## The Game of POP

No one knows how to live a funky life more than Prince:

Life, it ain’t real funky

Unless it’s got thatpop

Need a little extra pop in your life? Here’s a game you can play.

Create a game board consisting of *n* adjacent squares. Here’s a board for *n* = 10:

Still with me? Good.

The rules of POP are rather straightforward.

- Players alternate turns, placing either an O or a P in any unoccupied square.
- The winner is the first player to spell the word POP in three consecutive squares.

I first learned this game using O’s and S’s and trying to spell SOS, but for young kids, O’s and P’s are much better… the accidental occurrences of POO and POOP add a certain *je ne sais quoi*. (But not as much as foreign phrases add to a sentence about feces.)

Alex and Eli played this game tonight on the board shown above. After six turns, the game was decided. (As you can see, an accidental POO occurred in squares 6‑8. I mean an accidental occurrence of the *word* POO, not an actual occurrence of POO itself. If the latter had happened, the game would have ended immediately, and I wouldn’t be writing about it now.) It was Alex’s turn, and he realized that he lost: playing either an O or a P in squares 3‑4 would give Eli the win, and playing either an O or P in squares 9‑10 would just delay the inevitable.

“So, what’re you gonna do?” I asked.

Alex added an O to the third square, shrugged, and handed the pencil to Eli.

*A coward dies a thousand deaths; the valiant die but once.*

In that game, Alex went first and lost. So an immediate question:

- Will the second player always win when
*n*= 10?

This then leads to follow-up questions:

- Are there other values of
*n*such that the second player has a winning strategy? - Are there any values of
*n*such that the first player has a winning strategy? - Are there values of
*n*for which neither player has a winning strategy?

If you’d like to play a game of POP, then head over to **The Game of POP spreadsheet on Google Drive**, email the link to your friend, and start adding O’s and P’s. Feel free to change the size of the game board, too! Just please be a sweetie — when you finish, clear all your letters, reset the size of the game board to 10 squares, and be sure all the directions are retained at the top of the page.

Enjoy!

## P (NFL ∪ Math) > 0

John Urschel is an offensive lineman for the Baltimore Ravens and admits, “I love hitting people.” As it turns out, he loves hitting the books, too. He earned a masters degree in mathematics from Penn State, and he recently published a paper with the impressive title A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians in the *Journal for Computational Mathematics*.

Note that Urschel was the *lead author*, even though his three co-authors were an associate math professor from Tufts and two math professors from Penn State.

I have to wonder if the paper was fairly refereed. I mean, honestly, who in the math community is gonna tell a 6’3″, 308‑pound football player that he made an error?

A la Paul Erdös, Urschel doesn’t need much to be happy. In an essay published March 18, he wrote:

I drive a used hatchback Nissan Versa and live on less than $25k a year. It’s not because I’m frugal or trying to save for some big purchase, it’s because the things I love the most in this world (reading math, doing research, playing chess) are very, very inexpensive.

I was thinking about how Urschel has superior talent in two fields, when I saw this comment on an article on Deadspin:

Here’s the thing.

There are 1,596 players in the National Football League at any given time (32 teams with 53 players each). Throw in a few more who serve on practice squads and occasionally get a chance when someone else gets hurt, so maybe that number climbs to 2,000. Still, the chance of making it to the NFL is unbelievably remote. Recruit 757 claims that only 0.008% of all high school athletes get drafted by the NFL.

And if you can believe Wolfram Alpha, there are 2,770 mathematicians in the United States, or approximately 1/47,165 of the U.S. workforce.

Point is, the probability of becoming **either** a professional football player **or** a mathematician is ridiculously small. Becoming both is smaller still. Though John Urschel proved it’s greater than 0. The saving grace is that he seems like a down-to-earth guy who realizes how lucky he is.

To read a math article written by John Urschel, check out 1 in 600 Billion.

## Will the Real Steve Reinhart Please Stand Up?

I met Steve Reinhart when he was a presenter at a 2001 NCTM Academy in Branson, MO. I only met him that one time, yet he had a profound effect on my teaching philosophy. Read his article “Never Say Anything a Kid Can Say!” and you’ll see why.

Last week, I met a second Steve Reinhart who works for an educational publisher. I asked the second one if he knew the first one, and he told me a funny story about how he was in his hotel room at a conference, and the door to his room opened. He looked at the guy standing there, realized what happened and asked, “Steve Reinhart?” And the first Steve Reinhart said, “Yep,” paused for a second, then asked, “Are you Steve Reinhart, too?” And the second one said, “Yep.” With the same name, when the first one showed up but there was already one checked in, the hotel receptionist gave the first a key to a room already occupied by the second.

Now, that’s a funny coincidence.

What does a mathematician do when it starts to rain?

Coincide.

At the end of my senior year of college, I finished my last final on Friday afternoon. I then worked 8 hours, headed home and pulled an all-nighter cleaning our apartment before returning the key on Saturday morning, and then headed to work for another 12-hour shift. After work, I trekked to the on-campus hotel where my best friend and I would stay the night before going our separate ways. After nearly 50 hours awake and a 3-mile walk, I was delirious when I arrived at the hotel. I told the hotel clerk my name, and he handed me a key. I walked wearily to the elevator, exited at the second floor, looked at the key to check the room number (it was the olden days — the room number was etched on the brass key that I was given), and proceeded to Room 222.

When I opened the door, a naked, middle-aged woman lying in the bed quickly pulled the sheets over herself, and a naked, middle-aged man sitting on the toilet with the bathroom door ajar gave me a look I’ll never forget. I said, “Oh, my gosh! I’m so sorry,” then quickly exited and closed the door. I heard some indecipherable yelling come from the room as I made my way down the hall.

