history. We also looked for the earliest dated appearance in teaching material, online or in print. Our sources are listed at the end of the post.

What we found

No historical source supports any of the national names. We did not find the method in the historical record of Japanese, Chinese, Maya or Indian mathematics as described by the scholarly sources we used for the traditions series, and we found no scholar who claims it is there. The writers who have looked at the question say the same thing in different words:

• A mathematics teaching site, Make Math Moments, lists the names Chinese, Vedic, Indian, stick and line multiplication, floats the idea that the method "very well could have originated" in Ontario's own curriculum, and then concedes that they have no idea where it came from. It is honest speculation, not evidence.
• Giulio D'Agostini, a physicist at Sapienza University of Rome, wrote a dialogue about the method for the Italian mathematics teaching journal Progetto Alice (number 55, 2018; also on arXiv as 1802.02630). He calls it a "so-called Japanese method" that the media spread with videos under headlines like "Goodbye multiplication tables". His point is that the method does not remove the need for tables at all, but that it is a good way to show children the reasoning behind the column method they learned mechanically. He gives no origin for it, and says a Google search for "japanese method multiplication" returned about 80 million results in early February 2018.
• One of the Italian articles he cites, in the children's magazine Focus Junior, is dated 16 June 2016 and is headlined, in translation, "Goodbye times tables: the Japanese method for learning multiplication".

The earliest dated appearance we could find is from January 2011. A mathematics teacher's blog, Teaching Math with Technology, posted "Japanese Multiplication: How Does it Work?" on 17 January 2011, embedding a video of the trick and explaining it with expanded notation and the FOIL rule from algebra. Its worked example is 21 x 13. So by early 2011 the method, the "Japanese" label and the 21 x 13 example were already circulating together online, and the teacher found them "through the internet and friends", not in a textbook. We could not find the original video or any earlier printed source. If you know of a dated print source before 2011, we would like to hear about it, and we will update this post.

That gives our honest verdict: the names are folklore. "Japanese multiplication" is a label that came with a viral video, not with a historical source. The same is true of "Chinese stick multiplication" and "Mayan multiplication". The method itself is a sound piece of mathematics with a clear modern teaching purpose. Its national names are not.

What Japanese schools actually teach

The claim that "this is how Japanese children multiply" can be checked against Japan's national curriculum. The English translation of the Course of Study for elementary school arithmetic, posted by Japan's Ministry of Education (MEXT) in 2011 (the curriculum has since been revised, in 2017), sets out:

• Grade 2: learn the multiplication table up to 9 times 9 (the 九九 kuku) and multiply one-digit numbers accurately.
• Grade 3: multiply 2- and 3-digit numbers by 1- and 2-digit numbers, understanding that "those calculations are based on basic multiplication of 1-digit numbers", and learn "the way of calculation using algorithms in column forms". Grade 3 also introduces the soroban (算盤, the Japanese abacus) for simple addition and subtraction.

Nothing in that document mentions counting the crossings of lines. Japanese children learn the same column method as children elsewhere, on top of a times table they recite. That table has a very long history in Japan. About 80 ancient wooden tablets carrying kuku tables have been found there, including one from the Nara period (710 to 794) dug up at Kyōtango in Kyoto Prefecture, which officials probably used to speed up tax calculations. When C. G. Knott described "Japanese Arithmetic" to the Edinburgh Mathematical Society on 9 June 1893, his talk was about the abacus as used in China and Japan, with its multiplication and division tables; he had already written up the abacus for the Transactions of the Asiatic Society of Japan in 1886. Our Japanese mathematics post covers the soroban, anzan and wasan in detail. None of them involve counting crossings.

What about "Chinese stick multiplication"?

This name is the most plausible sounding, because Chinese mathematics really did use sticks. Counting rods (算筹 suanchou) were laid out on a counting board in place value columns from at least the 4th century BCE, with digits built from vertical and horizontal rods and the two forms alternating from column to column so that 3 and 21 could not be confused. But rods were a written number system, not a picture of a product. The Sunzi Suanjing (孫子算經) gives instructions for multiplying, dividing and taking square roots on the board, and multiplication there, like the column method, is built from single-digit products rather than from counting. We found no source in which Chinese mathematicians counted rod crossings to get a product. Our Chinese mathematics post works the same problems on a counting board, step by step.

And "Mayan" or "Vedic"?

Maya numerals were written with dots for ones and bars for fives in a base 20 place value system, so a Maya scribe also drew numbers as marks. We found no source showing that Maya arithmetic multiplied by crossing lines, and the bar and dot system groups marks by fives and twenties, which does not match a base 10 crossing count.

