Crazy Pólya's Volunteer Day on the Prairie

Let's say a new guy arrives in town from Hungary and joins Friends of the Blufflands which by this time has grown significantly and has expanded throughout the Upper Midwest. He calls himself Pólya. He is a rather odd fellow who is obsessed with numbers but is a hard, steady worker and falls in love with prairies. Here he is:

He eventually joins the board, becomes a site steward for one of the biggest prairies in the area, and schedules frequent work days. One day his enthusiasm bubbles over and he gets a group of 100 volunteers to work on his prize prairie. But this time he has a math trick up his sleeve and asks the group to buy into his scheme to make the day more "interesting". Pólya is rather charming in his own way, and he convinces all 100 volunteers to go along with his plan by offering them $50 each, costing Friends of the Blufflands $5,000. Other board members present that day started to object, but Pólya assured them that it would work out to be "a very good deal".

Then Pólya spells out the rules:

• Each volunteer starts out at the same point on the prairie and carries 4 marbles in his pocket which are identical except for color. One color represents north, one south, one east, and one west. At the starting point, the first volunteer cuts all unwanted woody stems within a square a half-step in each direction. He then reaches in his pocket, mixes up the marbles thoroughly, and pulls one out to determine which direction he should take his next step. At that location, he again cuts all the stems like before and continues doing this until he gets back to the starting point. Then he's done and can go home.

• The next volunteer starts the same journey. If the area he is on has already been cleared, he quickly moves to the next point.

• Each volunteer has to continue on his random walk until it brings him back to the starting point.

• Pólya, with a mischievous grin, guarantees each of the volunteers that they will eventually "get home", as long as they stick to it. All the volunteers agree to play his game and Pólya even gets them to sign a form saying they won't stop until they reach home base.

Now, Pólya is a pretty good math guy. He knows that he will lose many of his workers at the outset without getting much work out of them, but he also knows that some of his volunteers will be out there for a very long time.

Here are some questions that the volunteers should have thought through before they signed up and set off on their quixotic journey:

1. How many volunteers will be done after only two steps? Well, that's easy- after the first step, there would be 4 possibilities for the next step. If randomly selected, one fourth will take the step back to the starting point and be done for the day. That's 25 volunteers off to do what they want for the rest of the day with $50 for doing almost nothing. How about the rest?

2. How many steps will it take, on average, for the next 25 to reach home so that half will be done?

3. How many poor souls will still be out there after 100 steps? 1000? 1,000,000?

4. How many steps, again on average in this scenario, will it take for the 99th volunteer to be done?

5. Will that 100th volunteer ever reach home?

Okay, this might seem rather bizarre, but stay with me for a while- George Pólya was an actual person who was a renowned mathematician who immigrated to the U.S. from Hungary and contributed significantly to several areas of mathematics. Here is the actual Pólya:

And here is the Wikipedia version of his life: https://en.wikipedia.org/wiki/George_Pólya

His theory, called Pólya's Random Walk Theorem, describes a random walk in two dimensions like our work day on the prairie, and proves that everyone starting off on such a random walk will eventually get back to the starting point. Interestingly, if the dimensions are stretched to 3, adding an up and down, then the random traveler only gets back to the starting point about one third of the time- the rest wander away never reaching home. This led another well-known mathematician to famously say,"A drunk man will find his way home, but a drunk bird may get lost forever".

So what??, many of you might ask. What does this have to do with anything??? Well, like a lot of seemingly bizarre areas of thought, this theorem has had many important practical uses in physics, biology, financial calculations, and computer science. Remember the "Golden Fleece Awards" that ran from 1975 to 1988 under the direction of William Proxmire to mock studies supported by the federal government that were felt to be a waste of tax dollars? Many of those studies, however, yielded tremendous value and were followed up by the Golden Goose Awards, established by Congress in 2012, to specifically honor those studies previously ridiculed. Here are two well known examples that did just that:

The Sex Life of Screwworm Flies

• The Ridicule: Heavily mocked by politicians as a massive waste of taxpayer dollars after the USDA awarded a $250,000 grant

• The Value: The work led to the eradication of a cattle parasite that saved the U.S. livestock industry an estimated $20 billion dollars. Senator Proxmire later issued an apology for targeting this project.

The Gila Monster Lizard Saliva Study

• The Ridicule: Targeted as an "absurd animal study" by waste watch dogs.

• The Value: This study isolated the compound that eventually led to GLP-1 agonists like Ozempic which are widely used today for diabetes and weight loss.

Pòlya’s Random Walk Theorem might just have followed the same path.

Now, back to the prairie. Using Pólya's formulas, the first 25 get back to the starting point in 2 steps. The next 25 arrive in 28 steps. After 100 steps, 58 are home. Then after 1000 steps, 68 are home, 10,000 steps 74, 100,000 steps 81...but 19 are still out there working. How many steps for 99 to make it home? 10 to the power of 135! To put that in perspective, the Universe is 4.36 x 10 to the 17 seconds old. And that last volunteer? That poor fellow will have to keep working on and on approaching infinity, but "in principle" he will eventually make his way home!

Take a breath...if you've made it this far, thanks for persevering. I hope it stretched your mind a little bit for a final thought- for all endeavors, including those on the prairie, sometimes it's valuable to do a "random walk" outside the norms of the day, even risking becoming a subject of ridicule. And as you let your mind wander, think about this poem, The Path That Leads to Nowhere, by Corinne Roosevelt Robinson, sister of Theodore Roosevelt: https://www.potw.org/archive/potw260.html. The last stanza reads,

All the ways that lead to Somewhere

Echo with the hurrying feet

Of the struggling and the striving,

But the way I find so sweet

Bids me dream and bids me linger,

Joy and Beauty are its goal-

On the path that leads to Nowhere

I have sometimes found my soul!