I learned this decades ago from my middle school math teacher for 6th and 8th grades. ** I still use the hack to this day.** Does anyone know of any other interesting and useful math hacks?

I learned this decades ago from my middle school math teacher for 6th and 8th grades. ** I still use the hack to this day.** Does anyone know of any other interesting and useful math hacks?

Apparently, math enthusiasts on Twitter were interested this weekend in the math hack. So I thought Iβd share a proof of why it works.

To prove in math that something doesnβt work, you only have to show one case of failure. To prove that something always works, you need to generalize using variables.

How could we generalize 65*65, 75*75, etc?

It looks like ten times a number plus five, then squared:

(10π¦ + 5)^2

So letβs do the math and hopefully we will decipher the magic.

(10π¦+5)(10π¦+5)

= 100π¦π¦ + 50π¦ + 50π¦ + 25)

= 100π¦π¦ + 100π¦ + 25

It took a moment to realize that regrouping will reveal the magic:

= 100(π¦π¦ + π¦) + 25

Now to write this as a rule.

**Take a two digit number, last digit a 5, tens digit equal to****π¦**.I choose 35

**Multiply the π¦ digit times itself, add π¦ to the squared number.**3 times 3 is 9

9 + 3 = 12

**Multiply that number by 100.**1200

**Add 25**1225

********

In practice, with 35*35 I will mentally multiply 3*4 to get 12 and then tack on a 25 to the end of it.

Oh hmm, thatβs a neat trick.

I wonder what the basis is behind it. Something provable?