Posted December 21, 2011 By Presh Talwalkar. Read about me , or email me .

This week has already featured two puzzles about optimizing (Christmas trinket puzzle, auction theory puzzle), so I thought I’d keep the trend going today.

Today’s post is actually inspired by a problem I considered when I shave my head. Here is the story and my results.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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How frequently should you shave your head?

I don’t give any recommendations as this is ultimately a personal decision. I am not someone that needs a completely clean-shaven look. I can tolerate not shaving my head for up to 4 weeks.

That said, there is a tradeoff. If I let my hair grow for 4 weeks, then it will take me longer to shave because my hair is thicker.

I knew that if I shaved more frequently I could shave my head faster. But then again, I’d have to shave more frequently which means the total time I spend shaving might be higher.

So I ultimately wondered: what is the optimal frequency for shaving–that is the shaving interval that minimizes the time I spend shaving per 4 weeks?

I figured this would be a fun exercise, and so I collected some data.

Step 1: data on head shaving

To be completely scientific, I should time how long it takes to shave my head. And to be precise, I’d have to use the same shaving routine to avoid bias.

I decided to take a short-cut and go with my memory. I admit this is not perfect data, but this is how long I mentally allot myself to shave, meaning the data points are reasonable.

Here is how long it takes me to shave my head after letting it grow out for a given period of time:

7 days: 15 minutes

14 days: 20 minutes

21 days: 30 minutes

28 days: 50 minutes

You’ll notice that there’s a bit of a jump between 21 days and 28 days. This is because as my hair grows out it gets thicker, and hence it is more time-consuming to shave.

Armed with this data, we can then tackle the question of finding the relationship between shaving frequency and time required.

Step 2a: convert the data

If you just look at the data, you might think shaving 7 days is best at 15 minutes. But this is misleading because if I shave every 7 days, I have to shave 4 times in 28 days. This means the total shaving time would be 60 minutes per 28 days.

So to compare apples to apples, let’s convert everything into “shaving time per 28 days.” (This can be done by multiplying by 28/x where x is 7, 14, or 21 days shaving frequency)

Shaving time per 28 days 7 days: 60 minutes

14 days: 40 minutes

21 days: 40 minutes

28 days: 50 minutes

Now we have comparable data, so let’s get to the math.

Step 2b: find the best fit curve

We want to find a curve that connects these data points to make a meaningful relationship between shaving frequency and time required.

We can use a shortcut and use a numerical solver to figure out the best curve. Using WolframAlpha, we find the quadratic fit curve gives us an R2 of 98 percent.

(And I know, I’ve been using a lot of WolframAlpha this week…)

The curve looks like a pretty decent fit.

And one thing to notice: there seems to be a minimum point that is in between 14 days and 21 days. Let’s now figure out what that point is.

Step 3: solve for the minimum

We can plug in the best fit quadratic curve and get an answer on the minimum:

The mathematical answer is this: if I shave every 19 days, then I can cut my total shaving time required down to 38 minutes.

Step 4: conclusion

Although the statistical model states 19 days is optimal, it’s important not to be overly confident in the results.

The data were not perfect to begin with, and the minimum time is predicted to be 38 minutes–a whopping 2 minutes less than if I shaved every 14 days or 21 days.

Therefore, I think the answer is that shaving any time between 2 weeks and 3 weeks is ideal.