"Vedic" is the most interesting misattribution, because the structure really is the same as a Vedic Mathematics sutra. Urdhva-Tiryagbhyam ("vertically and crosswise") computes exactly the zone sums that the lines count: units times units, then the two cross products, then tens times tens. But that sutra comes from Bharati Krishna Tirtha's book Vedic Mathematics, published in 1965, and historians note that its sutras do not appear in the Vedic texts themselves. The sutra as taught gets each cross product from the times table, and we found no source that ties the line drawing to the 1965 book. Our Indian and Vedic mathematics post explains the sutra and its history.

The real ancestors

If the method has a family tree, it runs through documented algorithms rather than countries:

• The lattice or gelosia method, recorded in the Maghreb and in England around 1300 and spread by Italian printers, writes each digit product in a cell and adds along diagonals. Those diagonal sums are exactly the zone counts below. Our lattice, Napier's bones and slide rule post works the same problems on a lattice.
• Expanded notation and the area model, standard in modern primary teaching, split a number into tens and ones and multiply the parts. The 2011 blog explains the line method this way.
• Repeated addition, the oldest idea of all, which is what the dots really are: 8 lines crossing 9 lines is 8 groups of 9 points. The doubling method of the Rhind papyrus is another way to organise the same counting, covered in our Egyptian and Russian peasant multiplication post.

Why it works: the algebraic proof

It is not magic. It is the distributive law, drawn.

Step 1: two bundles cross in a x b points

Take a bundle of a parallel lines and a bundle of b parallel lines, and suppose the two directions are not parallel to each other. Any two non-parallel straight lines in a plane meet in exactly one point. So each of the a lines meets each of the b lines once, and the two bundles together make exactly a x b crossing points, as long as no three lines pass through the same point. (If you draw carefully that never happens, because lines in the same bundle are parallel and never meet each other.) This is the counting fact behind times tables too: a rows of b dots.

Step 2: write each number in expanded form

Write the first number with digits a₀ (units), a₁ (tens), a₂ (hundreds) and so on, and the second with digits b₀, b₁, b₂:

A = a0 + a1·10 + a2·10² + ... + am·10^m
B = b0 + b1·10 + b2·10² + ... + bn·10^n

In the drawing, bundle i of the first number has a_i lines and stands for a_i x 10^i. Bundle j of the second number has b_j lines and stands for b_j x 10^j.

Step 3: multiply out with the distributive law

Multiplying two sums means multiplying every term of one by every term of the other and adding the results:

A x B = sum over all i, j of  (ai x 10^i) x (bj x 10^j)
      = sum over all i, j of  (ai x bj) x 10^(i+j)

Each term a_i x b_j is exactly the number of crossings between bundle i and bundle j (Step 1). Its weight is 10^(i+j), which depends only on the sum i + j.

Step 4: a zone is all the pairs with the same i + j

Now group the terms by k = i + j:

A x B = sum over k of  c_k x 10^k,   where  c_k = sum over i + j = k of  ai x bj

The crossings of bundle i with bundle j sit where the ith group of the first number meets the jth group of the second. Because both numbers are drawn highest place first and in the same order, the groups with the same i + j line up along one diagonal of the picture. That diagonal is the zone. So the number of crossings in zone k is exactly c_k. This is the claim the method rests on, and it is now proved: the count in each zone is the coefficient of 10^k in the expanded product.

For 21 x 13 the coefficients are c₂ = 2 x 1 = 2, c₁ = 2 x 3 + 1 x 1 = 7, c₀ = 1 x 3 = 3, and 2 x 100 + 7 x 10 + 3 = 273.

Step 5: carrying turns the coefficients into digits

The c_k are not yet digits, because a zone can hold more than 9 crossings (the tens zone of 98 x 97 holds 135). Carrying fixes that without changing the value. Starting from k = 0, write t_k = c_k + carry_in, keep the digit d_k = t_k mod 10, and pass carry_out = (t_k minus d_k) / 10 to the next zone. Each step uses the identity

t_k x 10^k = d_k x 10^k + carry_out x 10^(k+1)

so the total never changes, and when the last carry is written down every place holds a single digit from 0 to 9. A number has only one such representation in base 10, so the result is the product.

The polynomial view, and why base does not matter

Steps 2 to 4 say something stronger. Replace 10 by a variable x. Then A(x) = a₀ + a₁x + a₂x² + ... and B(x) = b₀ + b₁x + b₂x² + ... are polynomials, and the zone counts are the coefficients of the polynomial A(x) x B(x). In computing terms, the zone counts are the convolution of the two digit lists. Carrying is only what happens when you set x = 10.

That means the crossing counts do not depend on the base at all. Draw 21 and 13 as before but read them as base 8 numerals: 21 in base 8 is 17 and 13 in base 8 is 11. The zones still hold 2, 7 and 3 crossings, and because none of them reaches 8 there is nothing to carry, so the product is 273 in base 8, which is 2 x 64 + 7 x 8 + 3 = 187 = 17 x 11. The lines are the same. Only the carrying rule changes.

