## Posts tagged ‘sigma’

### Σ Π :: The Sum and Product Game

This joke, or a close facsimile, has been taking a tour of email servers recently, and it’s now showing up on t-shirts, too:

…and it was delicious!

Appropriate for Pi Day, I suppose, as is the game my sons have been playing…

Eli said to Alex, “18 and 126.”

Alex thought for a second, then replied, “2, 7, and 9.”

“Yes!” Eli exclaimed.

I was confused. “What are you guys doing?” I asked.

“We invented a game,” Eli said. “We give each other **the sum and product of three numbers**, and the other person has to figure out what the numbers are.”

After further inquisition, I learned that it wasn’t just any three numbers but **positive integers** only, that **none can be larger than 15**, and that they must be **distinct**.

Hearing about this game made me immediately think about the famous Ages of Three Children problem:

A woman asks her neighbor the ages of his three children.

“Well,” he says, “the product of their ages is 72.”

“That’s not enough information,” the woman replies.

“The sum of their ages is your house number,” he explains further.

“I still don’t know,” she says.

“I’m sorry,” says the man. “I can’t stay and talk any longer. My eldest child is sick in bed.” He turns to leave.

“Now I know how old they are,” she says.

What are the ages of his children?

You should be able to solve that one on your own. But if you’re not so inclined, you can resort to Wikipedia.

But back to Alex and Eli’s game. It immediately occurred to me that there would likely be some ordered pairs of (sum, product) that wouldn’t correspond to a unique set of numbers. Upon inspection, I found eight of them:

(19, 144)

(20, 90)

(21, 168)

(21, 240)

(23, 360)

(25, 360)

(28, 630)

(30, 840)

My two favorite ordered pairs were:

(24, 240)

(26, 286)

I particularly like the latter one. If you think about it the right way (divisibility rules, anyone?), you’ll solve it in milliseconds.

And the Excel spreadsheet that I created to analyze this game led me to the following problem:

Three distinct positive integers, each less than or equal to 15, are selected at random. What is the most likely product?

Creating that problem was rather satisfying. It was only through looking at the spreadsheet that I would’ve even thought to ask the question. But once I did, I realized that solving it isn’t that tough — there are some likely culprits to be considered, many of which can be eliminated quickly. (The solution is left as an exercise for the reader.)

So, yeah. These are the things that happen in our geeky household. Sure, we bake cookies, play board games, and watch cartoons, but we also listen to the NPR Sunday Puzzle and create math games. You got a problem with that?

### A Nice (and Funny) Note

Professor Francisco Craveira de Carvalho from the math department at Universidade de Coimbra recently sent me two jokes.

Teacher: “… and this concludes the proof. PLOP!”

Student: “What was that?”

Student: “It’s his vocal version of the Halmos symbol.”

The *Halmos symbol*, which is also known as the *mathematician’s tombstone*, is a little black square (■) used to indicate the end of a proof. Named after mathematician Paul Halmos, it was created by Donald Knuth to be used in TeX to replace Q.E.D. when typesetting mathematics.

His second joke was graphical.

But most importantly, Professor Craveira sent me a message —

I bought your book in NY. Great fun! Congratulations!

— about which I was very excited! Despite its brevity, the message contained much information. Specifically,

- It proved that someone had actually read the introduction to
*Math Jokes 4 Mathy Folks*. (PLOP!) - Even if others had read the introduction, this was the first time someone used the email address in the introduction to send me a message.
- Professor Craveira told me that he bought my book at a Barnes & Noble on Fifth Avenue. To my knowledge, this is the first brick-and-mortar purchase of
*MJ4MF*.

Professor Craveira’s brief message of just 45 characters (not including spaces) made my day. Thanks, Professor Craveira!