Lattice Multiplication (also known as Venetian Squares, The Italian Method, the Chinese Method, amongst other terms) is a method of multiplying two multi-digit numbers that’s been around since at least the medieval period.
To create a lattice, you draw a grid with as many columns as the first number has digits. For example, for a 2-digit number (from 10-99), you’d have 2 columns:
The grid also has as many rows as the second number has digits. For example, for a 3-digit number (100 - 999), you’d have three rows. Multiplying a 2-digit number by a 3-digit number would result in a 2×3 grid, like this:
Arranging them the other way around (in this case, a 3×2 grid) is perfectly acceptable, too:
Now, through each cell, you draw a diagonal line from the bottom left corner to the top right: . This, then, results in our lattice.
Our 2×3 lattice now looks like this:
The alternate 3×2 lattice would appear so:
How A Lattice Works
When you multiply using a lattice, you write one number over the top of the lattice, one digit over each column, and the other down the right-hand side, one digit beside each row. I’ll use the 2×3 lattice to demonstrate (multiplying 54 and 321).
54321
Each cell contains the product of the two single-digit numbers that line up with the column and row that cell is in, with the tens digit going in the top left. (or a zero, if the product is less than 10), and the ones digit going in the bottom right. To show you what I mean, here are two single cells from the 2×3 lattice shown above.
Left Column, Top Row53Right Column, Middle Row42
To demonstrate every single possible combination, I’ve got a lattice below for the equation 1,234,567,890×1,234,567,890, or 1,234,567,8902
A 10×10 Lattice Showing All Possible Combinations
12345678901234567890
Looking closely at the above lattice, you’ll see that while there are 100 pairs of multiplicands (and thus 100 cells) there are only 37 distinct products. Here’s why:
Of the 100 possible pairs, there are 10 square numbers (that is, the number times itself), which leaves 90 pairs of differing multiplicands. Which multiplicand is over a column and which one is beside a row is irrelevant, the product is the same. This cuts those 90 products in half—i.e. 45; when you include the 10 squares, we are left with 55 distinct pairs of multiplicands and thus 55 distinct products (thus far).
0 times any number is exactly 0, which results in 10 distinct pairs of multiplicands having the exact same product: 0. Taking that into account drops our number of distinct products by 9, leaving us with 46 distinct products (again, thus far).
There are 9 other instances where two distinct pairs of multiplicands have the same product.
When you take the 9 overlapping products into account, that leaves us with exactly 37 distinct products.
So, with our 2×3 Lattice, this is what it would look like with the cells filled in.
Adding Up The Numbers
Notice that the diagonal lines line up with each other and seperate the digits into diagonal groups within the lattice. This is how you get your final answer: you add up those diagonals, going from the bottom right to the top left.
First Diagonal (Bottom Right)
4
In the bottom right diagonal, you only have the one number. Therefore, our equation is .
Second Diagonal
13
In the next diagonal, we have the numbers 8, 0, and 5, which gives use the equation . Since this is a two digit sum, the number in the tens position (in this case, 1) goes to the next diagonal up:
13
If this resulted in a 3 digit sum (possible with a fairly large lattice), the hundreds digit would go two diagonals up, and so on for 4-, 5-, etc-digit numbers.
Third Diagonal
3
In the third , we have the numbers 2, 0, 0, the 1 that we carried over, and 0 again; i.e. .
Fourth Diagonal
7
Fifth Diagonal (Top Left)
1
Read the number down the left side and across the bottom of the lattice for your answer: 17,334.
1,234,567,8902
Of course, I can’t just leave you hanging with that big 10×10 monster unfinished, so here we go: