of how you can multiply numbers just by drawing lines.

Let's say you want to multiply 12 by 13

For this one we will draw one line

and for the two we will leave a little bit of space

and we will draw two lines.

For the other number, we will draw lines in the other direction.

Now we will group together different lines and we will count the dots.

Here we have six different dots.

In the middle we have five different dots.

And on the other side we have one dot.

And that's the answer 156.

We will do another example.

For 15 by 21,

we start out by drawing one line.

Then we'll leave a little bit space and draw five lines.

For the other number we draw lines in the other direction.

We draw two lines here and one line here.

We will again group the different lines

Here we have five dots.

In the middle we have eleven dots.

And the other end we have two dots.

We will need to make one adjustment because we have 11.

We will need to carry this one over which will make 3.

And our answer is 315.

We can even do this for bigger numbers.

For 123 we start out with one line and with some space two lines. And with some more space three lines.

And we draw the other lines for the other number in the other direction.

We again group the lines.

We have three dots here,

eight dots here,

fourteen,

eight,

and three.

Once again we're going to have to carry this one over (from the 14).

So that's our answer.

It'll be 39,483.

You can even use this trick when a number includes 0.

But one thing is when you draw this line you should still draw the line, but

put it in a different color or different marking.

And the trick will be that we don't count any of the dots for that line

So when we make the grouping, we have three dots here.

For the second arrangement,

we're only going to count the dots for the non-zero lines.

So we have six dots here. We don't count any of the intersections for the zero line.

In the middle, we again ignore the zero line, which makes seven.

Here we have two dots.

And two dots here.

So you can even use this method when you have a number that includes zero.

The reason it works is you can even use it for algebraic equations.

Let's say you have x plus 2y times x plus 3y.

We will draw one line for the x,

and we will draw two lines for the 2y and we will draw the lines in blue.

We will put all the x lines in black and all the y lines in blue.

The other number we draw the lines in the other direction

And again we make groupings.

So here we have six dots and because

both of the lines are blue that means it's y-squared.

In the middle we have five dots,

and these are the intersections of a black and blue line,

which means it will be an x and a y.

On the end we just have one dot and that's for two black lines so it's x-squared.

So that's the answer.

I hope you enjoyed this trick. Please subscribe to my channel. I make videos on math and game theory.

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