I returned to the lobby, explained that I must have been given the wrong key, and told the receptionist what I had seen. “I’m very sorry, Mr. Vennebush,” he told me.

“No apology necessary,” I said. “But I don’t think I’m the one you need to be worried about.”

And then, as if on queue, the elevator door opened across the lobby, and an irate-but-now-clothed guest yelled, “What the f**k kind of place is this?”

That was **not** a funny coincidence.

This is just a funny mathematical coincidence:

And the crazy part? It’s accurate to within 0.00002.

But as far as coincidences go, this is good advice:

It’s far more likely for something to seem suspicious and turn out to be nothing, than for something to seem like nothing and wake up to a smoking crater where your city used to be.

## Shoestring Probability

March 15 is Shoe the World Day. And April 5 is One Day Without Shoes Day.

Shoes and math have a lot in common.

A shoe salesman consults a mathematician on what size shoes to keep in stock. The mathematician tells him, “There is a simple equation for that,” and shows him the Gaussian normal distribution.

The shoe salesman stares at the equation for a while, then asks, “What’s that symbol?”

“That’s the Greek letter π.”

“What is π?”

“The ratio between the circumference and the diameter of a circle.”

The shoe salesperson thinks for a minute. “What the hell does a circle have to do with shoes?”

As it turns out, there are at least 43 different ways to arrange the laces on your shoes. My favorite is the hexagram method:

But there are some fun things to do with your shoelaces other than lacing up your kicks. Here’s one.

Take the shoelaces out of your shoes. Fold the shoelaces in half and hold them in one hand so that the four aglets are exposed but the rest of the shoelaces are hidden in your palm. Like this:

Have a friend select two of the aglets and tie those ends together (I recommend a square knot). Then, have your friend tie the other two ends together. Finally, offer your friend the following wager:

You give me $1 if you formed one large loop.

I’ll give you $1 if you didn’t.

Is it a fair bet?

Too easy? Then try this. Take your shoelaces *and* your friend’s shoelaces, fold them in half, and then expose the eight aglets. Choose two at a time and tie them together. The wager remains the same.

Now is it fair?

Is it possible to create a fair wager with any number of shoelaces? If so, how many?

## Passwords, Age Restrictions, and Computer Silliness

My computer has been a bad boy recently.

First, it told me that my password is going to expire approximately 11 months *before I was born*…

Interestingly, the folks at www.timeanddate.com disagree with the number of days between March 31, 1970, and the date that screen capture was snapped (March 1, 2015). So much for the truism that, “Computers make very fast, very accurate mistakes.” I thought the difference could be explained by excluding the end dates, but that doesn’t seem to be the case, so I’m not sure what ADPassMon is doing. (Then again, I’m not sure why I’m wasting my time checking the calculations of a piece of software whose warning messages suggest the existence of time travel.)

Then, when attempting to register my sons for ski camp, it gave one of the craziest age restrictions I’ve ever seen…

An age of 5.925 corresponds to 5 years, 11 months, 7 days, and 15 hours, which seems quite an arbitrary cut-off for a ski camp. Further, an age of 7.999 years means that kids are eligible for ski camp so long as they are not within 15 hours, 14 minutes, and 24 seconds of their eighth birthday. The framers of the Common Core would be happy with the consideration paid to MP.6: Attend to Precision. Where else have you seen ages expressed to the nearest thousandth? Not even parents of newborns use this many decimal places.

Both of these issues remind me of a childhood friend who wanted to be a writer. He said he wanted to write stuff that would be widely read, cause an emotional reaction, and make people scream and cry. He now writes error messages for Microsoft.

Here’s wishing you an error-free day!

## Deliver Us Not Into Bad Math

What better way to celebrate National Pizza Day than sharing this sign, which hangs in our local Pizza Hut:

Admittedly, I’ve never been very good with proportions, but even I know that

.

Yet, that’s what’s implied by the statements for $3 and $5 in the sign. Further,

- For $1, you can feed 4 children for 1 day. That’s a
**daily rate of 25.0¢**per child. - For $3, you can feed 2 children for 7 days. That’s a
**daily rate of 21.4¢**per child. - For $5, you can feed 1 child for 30 days. That’s a
**daily rate of 16.7¢**per child.

Will the real **price per child per day** please stand up?

And then I took a look at that last statement — that $10 can feed a classroom for a day — and it really blew my mind. Daily rates of 16.7 to 25.0¢ per child imply that classrooms have 40 to 60 students. I don’t know where these hungry students are, but maybe there should be a secondary campaign to reduce class size?

Though let’s be honest. What really seems to be needed here is an entirely new campaign:

## Can I Get Your Digits?

Saw this on a t-shirt recently:

It made me think about this problem involving digits.

Consider the number

1234567891011121314151617181920obtained by writing the numbers from 1 to 20 in order side-by-side.

What’s the greatest number that can be obtained by crossing out 20 digits?

If a fetching lady or handsome gent catches your fancy by solving that problem, you might want to ask her or him…

How can I know so many digits of π and so few digits of your phone number?

And if he or she still hasn’t taken leave of you, then you could really press your luck with the following:

- Ask your new friend to write down a number with four or more digits.

459,163

- Then, have your friend add the digits.

4 + 5 + 9 + 1 + 6 + 3 = 28

- Subtract the sum from the original number.

459,163 – 28 = 459,135

- Have your friend cross out one of the digits, and then read the remaining number aloud to you.

45,935

- Then, miraculously announce the missing digit.

**1**

The secret to the trick? Simple. Just add the digits of the number that your friend reads aloud, and then figure out what number must be added to get the sum to a multiple of 9. Above, the digits of the number 45,935 have a sum of 26, which is 1 less than a multiple of 9, so the removed digit is 1.