A free check: total crossings and casting out nines

Every line of the first number crosses every line of the second, so the total number of crossings is (sum of digits of A) x (sum of digits of B). A number is congruent to its digit sum modulo 9, so this total is also congruent to A x B modulo 9. That is casting out nines, built into the drawing:

If your zone counts do not add up to the product of the digit sums, you have miscounted somewhere. Like any casting out nines check, this one cannot catch an error that changes a result by a multiple of 9, such as two transposed digits in the answer.

Same numbers, four pictures: lines, lattice, crosswise and columns

Because the zone counts are the convolution of the digit lists, every method that computes that convolution produces the same intermediate numbers. Here they are for 47 x 53.

The only difference is how each single-digit product is obtained. The line method gets 7 x 5 = 35 by counting 35 dots; the others recall it in one step. That is why the line method looks fast on 21 x 13 and slow on 98 x 97.

Problem 1: 98 x 97 = 9,506

This is the worst problem in the set for the line method and the best for some others. Every digit is large.

Lines to draw: 9 + 8 = 17 for 98, and 9 + 7 = 16 for 97, so 33 lines in all. Crossings to count: 17 x 16 = 272.

    |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |
----2--2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1----
    |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |
----1--1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0----
    |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |

The grid has four blocks. Top left is 9 lines crossing 9 lines, all printed 2. Top right (9 lines by 7) and bottom left (8 lines by 9) are both printed 1: together they are the tens zone. Bottom right is 8 lines crossing 7, printed 0.

Now carry, right to left:

Reading the written digits upwards gives 9,506. Note the carry of 14 out of the tens zone. Most children's examples never produce a two-digit carry, and this is where the method most often goes wrong in practice: the carry is the whole of 140 divided by 10, not just its first digit.

How the other methods compare. Vedic Nikhilam does 98 x 97 in two tiny steps: the deficits from 100 are 2 and 3, so the left part is 98 minus 3 = 95 and the right part is 2 x 3 = 06, giving 9506. The column method needs four single-digit products (8 x 7, 9 x 7, 8 x 9, 9 x 9) and one addition. The line method needs 33 lines and 272 counted dots to get the same three zone totals. On this problem it is not a contender.

Problem 2: 47 x 53 = 2,491

Lines to draw: 4 + 7 = 11 for 47 and 5 + 3 = 8 for 53, so 19 lines. Crossings to count: 11 x 8 = 88.

    |  |  |  |  |     |  |  |
----2--2--2--2--2-----1--1--1----
----2--2--2--2--2-----1--1--1----
----2--2--2--2--2-----1--1--1----
----2--2--2--2--2-----1--1--1----
    |  |  |  |  |     |  |  |
----1--1--1--1--1-----0--0--0----
----1--1--1--1--1-----0--0--0----
----1--1--1--1--1-----0--0--0----
----1--1--1--1--1-----0--0--0----
----1--1--1--1--1-----0--0--0----
----1--1--1--1--1-----0--0--0----
----1--1--1--1--1-----0--0--0----
    |  |  |  |  |     |  |  |

The answer is 2,491. (The tens zone holding 47 crossings when one factor is 47 is a coincidence.)

How the other methods compare. 47 x 53 is 50 minus 3 times 50 plus 3, so the difference of squares gives 2,500 minus 9 = 2,491 in one line of mental work. The crosswise sutra gets 20, 47 and 21 with three table lookups and two additions. The line method gets the same three numbers by counting 88 dots.

Problem 3: 386 x 524 = 202,264

With three digits on each side there are three bundles each way, nine blocks of crossings and five zones, from 10⁴ down to 10⁰.

Lines to draw: 3 + 8 + 6 = 17 for 386 and 5 + 2 + 4 = 11 for 524, so 28 lines. Crossings to count: 17 x 11 = 187.

    |  |  |  |  |     |  |     |  |  |  |
----4--4--4--4--4-----3--3-----2--2--2--2----
----4--4--4--4--4-----3--3-----2--2--2--2----
----4--4--4--4--4-----3--3-----2--2--2--2----
    |  |  |  |  |     |  |     |  |  |  |
----3--3--3--3--3-----2--2-----1--1--1--1----
----3--3--3--3--3-----2--2-----1--1--1--1----
----3--3--3--3--3-----2--2-----1--1--1--1----
----3--3--3--3--3-----2--2-----1--1--1--1----
----3--3--3--3--3-----2--2-----1--1--1--1----
----3--3--3--3--3-----2--2-----1--1--1--1----
----3--3--3--3--3-----2--2-----1--1--1--1----
----3--3--3--3--3-----2--2-----1--1--1--1----
    |  |  |  |  |     |  |     |  |  |  |
----2--2--2--2--2-----1--1-----0--0--0--0----
----2--2--2--2--2-----1--1-----0--0--0--0----
----2--2--2--2--2-----1--1-----0--0--0--0----
----2--2--2--2--2-----1--1-----0--0--0--0----
----2--2--2--2--2-----1--1-----0--0--0--0----
----2--2--2--2--2-----1--1-----0--0--0--0----
    |  |  |  |  |     |  |     |  |  |  |

