I Figured: The Empty Closet

from http://www.joshuacallaghan.com/Big%20Empty%20Closet.htm
from http://www.joshuacallaghan.com/Big%20Empty%20Closet.htm

Let’s say I’m starting over with my closet. I want to wear a different outfit – pants, shirt, coat/sweater – every day for 30 days. What’s the minimum number of articles of clothing I need to buy?

Okay, this is an interesting question, but not really. I posted it to my Facebook page and it took a few minutes to figure it out with the help of my friends Brian, Leah and Adam.

Here’s how we started off:

From Brian:

It’s basically a 3-dial combination lock. And you want 30 potential combinations, so we just have to solve for how many numbers are on each dial, and then multiply them together. 

For example, if each dial had 3 numbers, you’d have 27 combinations (3x3x3). If you had 3 pants, 3 shirts, and 3 coats, you could go  27 days without repeating. If you just add one more coat (4), you could go 36 days.

From Leah:

If you buy 3 of each (pants, sweater/jacket, shirt) then you would have 27 days worth. If you buy 4 of each then you would have 52 outfits I believe.

But what if I didn’t plan on buying equal numbers of each item?

From Leah:

So if you have equal number of 2 articles (3 shirts and 3 pants) then those make 9 unique outfits. Then you can multiply that by the number of the other article. So if you have 1 jacket you have 9 outfits. 2 jackets is 18 outfits and 3 jackets is 27 outfits… Or you could do 4 shirts and 4 pants and then 2 jackets would equal 32 outfits. I am having a hard time coming up with exactly 30.

From Adam:

2 pants, 4 shirts, 4 coats…2 pants, 5 shirts, 3 coats

So, after a few minutes of trying, it seemed like it would take at least 10 articles of clothing to wear a different combination every day for 30 days. But then, it occurred to me that there was a final variable we had not taken into account: weather.

What if it were warm for at least 10 days of the month? Common courtesy and social norms require me to wear both pants and a shirt, but a coat is dictated purely by weather.

So the answer is… 9 items.

Here’s how it works:

If I bought two pairs of pants (P1 and P2), five shirts (S1-5) and two coats (C1 and C2), I could create combinations as follows using the pants as the base:

P1

Shirt Only: P1S1, P1S2, P1S2, P1S3, P1S4, P1S5

Coat 1: P1S1C1, P1S2C1, P1S3C1, P1S4C1, P1S5C1

Coat 2: P1S1C2,  P1S2C2, P1S3C2, P1S4C2, P1S5C2

P2

Shirt Only: P2S1, P2S2, P2S2, P2S3, P2S4, P2S5

Coat 1: P2S1C1, P2S2C1, P2S3C1, P2S4C1, P2S5C1

Coat 2: P2S1C2, P2S2C2,  P2S3C2, P2S4C2, P2S5C2

 

How did we do? Can you come up with less than 9 items? Let me know in the comments or on twitter @cheimbuch. And if you have an equation that would make this simpler, please teach us how to express it.

 

 

 

I Figured is where I try to answer mildly interesting questions with the help of my friends, people I find online and with the occasional phone call, though, that takes more effort than I’d like.

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