Magically Multiplying with Lines

“Magic is just science that we don’t understand yet.” – Arthur C. Clarke

The same can be said of mathematics. Such enchantments can occur when a professor stands at the front of the room and magically finds the sum of an infinite series, the product of polynomials, or even the product of two numbers. When we stand on the other side of understanding, that which seemed magical can seem quite logical.

Math is a place where we can help each other bridge those gaps and see the magical become understood. To tap into this, let’s see an example of multiplying numbers that simply mystified me when I first saw it back in 2006. While more polished videos exist now, the simplicity of this video has its charm and the silence adds to its mystery.

Before unveiling the math (trick) behind this magic. Let’s diverge to another form of multiplication by using a table to review the process of multiplication. We’ll find the product of 21 and 13. To begin, we form the table as below:

103
20
1

Then each entry of the table is filled with the product of the numbers in its respective row and column as seen below:

103
2020060
1103

Now, to simplify the discussion below, let’s concentrate only on the entries that were products of row and column headers in the table above. This gives us:

20060
103

Adding the entries of the table gives the desired product to our problem, 273. However, notice for a moment that we could also add along the 3 diagonals that run from the upper right to the lower left. Our first diagonal has only one entry 200. The second diagonal has the numbers 60 and 10; so the sum is 70. Finally, the last diagonal has the entry 3. As you sum along each diagonal, the numbers have the same place value. The first is hundreds, the second tens and the last ones.

Let’s use this knowledge to look at the trick again. Below is my rendering of the trick with the intersection points delineated by color.

Compare this to the table and adding along those diagonals. Notice a similarity? If not, the colors of the dots may help. Each color corresponds to a different place value. Each purple dot is a hundred, each green ten and each orange one. Further, notice how adding the dots, particularly in the middle of this picture, corresponds to adding along the diagonals in the table.

Got it? Try the method of drawing lines to find the product of 32 and 12. When you get that, try 62 times 14. You might want to go back to the table, if you struggle to think through what to do with the sums you get with the points.

Magical? Possibly…but when we bridge that magical gap with understanding, such a trick, at most, is magical for mathematical muggles.