The Line Technique for Multiplication Will Blow Your Mind

Once you graduate from high school, there just isn’t much call for doing multiplication by hand anymore. We’re all carrying supercomputers around in our pockets to do our math for us — no need to pull out some scratch paper or your handy-dandy multiplication tables. But we just found out about a method of multiplication that’s not just fun to do by hand; it’s also kind of mind-blowing.

Draw Your Work

You remember how to do multiplication by hand, right? First, you place the two numbers on top of each other. Then you multiply the top digits by the digit below them in the ones place, then you multiply the top digits by the digit below them in the tens place (make sure you add a 0 to the end), and then multiply the top digits by the digit below in the 100s place, and so on. Finally, add all of those numbers up, and voila, you’ve got your answer. If only it weren’t so dull. Fortunately, there’s another way to look at it — one that involves drawing.

See How It’s Done

Let’s start off with an easy one. Say you want to multiply 4 and 3. First, draw four parallel diagonal lines starting from the lower left and going to the upper right. (In all of these examples, each subsequent line should be placed to the right of the last one.) Next, draw three more diagonal lines perpendicular to (and intersecting) the first ones, this time starting in the upper left and going to the lower right. Now, count the intersections. You should get 12 — and there’s your answer. Piece of cake! Now let’s up the ante with double-digit numbers.

Let’s do 31 x 22. For the 31, draw three diagonal lines from the lower left, leave a space, then one more diagonal line. Do the same thing for the 22 — two lines from the upper left, a space, then two more lines. The result should look like a diamond with three lines for the upper left side, two lines on the lower left side, one line on the lower right side, and two lines on the upper right side.

Next, imagine the diamond separated into vertical sections. The left corner is in a section by itself, the top and bottom corners are in a section together since they’re vertically aligned, and the right corner is all alone like the left one. Now you add up the intersections again. You should get 6 in the leftmost section, and 8 in the middle section, and a 2 in the rightmost section. There’s your answer: 682.

Scaling Up

Pretty incredible technique, right? It really blew our minds to see it in action. But we’ve only scratched the surface. Let’s try it again with the numbers 31 and 23. It’s just one number different, but it adds an interesting wrinkle. This time, you’ll get a 6 on the left side, an 11 in the middle, and a 3 on the right. That 11 is where the trouble comes in. Just like more familiar types of multiplication, you’ll have to carry the one — add it to the 6 on the left side and you’ll have your answer: 713.

The same method works on three-digit numbers, too, but we don’t have to go all the way through the steps again. Multiply 412 by 121 using this method and organize in those vertical columns again — you’ll get 4, 9, 8, 5, and 2 in order for a solution of 49,852.

Let’s try one more scenario. Let’s multiply 246 and 305. You know what to do. Start with two lines, then four lines, then six lines. Now do three lines, then … zero lines(?) … then five lines. Actually, the best way to make sure it adds up right is to put one line in a different color to represent the 0, and make a mental note not to count any intersections that include that color. For this problem, you’ll end up with 6, 12, 28, 20, and 30. That’s a lot of numbers to carry over, but the end result will be 75,030. Now that’s mathe-magic.

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