of how you can multiply numbers just by drawing lines.

教你怎麼用畫線來算乘法

Let's say you want to multiply 12 by 13

假設你要算 12乘以 13

For this one we will draw one line

我們先畫一條線（代表 12的十位數）

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

然後稍留點空間，代表個位數的2

and we will draw two lines.

再畫兩條線，

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

至於另一個數13，則畫在另一個方向

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

現在，把這些線加以分組，然後計算交叉點

Here we have six different dots.

這裡有6 個點（代表個位數）

In the middle we have five different dots.

中間有五個點

And on the other side we have one dot.

另一端，我們有五個點

And that's the answer 156.

所以答案是156

We will do another example.

再算另一題

For 15 by 21,

15乘以 21

we start out by drawing one line.

先畫一條十位數的線

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

然後一樣留點空間再畫五條線

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

至於另一個數 21，則從另一端畫起

We draw two lines here and one line here.

畫出十位數的兩條線和個位數的一條線

We will again group the different lines

再把不同的線加以分組

Here we have five dots.

這裡有五個交叉點

In the middle we have eleven dots.

在中間，可以算出有十一個點

And the other end we have two dots.

另一端則有兩個點

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

這裡要稍微調整一下，因為有兩位數 11

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

需要進位，移項後百位數變成 3

And our answer is 315.

得到答案 315

We can even do this for bigger numbers.

接著我們可以來算更大的數目

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

123，從百位的 1條線畫起，稍遠處則畫兩條線代表 10位數的 2，接著留點空間畫個位數的 3

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

另一個方向則分別畫出321

We again group the lines.

將這些線分組

We have three dots here,

這裡有三個點

eight dots here,

這裡有八個

fourteen,

中間十四個

eight,

八個

and three.

以及三個

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

必須再一次從中間的十四，進一位

So that's our answer.

得出答案

It'll be 39,483.

39,483

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

這個小把戲一樣可以用在數字中包含 0的

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

當你畫線時還是要把它畫出來，但是

put it in a different color or different marking.

要用不顏色或不同記號

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

但是在計算交叉點時，與這條淺色線相交的不予計入

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

所以分組後，這裡有3個點

For the second arrangement,

而在倒數第二個分組

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

我們只計算非「零」的線

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

所以這裡有六個點，任何與「零」線交叉的統統不算

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

中間零線全部可以跳過，所以算出七個點

Here we have two dots.

這裡有兩點

And two dots here.

這裡也有兩個點

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

因此即便數字中包含零的，一樣可用這個方法來計算

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

同理，你也可以運用它來計算代數方程式

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

假設你要算 x+2y 乘以 x+3y

We will draw one line for the x,

我們先畫一條線代表x

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

再畫兩條線代表2y，特別用藍線代表

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

所有的x 都以黑線表示，y則用藍線表示

The other number we draw the lines in the other direction

另一個刮號內的數，畫在另一個方向

And again we make groupings.

再次分組

So here we have six dots and because

這裡有六個點，而因為

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

兩條線都是藍色的，意謂它是y平方

In the middle we have five dots,

中間部分則有五個點

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

交叉部分則是各一條黑、藍色線

which means it will be an x and a y.

代表一個x 一個 y

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

在左端，有一個點以及兩條黑線，代表x的平方

So that's the answer.

得出答案

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

希望你能享受這個數學小把戲，未來我還會錄製數學和遊戲理論的短片，請訂閱我的頻道

You can follow me on Twitter @preshtalwalkar, and get my ebooks on Amazon.com

可以在 Twitter @preshtalwalkar 追蹤到我，或者上亞馬遜購買電子書