## 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.

## And the Oscar Goes To… a Mathematician?

When you sit down to watch The Oscars on Sunday, February 22, you’ll be witnessing history when a mathematician takes home one of those 13.5″ tall gold statuettes.

Math professor Robert Bridson from the University of British Columbia will receive an Academy Award for Technical Achievement, in particular, for his “pioneering work on voxel data structures and its subsequent validation in fluid simulation tools,” according to the Academy of Motion Picture Arts and Sciences. Less technically, he figured out the mathematical software used to create breathtaking scenes for *Gravity*, *Avatar*, *The Avengers*, and *The Hobbit*.

Incidentally, Bridson is also the first person from Newfoundland to win an Academy Award. He said he hopes his win will inspire kids who like movies to consider a career in mathematics.

Bridson may be the first mathematician honored by the Academy, but his work is certainly not the first math to appear in a movie. The following gem, uploaded to YouTube by AlonzoMosleyFBI, shows 100 different movie clip in which each number 1‑100 is used in a quote. Awesome.

## Colored Pyramids and the Mind of a 7-Year-Old

This is what kept me up last night. Literally.

Form a row of ten squares, with each square randomly colored red, green, or yellow. Call this Row 1. Then place nine squares in Row 2 slightly offset above Row 1, and color the squares in Row 2 according to the following rules:

- If two side-by-side squares have the same color, the square between them in the row above has the same color.
- If two side-by-side squares have different colors, the square between them in the row above has the third color.

Continue in this manner with eight squares in Row 3, seven squares in Row 4, and so on, with just one square in Row 10.

Here’s an example of a pyramid constructed in this manner:

The 1’s, 2’s and 3’s in the diagram appear because this image was created with Microsoft Excel. Formulas were used to determine the number in each square, and conditional formatting was used to color the squares.

So far, this is just a test of how well you can follow rules, and it isn’t much fun. But here’s where it gets interesting.

**How can you predict the color of the lone square in Row 10 after seeing only the arrangement of squares in Row 1 (and without constructing the rows in between)?**

That’s right…

*This shit just got real.*

I found this problem last night on the Purdue Math Department’s Problem of the Week website. I lay awake in bed longer than I should have, but no solution came to me either while laying there awake or while I was sleeping. When I woke up, I shared it with my sons. I suspected they’d have fun coloring squares; I never suspected what actually happened.

I explained the problem to Alex and Eli, and I showed them an example of how to generate Row 9 from an arbitrary Row 10. They then pulled out their box of crayons and constructed rows 8, 7, 6, …, 1. Alex then looked at his pyramid for about 8 seconds and said, “Oh, I get it. You can find the color of the top square by _________.” (*Spoiler omitted.*)

“Is that your conjecture?” I asked.

“What’s a *conjecture*?” he replied.

“It’s a guess,” I told him. “It’s what you think the rule is.”

“No,” he said. “It’s *not* a guess. That’s the rule.”

He was pretty cocky for a seven-year-old.

So we tested his rule for another randomly-generated Row 10. It worked. So we tested it again, and it worked again. We tested it for six different hand-drawn pyramids… and it worked for every one of them.

That’s when I generated **this Excel file**. We used it to test Alex’s conjecture on 100+ other pyramids. It worked every time.

I still have no idea how he divined the rule so quickly.

For me, though, the cool part came when I was able to extend the puzzle with the following:

**For what values of k can you predict the color of the lone square in Row k when there are k blocks in Row 1?**

Trivially, if I gave you just two blocks in Row 1, you could most certainly predict the color of the square in Row 2.

But if I gave you, say, five blocks in Row 1, could you predict the color of the lone square in Row 5?

The final part of the Puzzle of the Week description from the Purdue website says exactly what you’d expect it to say:

Prove your answer.

I have a proof showing that Alex’s conjecture holds. Incidentally, that proof can be extended to prove the general result for the extension just posed.

Alex, however, has not yet generated a proof of his conjecture.

Then again, he’s only seven.

## All You Need is LOVE

Valentine’s Day is almost here, but maybe you’ve been looking for love in all the wrong places. One possibility is to pop over to Wolfram Alpha and ask:

Or, with a little mathematical creativity, you might be able to find some over at Desmos:

Or perhaps you’ve already found a special someone. If so, you might want to tell her how beautiful she is, using this (paraphrased) mathematical gem from Woody Allen:

Your figure describes a set of parabolas that could cause cardiac arrest in a yak.

(No, it’s not sexist of me to imply that readers would have girlfriends. It’s just that a compliment about a paramour’s curves doesn’t work so well when directed at a male.)

Perhaps your special someone makes your heart skip a beat.

If so, this graph can help you get your beat back:

**https://www.desmos.com/calculator/mhnm66dl2o**

Wherever you look for love on this Feast of St. Valentine, I hope you find it — or at least stumble on a couple of great problems to distract you.