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### The Math of Lawn Mowing

Google and Yahoo have apparently solved their lawn mowing needs by hiring a herder with 200 goats to spend a week, eating the grass and fertilizing at the same time. If you are fortunate enough to own a lawn, but cannot find any goats in the neighborhood, and local rabbits cannot handle this amount of work, you may need to take care of the lawn yourself. Probably, by means of mowing rather than eating the grass. Now, a natural question comes to mind: where to start and in what pattern should you mow to be most efficient?

Consider these three potential mowing patterns.

Which one should you pick and why? Obviously, you want to walk less. More importantly, you want to consume less gas and make fewer sharp turns, as it is rather inconvenient with a bulky mower.

Assume your lawn is 50' by 100', and the blade of your mower is 2' wide.

In the mowing pattern of vertical parallel lines (left), you will walk 25 stripes of 100' each, 2500' total. You will turn half a circle (180 degrees) 24 times.

In the second scenario, horizontal lines (center), you will walk 50 stripes of 50' each, 2500' total. You will turn half a circle 49 times, twice more than in the first case! (You should try turning 180 degrees with a heavy mower to understand how un-fun this is going to be.)

In the third case (right), you will be moving in concentric motions, and you will walk:
100' + 48' + 98' + 46' + 96' + ... + 56' + 4' + 54' + 2' + 52 '=
(100' + 98' + ... + 56' + 54' + 52') + (48' + 46' + 4' + 2') =
152' x 12.5 + 50' x 12' =
1900' + 600' =
2500'.
Same distance! But you will make 48 90-degree turns, which are much friendlier than 180-degree turns.

Overall, the concentric pattern is your best choice. It does not save you any distance, but it allows you to avoid pirouetting with your mower. Only sharp parade turns. Better yet, forget about lawn corners, smooth your mowing angles and turn concentric rectangles into elegant ovals. No sharp turns at all! Just like Google's goats.

Read more stories about math of cooking, dating, parenting, home management and travel from The Math Mom: www.TheMathMom.com
Planning any air travel? See why airplane flight route looks suspiciously curved while the shortest distance between two points should be a straight line.

#### 57 comments:

1. Maria

Love your passion and articles but this time you may have erred. It think it will take less time and energy for the first method because your walking gait will only be interrupted and slowed 24 times versus 48 and thus you will use less gas because the first will be faster. Add to this that sharp 90 degree turns often require repositioning, backing up and aligning to get the corners right. Forget about trying to do ovals it’s extremely difficult with a lawn mower of any type other than one of those monster robot like things the pros use.

As you stated the amount of feet the same in all three, but the total amount of degrees turned is the same for methods 1 and 3. Method 2 is way out of the running.
#1 24 x 180 degree turns = 4320
#2 49 x 180 degree turns = 8820
#3 48 x 90 degree turns = 4320

That said, as a stay at home father, cook, husband, world traveler, former member of Mensa and Mu Alpha Theta, one of the Architects of Lotus 123, Wang and Leading Edge Word Processing , a gardener who mows his own lawn and partially put himself through college mowing lawns, I believe #1 is the best of your three patterns. Actually, I like a fourth method that is even better in situations where the lawn edges do not allow for easy turning. In that case you would mow a band around the circumference once or twice (depending upon space needed for easy turns), followed by the remainder using pattern #1. In this pattern you can literally run with the mower the length of the longer sides because when you do the turns your feet will still be on lawn and not tangled up in some shrubs.

Obviously you don’t mow lawns much and “friendlier” isn’t a math term but I love what you do!!

Dr Neutrino

2. Dear Dr Neutrino,
Thank you so much for the note.
Your background looks fascinating and you obviously have a lot of mowing experience.
You are right that I don't do much mowing and relied mostly on my husband's and friends experiences as well as my own experience navigating with stroller filled with kids and groceries that may be of relevance :)

I agree with your math, and therefore with your permission posted your email as a comment above.

Your point of being interrupted and slowed 24 times versus 48 is a very good one.
Regarding the total degrees. It all depends on what "weights" we will assign to the turns. And this becomes subjective. Is 90 degree turn half as difficult as 180 degree turn? My husband and a few men I queried think that 180 degree is a hell of a turn with a mower, while 90 degree requires only a small fraction of this energy. However it obviously depends on what type of mower it is and how strong and experienced you are. You definitely have a lot of experience and I therefore I am thrilled to post your opinion. This all is not about who is right and who is wrong but rather bringing math thinking into the real lives. Playfully.

3. I did forget to tell you that the 180 degree pivot is done by pushing down on the mower handles pivoting on the rear wheels and thus lifting the front wheels up in the air to reduce friction. I find it easier and faster than a 90 degree turn because you actually get to use your forward momentum.

Dr Neutrino

4. I have been amused by the thought that people are putting into lawn mowing patterns! I use the 3rd pattern you have described and sometimes a mitxture of patterns 1-3 I never considered it before but having read your post I think I should maybe give it some more thought! I mainly use the 3rd pattern as I think it looks best once finished! Thanks for the interesting and informative post!

5. Haha let's do the math when do the gardening
I love Mowing my Lawn <3

6. An interesting observation on lawn mowing. There is some embedded geometry here as well.

7. I did neglect to tell an individual that the 180 degree pivot is performed by pressing down regarding the mower handles pivoting with the rear end wheels and so lifting the front side tires up within the air to reduce friction. I think it is much easier and also at a faster rate than just a ninety level shift since you actually reach make use of your forward momentum.

8. I agree along with your math, and therefore along with your permission posted the e-mail as a comment above.

Verti Draining

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