10,000 Left Shoes

Who would ever want to have 10,000 left shoes?

Well, here is a story that has stuck in my head for many years. I may be off on the specifics but I believe the just of it is true. And, if not, it could have happened or, in storytelling parlance, it’s plausible.

As I recall, in the mid 1960’s in Israel I read in the paper that a shipment of 10,000 left shoes had arrived in the port of Haifa from Italy. At the time, custom duty rates in Israel were extremely high, especially on such luxury imports. So the importer did not claim the shoes. Besides, what would he do with 10,000 left shoes? Stuck with this load, the custom office did what most custom agencies do with abandoned stuff, it auctioned it off. But no one had any use for 10,000 left shoes so no one made any bid. Well, eventually the original importer made a very low bid and the custom office was happy to get rid of the shoes that took space in its warehouse. So the importer got his 10,000 left shoes practically paying no duty.

A month or two later a shipment of 10,000 right shoes arrived in the port of Haifa and the same importer did not claim them because he did not want to pay the duty…

Surely, you have figured out by now the rest of the story.

Where is the math in this story?

It has to do with sets and how sets can be split, added up and rearranged. A pair of shoes is a set, a very useful set. It can be sold. 10,000 pairs is a set of 10,000 such useful sets. Splitting these pairs into 10,000 left shoes and 10,000 right shoes generates 2 sets containing 10,000 of useless shoes, which no one wants to buy. But then, whoever has both sets can recombine them to form the original 10,000 sets of sellable pairs of shoes.

Visual Statistics - a Powerful Mathematical View of the World

Hans Rosling: Watch the end of poverty

With the right mathematical tools, starting with numerical data and using visual representation, Hans Rosling, a Swedish statistician, present an alternative view of the geopolitical world.

The I Divide and You Decide Method for Splitting Ice-cream, Slicing Cake and Dividing All Sorts of Things Without Losing Your Friends

Editorial note. This posting was inspired by Math Doesn't Suck by Danica McKellar. I had written notes about this subject for this blog a while ago and I have taught it to students. As a matter of fact it is a popular "trick" parents use to squelch feuding kids. But the "Cake Cutting Problem" [as described in Wikipedia] is a very important mathematical subject [see also Math World's "Cake Cutting".] But when I read Math Doesn’t Suck and, in particular, the Chapter 7, "Is Your Sister Trying to Cheat You out of Your Fair Share? Comparing Fractions", I got the idea to tell it as a situation-story for kids. Danica writes for girls, especially of middle-school age. I write for everyone, especially kids, girls and boys, and adults too. So the situation I describe here is ageless and genderless.

One weekend you have your best friend for a sleepover. The two of you make a tent and crawl into your sleeping bags in your bedroom and as usual you stay up late. You go silent and pretend you’re asleep till your parents go to sleep. Then you turn on your flashlights and whisper stories and jokes. You get hungry and decide to raid the fridge. In the freezer you discover the partially-eaten ice-cream. So you decide to split it. Silently you get the ice-cream scoop and two bowls and start to divvy it up. But soon the two of you get into an argument. “Not fair,” your friend whispers in protest. “You got more.” You try better but now your share looks too small to you. But you don’t want to admit it. You return all the ice cream from the bowls to the carton, give the ice-cream scoop to your friend and say, “You do it.”

Your friend tries and now you feel you don’t get your fair share.

The argument threatens to wake your sleeping parents. Worse, the ice cream is melting and both of you are craving it. What to do?

You have an idea. Since when one of you does it all the other person feels cheated, why not split the job? So you tell your friend, “I got it! I know what to do. I’ll divide the ice cream into the two bowls the best I can and then you will select which bowl is yours. This way neither of us has a reason to complain.” Your friend thinks about it for a moment. So you add, “It’s okay if we switch jobs. You can split the ice cream as best as you can and I’ll select my bowl.” Right away your friend says, “Okay.” And after a second adds, “that makes no difference.” Your friend got it. The ice cream is delicious and you and your friend sleep in late.

A few months later a friend has a birthday party. After the party the friend invites you and your best friend for a sleepover. In the middle of the night the three of you get a strong craving for the leftover birthday cake. You sneak to the kitchen and get the cake. Your friend, whose birthday cake it is, tries to slice the cake evenly between the three of you. But each time he cuts it one of you feels cheated. After a while your friend gives up. You recall how you split the ice cream so you say, “I know how to divide anything between two people so no one can complain but I don’t know what to do when there are three of us.”

Your best friend says, “Hey, I know. One of us will slice a 1/3 of the cake. Then each of the other two get a turn to say if the cake is fairly divided into a 1/3 and 2/3 piece. If not, the person gets to adjust the slices and we start over. Once we have an agreement, one of the two who did not make the final cut gets the slice that is the 1/3. Then we have to divide what is left among 2 of us and this we know how to do.