The diagonal structure is easiest to see here. Read the nine blocks as a 3 by 3 table: the top left block is zone 4, the two blocks touching it are zone 3, the three blocks on the main anti-diagonal are zone 2, then two blocks of zone 1 and the bottom right block, zone 0.

The answer is 202,264. Every zone produced a carry, and the middle zone combines three separate blocks of crossings that are not next to each other in the classic slanted drawing. This is where people lose count: in the slanted version the hundreds zone is a strip through the middle of the diamond that contains three differently shaped clusters.

How the other methods compare. The column method and the lattice each need nine single-digit products, the same nine pairs counted here. The crosswise sutra for three digits needs the same nine products grouped into five steps. The line method replaces each of those nine lookups with counting: 15, 6, 40, 12, 16, 30, 32, 12 and 24 dots, 187 in all.

Problem 4: 85² = 7,225

Squaring is multiplication of a number by itself, so the drawing is symmetric: the same bundles go both ways.

Lines to draw: 8 + 5 = 13 each way, so 26 lines. Crossings to count: 13 x 13 = 169.

    |  |  |  |  |  |  |  |     |  |  |  |  |
----2--2--2--2--2--2--2--2-----1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1----
    |  |  |  |  |  |  |  |     |  |  |  |  |
----1--1--1--1--1--1--1--1-----0--0--0--0--0----
----1--1--1--1--1--1--1--1-----0--0--0--0--0----
----1--1--1--1--1--1--1--1-----0--0--0--0--0----
----1--1--1--1--1--1--1--1-----0--0--0--0--0----
----1--1--1--1--1--1--1--1-----0--0--0--0--0----
    |  |  |  |  |  |  |  |     |  |  |  |  |

The answer is 7,225. Because the drawing is symmetric, the two tens blocks are mirror images and always hold the same number of crossings, so you can count one and double it. That halves the work in the tens zone, and it is the line method's version of the algebra identity (10a + b)² = 100a² + 2 x 10ab + b².

How the other methods compare. For any square ending in 5, the Vedic Ekadhikena Purvena shortcut multiplies the leading digit by one more than itself and appends 25: 8 x 9 = 72, then 25, giving 7225. That is one table fact. The line method counts 169 dots, or 64 + 40 + 25 = 129 if you use the symmetry.

Zeros: phantom lines, and why they matter

A zero digit has no lines, but it still holds a place. If you simply leave it out, the bundles on either side slide together and every zone after it shifts by one place value. 203 x 12 drawn without the zero would be read as 23 x 12 = 276, which is wrong.

The fix used by teachers is a phantom line: draw one dashed or dotted line where the zero bundle would be, so that the zones stay in the right places, and never count its crossings. Here is 203 x 12 = 2,436 with a phantom line for the 0 in 203 (the dotted row; its crossings are printed o and not counted).

    |     |  |
----3-----2--2----
----3-----2--2----
    |     |  |
....o.....o..o....
    |     |  |
----1-----0--0----
----1-----0--0----
----1-----0--0----
    |     |  |

No zone exceeds 9, so there is nothing to carry and the digits read straight off: 2,436. In the algebra, the phantom line is simply a₁ = 0: the term is still in the sum, it just contributes nothing. In the picture it is what keeps the hundreds crossings of 2 x 2 out of the tens zone.

Problem 5: √7,744 = 88, and why lines cannot find it

There is no line method for square roots. The method is a one way machine: it takes two digit lists and returns their product. A square root asks the opposite question: which digit list, crossed with itself, gives 7,744? Answering that means searching, and the picture gives you nothing to search with. Methods that do extract roots, such as the digit by digit method on a counting board or soroban, the duplex method of the Vedic book, or Napier's square root rod, all work digit by digit from the left with a subtraction at each step. Line multiplication has no subtraction step and no way to work from the left.

