30 million three-mile walks

Five or six days a week, I walk for an hour. I always start and end at my house. And I never do the same walk twice.

How is this possible?

An hour’s walk for me is about three miles. In my suburban neighborhood, I encounter an intersection about once every 200 yards. So I pass approximately 25 intersections on each walk. There are typically at least two potential choices at each intersection that will take me further away on the way out, and back towards home on the way back.

So there are at least 225 (two to the twenty-fifth power) different walks. Two to the twenty-fifth is more than 30 million.

I have been in front of every house within a mile of here. I have explored every shortcut, walked through every bit of parkland, and seen every yard sign. I have walked every sidewalk in both directions, on either side of the street or down the middle of the street. All these places are all familiar to me.

And yet every walk is different. Every turn is an opportunity to see something different. Things look different depending on the weather or the time of day or the season — or what’s on the curb on trash day. There are always a few people about — most walking their dogs — and they’re friendly behind their masks. They typically swing a few feet to the side if we’re going in opposite directions.

Every walk is familiar. Every walk is an adventure. Walking is always the same. Walking is always different.

Life is like this

Walks are an example of the power of combinatorics. A small number of choices at each turn, applied to a small number of turns, yields a huge number of possible outcomes.

Actually, everything is like this.

Everything you write and everything your read is made up of the same letters and punctuation and a similar set of words to choose from. And yet every piece of writing is different. How can so few elements create so much variety? Creativity and the power of combinatorics. Each choice leads to the next, and all the choices together create a unique piece of writing.

Every piece of music is made from the same notes and sounds. And yet there is infinite variety in how they can be combined and experienced.

Every person’s face is made of the same parts, with a limited amount of variation in those parts, but every face is different. How can you fall in love with one twin and not the other? How does your child’s face give you joy that is not there when you see another child’s face? Combinatorics again.

Life is like this.

You have a hundred decisions to make a day. You choose what to eat, who to call, what to say, where to go. You don’t have an infinite number of choices.

But you do have countless possible paths and outcomes.

Today, when you encounter one of those choices, try something different. Whenever you have the opportunity, try something different.

There are so many different paths to take. Why take the same one over and over?

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  1. Great post Josh. It’s truly remarkable how most people are tied to their rituals. Even to the point where they walk and go, following the same routes. I find when you drive in a car, vs. bicycling or walking, each gives you a different perspective on what you see and how you see it. Just by alternating methods of mobility, you can change your day and mind set. Taking a look at bicycling, there are so many speed bikers that just have their head down and when going through a forest or nature trail will miss all the scenery. Where a hobby bicyclist, taking their time, can appreciate the things around them, birds chattering, frogs splashing in a stream, or the flowers blooming. Net the pace we do anything can dictate how much or how little we observe.

  2. One of the popular thoughts of the day is that one ought to reduce life to habits and reduce the number of decisions one has to make; some take it as far as to buy multiples of one outfit and wear the “same” outfit every day, for example. (Folks like this claim to save the brainpower for “real” decisions.)

    Thoughts on that extreme or habits, in general?

  3. I’m not sure that I have a comment other than to say I really enjoyed this…Stay safe and be well to all here.

  4. Besides the point, but I find the math suspect and now I want to find a formula for the number of unique paths in n^2 blocks of length d. There’s got to be a closed form!

    Also, probably not the point, but how do you guarantee the walks are unique? Do you randomize? Do you have an algorithm?


    1. Here is the algorithm:

      Once you reach an intersection, you can go one of two ways (can’t go back). That means two choices.

      Then you reach the next intersection. Again, you can go one of two ways. Now four possible paths.

      Repeat for the first 12 or so choices. You will have gone 1.5 miles, on 2^12 possible routes.

      Now start back. At each intersection, you have two choices that will lead approximately back (or at least not further away). Same deal as before on number of paths.