## Posts tagged ‘difference’

### A Father’s Day Gift Worth Waiting For

Alex made a Father’s Day Book for me. Because the book didn’t make it on our trip to France, however, I didn’t receive it until this past weekend. It was worth the wait.

The book was laudatory in praising my handling of routine fatherly duties:

I loved when you took me to Smashburger.

I appreciated when you helped me find a worm.

I love when you read to me at night.

I love when I see you at the sign-out sheet [at after-school care]. It means I can spend time with you.

But my favorite accolade — surprise! — was mathematical:

I liked the multiplication trick you taught me. Take two numbers, find the middle [average], square it. Find the difference [from one number to the average], square it, subtract it. (BOOM! Done!)

Priceless.

The trick that I taught him was how to use the difference of squares to quickly find a product. For instance, if you want to multiply 23 × 17, then…

- The average of 23 an 17 is 20, and 20
^{2}= 400. - The difference between 23 and 20 is 3, and 3
^{2}= 9. - Subtract 400 – 9 = 391.
- So, 23 × 17 = 391.
**BOOM! Done!**

This works because

,

and if you let *a* = 20 and *b* = 3, then you have

.

In particular, I suggested this method if (1) the numbers are relatively small and (2) either both are odd or both are even. I would not recommend this method for finding the product 6,433 × 58:

- The average is 3,245.5, and (3,245.5)
^{2}= 10,533,270.25. - The difference between 6,433 and 3,245.5 is 3,187.5, and (3,187.5)
^{2}= 10,160,156.25. - Subtract 10,533,270.25 – 10,160,156.25 = 373,114.
- So, 6,433 × 58 = 373,114.

Sure, it works, but that problem screams for a calculator. The trick only has utility when the numbers are small and nice enough that finding the square of the average and difference is reasonable.

Then again, it’s not atypical for sons to do unreasonable things…

Son: Would you do my homework?

Dad: Sorry, son, it wouldn’t be right.

Son: That’s okay. Can you give it a try, anyway?

I’m just glad that my sons understand math at an abstract level…

A young boy asks his mother for some help with math. “There are four ducks on a pond. Two more ducks join them on the pond. How many ducks are there?”

The mother is surprised. She asks, “You don’t know what 4 + 2 is?”

“Sure, I do,” says the boy. “It’s 6. But what does that have to do with ducks?”

### What’s the Difference?

Today is 8/20/12, which makes it a *difference day*, because the difference between the date and the month is equal to the year: 20 – 8 = 12. In general, a day in the form *mm*/*dd*/*yy* is a difference day if *dd* – *mm* = *yy*.

Dates of this type are relatively rare. There will be exactly 12 per year through 2019, but from 2031 through 2099, there won’t be any. So here’s a question:

How many difference days will there be during the 21st century?

The answer is below.

Speaking of differences, here are some math jokes about differences.

**What’s the difference between a narcoleptic and a math professor?**

The narcoleptic is a slumber nut.

**What’s the difference between a math Ph.D. and a large pizza?**

A large pizza can feed a family of four.

**What’s the difference between a lemma and a proposition?
**You’ll never receive a lemma at a bar.

**What’s the difference between a mathematician and a chocolate muffin?
**One is a mathematician, and the other is a chocolate muffin.

Though I believe mathematicians are useful, I would much prefer a machine for turning theorems into coffee.

(Admittedly, that last one is in the wrong format. But it seems weird to ask, “What’s the difference between a mathematician and a machine for turning theorems into coffee?” The answer would be, “Nothing.”)

There will be 281 difference days during the 21st century. There are 12 per year for 2000–19, but then the number per year starts to decrease. You might expect there to be 11 difference days in 2020, 10 difference days in 2021, and so on, with the number decreasing by 1 each year. But February, April, June, September and November cause problems because they have fewer than 31 days. So the total number of difference days during the 21st century is:

20(12) + 10 + 10 + 8 + 8 + 7 + 5 + 5 + 3 + 3 + 1 + 1 = 281