What the lines can do is check a guess. 7,744 lies between 80² = 6,400 and 90² = 8,100, and a square ending in 4 must have a root ending in 2 or 8. Try 88:

    |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |
----2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1--1----
----2--2--2--2--2--2--2--2-----1--1--1--1--1--1--1--1----
    |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |
----1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0--0----
----1--1--1--1--1--1--1--1-----0--0--0--0--0--0--0--0----
    |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |

The grid confirms 88² = 7,744, at a cost of 32 lines and 16 x 16 = 256 crossings. The neighbours 87² = 7,569 and 89² = 7,921 rule out the alternatives. Note that the guess came from reasoning about sizes and last digits, not from the lines.

Problem 6: 7,429 ÷ 37 = 200 remainder 29, and why lines cannot divide

There is no line method for division either, for the same reason. Long division asks, at each step, how many times the divisor fits, and then subtracts. The lines compute a product only after you have chosen both factors.

Again the lines can check an answer. If the quotient is 200 with remainder 29, then 37 x 200 + 29 must equal 7,429. Here is 37 x 200, with phantom lines for the two zeros in 200:

    |  |     :     :
----3--3-----o-----o----
----3--3-----o-----o----
----3--3-----o-----o----
    |  |     :     :
----2--2-----o-----o----
----2--2-----o-----o----
----2--2-----o-----o----
----2--2-----o-----o----
----2--2-----o-----o----
----2--2-----o-----o----
----2--2-----o-----o----
    |  |     :     :

Carrying the 1 from the hundreds zone (14 = 4 carry 1) gives 6 + 1 = 7 thousands, so 37 x 200 = 7,400, and 7,400 + 29 = 7,429. To show 200 is the largest possible quotient, try 201: the units bundle of 201 becomes one real line, adding 3 crossings in the tens zone and 7 in the units zone, so 37 x 201 = 7,437, which is more than 7,429. The quotient is 200 remainder 29, but the lines only confirmed it. The division itself happened in your head.

Problem 7: 1/19 as a decimal, and why lines cannot get near it

1/19 = 0.052631578947368421 repeating, an 18 digit cycle. Finding those digits is a division, which the line method cannot do. Checking them is a multiplication, and here the line method collapses under its own weight.

The check is that 19 x 052,631,578,947,368,421 = 999,999,999,999,999,999 (eighteen nines), which is why the 18 digits repeat. The digits of 52,631,578,947,368,421 add up to 81. As a line drawing, the check would need 1 + 9 = 10 lines for 19 and 81 lines for the long number (it has no zero digits), and 10 x 81 = 810 crossings spread over 18 zones. Nobody should do that. The column method does the same check with 34 single-digit products, and the Ekadhikena Purvena method in the Vedic book generates the digits directly by multiplying by 2 from the right. This problem marks the end of the line method's range.

Where the method stops being fast: 8 x 9 is 72 dots

The line method has one real strength: it never asks for a times table fact. It replaces every single-digit product with counting. That is wonderful when the digits are 1, 2 and 3 and terrible when they are 8 and 9. Here is a single pair, 8 x 9 = 72, as lines:

    |  |  |  |  |  |  |  |  |
----0--0--0--0--0--0--0--0--0----
----0--0--0--0--0--0--0--0--0----
----0--0--0--0--0--0--0--0--0----
----0--0--0--0--0--0--0--0--0----
----0--0--0--0--0--0--0--0--0----
----0--0--0--0--0--0--0--0--0----
----0--0--0--0--0--0--0--0--0----
----0--0--0--0--0--0--0--0--0----
    |  |  |  |  |  |  |  |  |

That is 17 lines and 72 crossings to produce one fact that a times table gives instantly. The rest of the method has nothing left to do: one zone, 72 = 2 carry 7, answer 72.

The cost grows with the size of the digits, not just the number of digits:

And the totals for whole problems:

Two costs hide in those counts. Time: counting 272 dots is slower than recalling four facts. Accuracy: a dense block of 8 lines by 9 is easy to miscount by one, and the total crossings check tells you that something is wrong, not where.

We did not find any peer reviewed study that times the line method against the column method or measures its error rate, so we make no claim beyond these counts. If you want to measure it yourself, the counts in the table are the right thing to time.

Classroom value versus speed

D'Agostini's conclusion in Progetto Alice is the fair one: the "revolutionary" value of the method dissolves on contact with large digits, but it has real teaching value because it shows why the column method works. Here is how it compares on the dimensions that matter.

Every method except the line method assumes the times table. The line method trades it for counting: a good first picture for children who do not yet know their tables, and a poor tool for anyone who does.

Where it earns its place in a classroom:

• As a first proof. Drawing 21 x 13 and seeing 200, 70 and 3 appear as three groups of dots makes the distributive law visible. Children can count the dots and check the answer before they can recite the 7 times table.
• As a bridge to the column method and to algebra. The zones are the columns. The two tens blocks are the "outer" and "inner" of FOIL. Moving from a drawing of (2x + 1)(x + 3) to the line picture of 21 x 13 is a short step, and the 2011 teacher's blog made exactly that connection.