It works and the three of you have a wonderful slumber party.

The following winter a group of friendly families go on a camping trip. One day you want a change from snowboarding and skiing. You decide to have a snowball fight. There is plenty of snow especially where the snowplow left a huge clean mound. But to be fair each fighter should have the same pile. People start to argue. But this time you know what to do right away.

You arrange your twelve friends in a circle, ordering them alphabetically by name, in front of the large snow mound. “Alice, you will take as much snow as you think it is fair for one person from this mound. When you are done, Carol, you get to decide if Alice’s pile is just right, too small or too large. If it is too small add to it and if it is too large take away from it. All snow is taken from or returned to this mound. Once you are done it’s your turn David to do as Carol did. We’ll go around all of us until no one makes any more changes to the pile that Alice started. Once that happens, the person after the last person to adjust the pile size gets this pile. For example, if Jeff is the last person to make any changes to the pile then Michelle gets it. Then we repeat the whole process again without Michelle.”

“It’ll take for ever,” Diane says.

“But it’s fair,” your best friend comes to your defense.

“If we want to play, we’ll do it fast,” you say.

So it’s agreed to give it a try and Alice starts. Soon no one can hold their hands idle any longer. There is simply too much snow around and the balls start flying. Who said everyone should get an equal pile when there is endless snow.

Congratulations. You and your friends discovered important principles of mathematical thinking: induction and recursion. Say you have a collection of things that you can order. In this collection some property is true for the first item. (We call this Rule A.) Also you know that any time this property is true for an item in the group, it is also true for the next item. (We call this Rule B.) This is called induction. Because, due to Rule A the first item has the property and therefore, because of Rule B, the second item has it too. But then by repeating Rule B so does item no. 3 and so on and so on… every item till the last item gets this property.

Recursion is when you repeat the same action again and again on a collection of things that grows smaller with each iteration till you must stop because you run out of those things.

You and your two friends recursively divided the birthday cake. It took only two steps but when the recursion ended all of you had no reason to feel cheated. And the reason it was so was induction. Then, for your snowball fight, you generalized your method and came up with your own procedure, or algorithm in geek-speak.

In Chapter 7 of her Math Doesn't Suck, Danica McKellar shows us how to avoid problems when things can be counted or measured precisely. Now you discovered that mathematical thinking is also helpful when things cannot be counted or measured. For if you could do that, you could have measured how much ice cream or cake each of you got and make the division by measuring. But that would have been too messy. Besides being fair and satisfied means different things to different people. Math came to the rescue with a simple, common sense method — I Divide and You decide!

Awesome.

Friendly Gifts/Favors and Mathematical Reasoning

It is common among friends and relatives of professionals to expect a favorable treatment, that is, a discount or a freebie, when they need the service or product of their professional friend. And it is almost just as common practice for the professional to oblige with such an expectation. For example, if you have a friend who is a plumber, I dare presume that, when your toilet is plugged and you urgently need a plumber, you might call your plumber friend, ask his assistance and expect him to give you a discount or perhaps even a freebie. You may then reciprocate buying him a dinner or a bottle of wine but the value of this thank-you gift is much lower than the value of the service.

Or, say, your friend is an author who just published a new book. You probably expect her to give you a free copy of the book, perhaps even an autographed one with a personal dedication.


My question is this:

In mutual relationship, why it is the pro who has to favor you? Why don’t you favor the pro?

Consider the example of your author friend. Why should she give you a free copy of her book? Why shouldn’t you buy her book and pay double its price?

OK, I know, the bookseller can’t take a payment larger than what they sell it for but you get the idea. Beside, we can figure out a workaround this formal limitation. For if you truly like your friend and want to (a) encourage her writing and/or (b) encourage her publisher to publish more of her books and/or publish more books of this kind, then you can send the extra payment with an explanation to either the publisher or your friend the author. Or at least, buy the hardbound book at a full-price retailer, not a paperback at discounter, and then, when the paperback comes out, buy it too.


Where is the math here?

Think of positive and negative numbers and especially think of the duality between the positives and negatives. In this case, why the positive should be a discount for you and the financial negative to your friend and not the other way around?

This is a clear example how positive and negative numbers are often set by the relevant context. For example, if I owe you money, then, as far as I am concerned, my debt to you has a negative value while, from your perspective the debt has a positive value. Similarly, for pilots going up is a positive experience and ascending is indicated by positive numbers and descending by negative numbers. On the other hand, for a scuba diver going into the ocean depth is a most positive experience, so for her descending is measured by positive numbers, which also indicate the increase in pressure, while ascending is measured by negative numbers.

When I talk to students, teachers and others about negative and positive numbers, I like to say:

There is nothing negative about the negative numbers.