Archive of posts filed under the Application category.

## How Much Would You Pay for a 20% Discount?

A local Office Max is going out of business and is having a very interesting sale.

I’m not sure I’ve ever seen a sale where you earn a discount by purchasing a certain number of items.  Of course, I immediately began exploring the mathematical consequences of the policy.

The first thing that occurred to me was that you can essentially purchase a 20% discount.  Say you need to buy n items.  Simply buying another 20 – n items earns you a 20% discount.  The natural question is thus, “Under what circumstances would buying an additional 20 – n items be worth a 20% discount?”

There are a variety of factors to consider.  For example, if you can just find an additional 20 – n items that you are happy to buy, it’s definitely worth it:  you get the 20% discount, and you get items of value to you.  Also, the answer likely depends on n:  if you are only 1 item short of the discount, it’s easier to justify an unnecessary purchase than if you are, say, 19 items short.

As an extreme case thinker, I considered the following scenario.  Suppose I wanted to buy one item; under what circumstances would I buy 19 items I didn’t want in order to get a 20% discount?

Obviously, the key to this strategy is finding a cheap item to purchase 19 times.  I thought I had found the cheapest possible item here:

Nineteen composition books would cost me $14.06. If the 20% discount saved me more than$14.06, this strategy would be worth it.  This sets the bar for my one item at $70.30. However, I later realized I could do better here: These paper folders cost more per item, but unlike the composition books above, the folders are themselves eligible for the 20% discount! Nineteen folders would cost$16.91, but they’ll be discounted 20% to $13.53. This means if my single item cost more than$67.65, this strategy would save me money.

I could have done a lot better if these Slim Jims were sold here, or these 10-cent envelopes!  But this is the best I could find in the store.

Another interesting question to consider is “For what range of prices would buying nine additional items, to receive a 10% discount, be a better strategy than buying 19 additional items, to get the 20% discount?”

In any event, I appreciate Office Max giving me something interesting to think about as I waited in line.  And as usual, I waited a very long time.  Let’s just say it’s no surprise they are going out of business.

## Hands on a Subway Pole

When I describe the role mathematics plays in my life, I often say that it gives me a set of tools to process and understand the world.  One way that manifests itself is that I see graphs everywhere.

For example, when I look at a pole on the subway, I see the distribution of hands that have been on the pole.

I think about things like this because they are interesting, but also because they are practical.  Where is the pole the dirtiest, and cleanest?  Where are germs most likely to reside?  New Yorkers know instinctively to touch as little as possible, but sometimes you have no choice.  Best to know your probabilities ahead of time.

## 3D Printing a Cube Frame

I’ve been having a lot of fun exploring mathematics through 3D printing.

Recently, I had the idea to 3D-print a “cube frame”, that is, the edges of a cube.  My first mathematical task was to figure out an equation whose graph was such a cube frame.

It took a little work, but I ended up with this.I exported the graph into the appropriate file format and successfully printed my cube frame!

Just as fun as the mathematical challenges of producing the graph are the engineering challenges, and mysteries, of the 3D printing process itself.  For example, notice the scaffolding that the software automatically adds in order to print the top square of the cube.

There are some interesting mathematical and structural consequences of the scaffold-building algorithm, but even more amazing was that one “face” of the cube contained no scaffolding at all.  This meant the 3D printer printed one edge of the top square into thin air!  And it succeeded!

I’m excited and inspired by 3D printing, and I’m looking forward to finding more ways to integrate into our math classrooms.

## An Ancient Tiling Problem

This menacing creature is Glyptotherium texanum.  Well, was Glypotherium texanum.

The armor caught my attention.  It’s made of small plates that create an interesting tiling of a curved surface.

Notice how the tiles slowly change from quadrilaterals to pentagons to hexagons, and back again!  I wonder what factors determine the shape of each tile as they grow.

## Field Goals, the Super Bowl, and Mathematical Models

There has been a tremendous increase in the use of mathematical analysis to make policy, inform organizational decisions, and explain news and events.  I generally think this is a positive thing:  I understand that math gives us a powerful set of tools for understanding and processing the world.  But I also understand the limitations inherent in mathematical modeling.

All mathematical models rely on assumptions that limit their impact.  Mathematicians are typically aware of the assumptions about objects, relationships, and contexts that their models make.  Politicians, journalists, and others who invoke mathematics to make a point seem less aware.  This often leads to bold, unjustified claims based on what “the math” has told them.

An inconsequential but illustrative example of this occurred during Super Bowl 49.

At the start of the second half with the game tied at 14, Seattle drove into scoring position, and faced a 4th-and-1 at New England’s 8 yard line.  Seattle basically had two options:  kick the field goal (a high-percentage play for 3 points), or try to make a 1st-down, and ultimately a touchdown (a lower-percentage play for 7 points).  Seattle opted for the field goal and went ahead 17-14.

Lots of people on Twitter second-guessed the decision, including The Upshot’s David Leohnardt.

Here, David Leonhardt is applying a simple expected value argument.  “Going for it” on 4th-and-1 at the opponent’s 8-yard-line likely produces more total points in the long run than kicking field goals, which suggests that Seattle should have gone for it.  It’s not a bad argument; in fact, I used a similar analysis on the NFL’s new overtime rule.

But in order to apply this argument, it’s important to understand what assumptions this model makes.  For example, this model assumes that the amount of time remaining in the game is irrelevant.  Of course, it’s not:  it’s easy to construct a situation in which “time remaining” is the determining factor in the kicking a field goal (say, the game is tied, and only seconds remain).

This model also assumes that all points are of equal worth.  But they aren’t.  Depending on the game situation, the extra four points a touchdown gives you may be irrelevant, or of significantly less value than the three points the field goal gives you (imagine a team up by six points late in the game).

There are lots of factors this analysis does not consider.  This doesn’t mean that the expected value argument is invalid.  It just means that, like all mathematical models, what it says depends on the assumptions it makes.  And the more we use mathematical models to drive our decisions, the more important it is to be clear about the assumptions that are made and the consequences they entail.