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## A short theorem about parking

Where should you park relative to the grocery store, if you’re conscientious and intend to return your shopping cart to the “shopping cart docking bay”?  Surprisingly, under a particular set of (ordinary) assumptions, it doesn’t matter.

Assumption 1.  The shopping cart docking bay is closer than the store itself, no matter where you park.

Assumption 3.  You’d like to minimize walking distance in total, including both before shopping and after.

Assumption 4.  You park between the store and the docking bay.

Consider the following diagram:

Assumption 1 means that we know L > x, no matter where the car is.  (Without this assumption, you might be tempted to return the cart to the store itself, which messes things up.)  So, you park the car anywhere you like.  Before you shop, you walk to the store (distance L).  Afterwards, you walk back the car (L) to unload then walk to the docking bay (x) to leave your cart, then walk back to the car (x).  Then:

Total distance walked = L+L+x+x = 2L+2x = 2(L+x)

Here’s the kicker: the distance (L+x) is a constant (i.e. it’s the distance from the store to a docking bay).  So:

No matter where you park, you will always travel twice the distance between the store and the docking bay.

If you park closer to the store, you have less distance to walk before you shop, but more distance afterwards.  If you park right next to the shopping cart docking bay, the reverse is true; you walk more at the beginning but less distance after returning the cart.  Of course, had you parked beyond the docking bay, this analysis fails.

My thanks to my friend Dr. William Hodge, who came up with this theorem in his head one day while walking into a Harris Teeter.

### 3 Responses

1. If you are shopping with your spouse, in the middle of August in the Northern Hemisphere, after you finish your shopping and unload your groceries, you should also make the assumption that you want to minimize the time your spouse will be sitting in the car waiting for you to turn the A/C on, thus you want to strive for the variable ‘X’ to be as short as possible. Hence, under these circumstances, park as close to the cart return as possible, even if on the opposite side!

2. You didn’t account for an additional x: you need to pick up the cart between parking and going to the shop. This breaks the symmetry and you should minimize x.

• I never take a cart from that area. I always get one inside the store.