Where it does not belong: in timed tests, in anything with digits above 5, and in any claim that it lets children skip their times tables. The headline "Goodbye multiplication tables" is exactly backwards. The method is a way to understand what the tables are for.

Faster ways to do the same problems

If speed is the goal, other methods in this series beat lines on every problem in the set:

• Near a base: Vedic Nikhilam does 98 x 97 in two small steps (95 and 06).
• Symmetric pairs: the difference of squares does 47 x 53 as 2,500 minus 9.
• Squares ending in 5: 85² is 8 x 9 = 72 with 25 appended.
• General multiplication: the crosswise sutra or the column method, both of which need the same zone sums but get each product in one recall. On a soroban, with training, the same products become bead movements, and with anzan they become mental images of the beads. Our comparison of Vedic, Chinese and Japanese methods counts the steps for each.
• Without any table at all: if the goal is to avoid memorised products entirely, the doubling and halving of Egyptian and Russian peasant multiplication needs only the ability to double and add. For 386 x 524 it needs 8 halvings, 8 doublings and 2 additions (1,048 + 67,072 + 134,144 = 202,264), far fewer operations than counting 187 dots.

Variants you will see online

Decimals. The lines ignore the decimal point. Draw 4.7 x 5.3 as 47 x 53, get 2,491, then place the point by counting decimal places in the factors (one each, so two in the answer): 24.91.

Colour coding. Teachers often colour each bundle and circle each zone; the 2011 blog used green, orange, blue and red bundles.

Three numbers. Line multiplication only crosses two sets of lines at a time, because crossings are points where two lines meet. To multiply three numbers, multiply two, then draw the product against the third.

Walkthrough: reproduce every number in Python

This is the core of the script used to compute every zone count and carry step in this post. It prints each carry step and the zone sums, and checks them against Python's own multiplication.

def digits(n):
    return [int(c) for c in str(n)]

def zones(a, b):
    """Crossings per zone: zone k collects digit pairs whose place values add to k."""
    A, B = digits(a)[::-1], digits(b)[::-1]     # A[i] is the digit at 10**i
    z = [0] * (len(A) + len(B) - 1)
    for i, x in enumerate(A):
        for j, y in enumerate(B):
            z[i + j] += x * y                    # x lines crossing y lines
    return z

def carry(z):
    out, c = [], 0
    for k, v in enumerate(z):                    # right to left
        t = v + c
        out.append(t % 10)
        c = t // 10
        print(f"zone 10^{k}: count {v}, carry in {t - v}, total {t}, write {t % 10}, carry out {c}")
    while c:
        out.append(c % 10)
        c //= 10
    return int("".join(map(str, out[::-1])))

for a, b in [(21, 13), (98, 97), (47, 53), (386, 524), (85, 85), (203, 12), (88, 88), (37, 200)]:
    z = zones(a, b)
    s = lambda n: sum(digits(n))
    assert sum(z) == s(a) * s(b)                 # total crossings = digit sum product
    result = carry(z)
    assert result == a * b
    print(a, "x", b, "zones high to low:", z[::-1], "=", result)

Run it with python3 lines.py. The two assert lines are the proof in miniature: the first checks that the crossings add up to the product of the digit sums, and the second checks that carrying the zone counts gives the true product.

Common mistakes

• Leaving out a zero. Without a phantom line, every later zone shifts by one place and the answer is wrong by a factor of ten or more. Always draw a dashed line for a 0 digit and never count its crossings.
• Drawing the bundles in different orders. Both numbers must go highest place first in the same direction, or the zones no longer line up along diagonals and the hundreds and units crossings swap places.
• Merging the two tens blocks wrongly. In a 2 digit by 2 digit drawing the tens zone is two separate clusters of crossings. Count both and add them; do not count one and forget the other. In squares, the two clusters are equal, so you can double one.
• Carrying only the first digit of a big zone. In 98 x 97 the tens total is 140, so the carry is 14, not 1. Any zone total of 100 or more produces a two digit carry.
• Miscounting dense blocks. A block of 8 lines by 9 lines has 72 points close together. Count it as 8 rows of 9, not dot by dot, and check the total against the product of the digit sums.
• Calling it Japanese. There is no evidence for the name, and Japan's curriculum teaches the times table and column multiplication instead. "Line multiplication" or "stick multiplication" is accurate.
• Expecting it to divide or take roots. It only multiplies. Use it to check a quotient or a root you found another way.

FAQ

What is line multiplication?

Line multiplication, also called stick multiplication, is a way to multiply two numbers by drawing each digit as a bundle of parallel lines, crossing the bundles of one number with those of the other, and counting the crossing points in each diagonal zone. Each zone is one place value. Carrying from right to left turns the zone counts into the answer, for example 2, 7 and 3 for 21 x 13 = 273.

