Nike combinatorics

Nike combinatorics


To illustrate the Fundamental Counting Principle with Nike's interactive shoe customization website.

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The Fundamental Counting Principle is usually introduced with examples such as (1) school lunches, and (2) outfits.

I was recently made aware of a lovely application on the Nike website. Karim Ani of Mathalicious introduced me to it and the context is much more interesting than either of those cited above. If you are a middle school teacher, go to Mathalicious right now and download their lesson on this topic, including teacher support materials.

Designing a shoe

So we start with this blank, boring shoe:

There are several things about the shoe that we can customize:

If we start with the outsole, we can choose to make our shoe appropriate for either (1) indoor or (2) outdoor.

Moving to the midsole, we get these choices:

Got that? Twelve colors for the top line paint, three for the rest of the midsole.

Pressing onwards, we have:

Overlay: 11 colors

Base mesh: either breathable (7 colors) or shield (3 colors)

Swoosh: 14 colors

Underlay: 10 colors

Lining: 12 colors

Laces: 14 colors (choose 2 sets of laces)

Heel ID: 8 character max on each heel (A-Z all caps and 0-9) in any of 13 colors, and finally...

Outsole again: If we choose indoor outsole, we get a choice of 8 colors (if we choose outdoor outsole, there's only one color-black).

So how many different pairs of shoes can we make?

Source: nike.com

My shoe

Here's the shoe I designed. Don't be jealous.

Source: Nike.com

Applying the Fundamental Counting Principle

The original statement of the Fundamental Counting Principle is about making two decisions-the first with A options and the second with B options. Then there are A times B total ways to make the two decisions.

But the Fundamental Counting Principle generalizes to more than two decisions. And it's this more general idea we need in order to figure out how many different Nike shoes we can design. In particular, we have nine decisions to make:

  1. Outsole (A)
  2. Midsole (B)
  3. Overlay (C)
  4. Base mesh (D)
  5. Swoosh (E)
  6. Underlay (F)
  7. Lining (G)
  8. Laces (H)
  9. Heel ID (I)

So once we have identified how many ways there are to make each decision, we will multiply:

Before we proceed, about how big do you think this number will be? Greater or less than 100? Greater or less than 1,000,000? Could each person in the US have a different pair of these shoes? Could each person in the world?

Notice that this calculation is based only on a numerical argument. Some people prefer tree diagrams to demonstrate the relationships involved. If you would like to see tree diagrams in action, consider spending some time with this packet.

How many pairs?

What about sizes?

Uh oh...

Size matters, doesn't it?

What if we include all possible shoe sizes?