Is More Always Worse?

Is it always true that if you have a good thing, then having more of it is better and, conversely, if you got something bad having more of it is worse? Case in point, consider the following situation, which is usually told as a joke:

"What is worse than finding a worm in the apple you are eating?"
"I don't know... Two worms."
"No. Half a worm!"

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.

Strange Numbers, Mixing Decimal Fractions with Ordinary Fractions

Although such number mix-ups are not common, I have seen them several times on gas stations. Finally I took some photos.

A few observations:

• I suspect that, if a student wrote an answer to a math problem in this form, the teacher will not accept it as valid.
  
• My son pointed out that these numbers are like having a second decimal point, such as 3.38.9, which is wrong.
• The values that these numbers represent are obviously clear; adding the decimal point, if noticed at all, causes no confusion: The ordinary fraction stands for 9/10 of a cent, not of a dollar.
• This can be viewed to be an example of mixed bases in the place-value columns, which is very acceptable and common practice but requires special notation. For example, the time 7:49:25.83 is correct and unambiguous. The different place values are separated by colons and a decimal point. The leftmost place value is either 12- or 24-base and may consist of a single digit or two; the next two place values are based 60 and the rightmost one is a decimal fraction of the preceding one (seconds), separated from it by a decimal point and, in this example, contains two decimal place values.
• The prices on the pumps at these gas stations are completely decimal, such as $3.389 and $3.499.

Does the inclusion of the decimal point add any more information? Does it serve any purpose? What did the people who placed the decimal points with the fractions tried to accomplish? Or is it the result of lack of understanding of how to express numerical values? (I did not ask any station attendant about their signs.)I am not questioning the presence of the 9/10 fractions because having them is a well-established practice in retail-gasoline pricing.

Question. Has anyone seen similar mixing of decimal fractions with ordinary fraction anywhere other than on gas-stations' price signs?