Is line multiplication really Japanese?

We found no historical source that supports the name. Japan's national curriculum teaches the 9 x 9 multiplication table in grade 2 and column multiplication in grade 3, and Japan's traditional calculating tool is the soroban. The earliest dated use of the name we found is a teacher's blog post from 17 January 2011 that embedded a viral video. The name is folklore.

Is it Chinese or Mayan multiplication?

There is no source for either name. Chinese mathematicians did calculate with counting rods, but rods were a place value number system and multiplication on the counting board used times table products. Maya numerals used bars and dots in base 20, and we found no record of a crossing line method in Maya arithmetic.

Why does line multiplication work?

Two bundles of a and b non-parallel lines cross in exactly a x b points. The crossings of the bundle for 10^i with the bundle for 10^j are worth 10^(i+j), and a zone collects all pairs with the same i + j. So each zone count is the coefficient of one power of ten in the expanded product, which is the distributive law. Carrying turns those coefficients into single digits.

Is line multiplication faster than normal multiplication?

Only for small digits. It needs no times table, but it replaces every table fact with counting. 21 x 13 needs 12 crossings, 98 x 97 needs 272 and 8 x 9 alone is 72. The column method does any of these with a handful of remembered facts. We found no peer reviewed study timing the two.

How do you handle zeros in line multiplication?

Draw one dashed or dotted phantom line where the zero bundle would be, so the zones keep their place values, and do not count any crossings on it. Without it, 203 x 12 turns into 23 x 12 and the answer comes out wrong.

Can you divide or find square roots with lines?

No. The method only turns two known factors into a product. Division and square roots need a search with subtraction at each step, which the picture does not provide. You can use lines to check a result, such as 37 x 200 + 29 = 7,429 or 88² = 7,744.

Is line multiplication the same as Vedic multiplication?

It computes the same numbers as Urdhva-Tiryagbhyam, the "vertically and crosswise" sutra from Bharati Krishna Tirtha's 1965 book Vedic Mathematics: the zone counts are exactly the crosswise sums. The difference is that the sutra gets each digit product from the times table, while the line method counts it as dots.

Should children learn line multiplication?

As a picture of why multiplication works, it is useful: children can see place value and the distributive law before they know their tables. As a replacement for the times tables, no. It becomes slow and error prone as soon as digits pass about 5.

Deep dives in this series

• Egyptian and Russian Peasant Multiplication: Doubling and Binary
• Lattice Multiplication, Napier's Bones and the Slide Rule Explained
• Indian and Vedic Mathematics Explained: the 16 Sutras, Śulba Sūtras, Zero, Āryabhaṭa, Brahmagupta and the Kerala School
• Chinese Mathematics Explained: Counting Rods, the Nine Chapters, Fangcheng, the Remainder Theorem and Horner before Horner
• Japanese Mathematics Explained: the Soroban, Anzan, Wasan, Seki Takakazu and Sangaku Temple Geometry
• Vedic vs Chinese vs Japanese Math: Same Problems, Three Systems

Checking a viral claim, checking a dependency: an honest bridge

Safeguard does not do mathematics, and nothing in this post was computed by a Safeguard model. The parallel to our work is in how this method spread: a video with a confident name was copied from site to site for years, and very few copies checked where it came from. Software components travel the same way. Most of an application is open source code pulled in from somewhere else, and it is safer when every component's origin is known, every version is checked against published vulnerabilities, and the check can be rerun later, the way the two assert lines above can be rerun on any numbers. If you build a classroom tool around this material, from a line drawing app to a notebook like the script above, its dependencies deserve the same checking and timely fixes as the code in a production service.

More in this series

Other posts in our fast calculation methods series:

• Babylonian Mathematics Explained: Base 60, YBC 7289 and Plimpton 322
• Chisanbop Explained: Korean Finger Math to 99, With Worked Examples
• Egyptian and Russian Peasant Multiplication: Doubling and Binary
• Fastest Mental Math Methods Compared: 14 Systems, Same 7 Problems
• Lattice Multiplication, Napier's Bones and the Slide Rule Explained
• Maya Mathematics Explained: Base 20, Bar and Dot Numerals and Zero
• Mental Calculation Techniques of World Champions: Left to Right, Criss-Cross, Squares and Calendars
• Trachtenberg Speed System Explained: Rules for 11, 12 and Any Number

Fix it without waiting for upstream

Whatever the vulnerability, most of the risk sits in two gaps. The first is the time between a fix being published and that fix actually running in your production builds. The second, and worse, is the time before any fix exists at all, while a maintainer works out a patch and a release schedule. Waiting through both gaps is a habit, not a requirement. Safeguard gives teams three ways to close them.

1. Autonomous remediation: pull requests, not tickets

When a fixed version exists, Safeguard's Griffin models open a tested, policy gated pull request that moves the affected dependency to the safe version. You can run it on one repository or in bulk across every project that carries the vulnerable version. Remediation covers eight categories, including dependencies, licenses, security posture and findings. Reachability analysis ranks the vulnerable code paths your application actually calls first, so the first pull requests you merge are the ones that matter most. Autonomous remediation ships switched off, with every category set to Manual, so your team decides how much runs on its own and what still needs a human approval.

2. Gold Open Source: the safe version in seconds, free

gold.safeguard.sh is free and needs no login. Search any CVE or package to see affected and fixed versions, CVSS, EPSS, CISA KEV status and a recommended safe version, or use the "Am I affected?" checker. The same data is available through a free read API, README status badges and RSS threat feeds. The safeguard-sh/gold-scan-action GitHub Action can fail a build on a known exploited vulnerability, on any known vulnerability, or never, so a version you have just removed cannot quietly come back in next week's merge.

3. Gold Registry: start from components that are already fixed

registry.safeguard.sh provides hardened, drop in replacements for upstream packages and container images. Every Gold artifact is held continuously to zero exploitable critical or high CVEs, zero malware and a completed license check, with SLSA L3 or higher provenance, a Sigstore signature and a public evidence page you can hand to an auditor. An artifact loses its Gold badge if it regresses. Teams can point their own package directory at the registry so hardened equivalents resolve without changing how developers install anything.

When upstream has no fix yet, or will not ship one, Safeguard first works to get the fix accepted upstream. If the issue is severe or upstream will not move, Safeguard builds a fork, fixes the vulnerability there and publishes a hardened Gold build, so you are not stuck on a maintainer's release schedule. You can also request a hardened, compatibility tested build of a specific package. This fork and patch flow is in early access today, so ask which packages it covers before you plan around it.

One honest limit: when the vulnerable code lives in a vendor appliance, a hosted service or an operating system runtime you do not build yourself, the fix still has to come from that vendor or distribution. The steps above apply to the open source dependencies and container images you build and ship, which is where most teams carry the largest and least tracked share of their exposure.

See how Safeguard would help

None of these demos involve any arithmetic; they show the software side described in the bridge above.

The first demo is a real Safeguard CLI session. It installs the CLI, opens the interactive terminal and asks it to fix a lodash CVE in a repository. It reads the finding and the advisory, asks before any command that changes something, edits package.json to lodash 4.17.21, runs npm install and npm test with approval, prints a summary and pushes nothing. The same flow applies to a web app that draws line multiplication diagrams for a classroom: scan the repository, then let the assistant propose the dependency upgrade under your approval.

Click through it at your own pace, or open it full screen.

The second demo sets up the Safeguard runner on your own machine, approves it with its device code and then scans from the terminal. That suits teachers and hobbyists who keep scripts like the walkthrough above on their own laptops and want the scan to run where the code lives.

Click through it at your own pace, or open it full screen.

The third demo connects Claude to the Safeguard MCP server with OAuth and lets you choose which tools Claude can use. If you already work with Claude while building an education app, it brings dependency findings into the same conversation, limited to the tools you allow.

Click through it at your own pace, or open it full screen.

To try it on your own repositories, download Safeguard and follow the CLI documentation.

Quick check

Sources

• Teaching Math with Technology (Sue Glascoe), "Japanese Multiplication: How Does it Work?", 17 January 2011, archived copy
• G. D'Agostini, "On the Japanese Multiplication Method. A father-and-daughter dialogue", arXiv:1802.02630, published in Progetto Alice 55 (2018)
• Make Math Moments, "Japanese Multiplication? Chinese Multiplication? Line Multiplication?"
• MEXT, Course of Study for Elementary School, Section 3 Arithmetic (English translation)
• nippon.com, Nara period wooden multiplication table found in Kyōtango (in French)
• C. G. Knott, "Japanese Arithmetic", Proceedings of the Edinburgh Mathematical Society, meeting of 9 June 1893
• MacTutor History of Mathematics, Chinese numerals

Further reading, cited in the text but not fetched for this post: L. Perego, "Addio tabelline: il metodo giapponese per imparare le moltiplicazioni", Focus Junior, 16 June 2016 (cited by D'Agostini); Bharati Krishna Tirtha, Vedic Mathematics (Motilal Banarsidass, 1965); Kim Plofker, Mathematics in India (Princeton, 2009). For the lattice history and the Vedic, Chinese and Japanese background, see the sources listed in the linked posts in this series.

Last reviewed: 25 October 2026.