The Math of Trail Running

Mathematicians like to remind us that their field of study can be used to describe much of what we see around us—the flow of water, the speed of a falling object, the calories we burn during a trail race as we increasingly hate ourselves.

What they’re usually talking about is calculus. Which means that most of us stand virtually no chance of doing the math.

In our trail-running world, however, simple algebra applies.

If you’ve ever wondered how many days of pastries you’re entitled to inhale after a hard run, or the likelihood that you’ll find yourself saying, “Oh, hell, why not?” to repeating a brutal trail or race, you’re in luck.

Hansel Post-Race Pastry Algorithm

Days of unbridled eating = (# of miles in a race/10) + (amount of vertical in feet/3,000). 

With this equation, you can easily calculate the number of days after a trail run that you’re justified in shoveling down pints of super-premium ice cream, pastries and desserts. It’s named for the U.S. trail runner Sarah Hansel, who last fall raced Italy’s epic 330-kilometer Italian Tor des Géants, with over 75,000 feet of climbing. Using this math, Sarah was justified in eating anything she wanted for—get this—50 days.

Strava-Free Pacing Calculator

Non-Strava Trail-Running Time in minutes = 1.1 x (Strava time) + (number of beautiful views) + (0 to 1 minute per mile self-consciousness factor).

If you have an app recording your every move while you trail run, you may feel peer pressure to rack up some strong numbers. Those of us who can be awkwardly self-conscious are even more susceptible to this Strava influence.

When the app’s off, though, so’s the heat. Our handy calculator computes your time when no one’s watching. It factors in old-school qualities like relaxing and actually enjoying the run, stopping to take in the views and dismissing your friends’ Strava comments, such as, “Was this with tire chains in your shorts, or were you just lazy?”

Meltzer’s Axiom of Head Bashing

Likelihood ratio of your uttering, “Sure, I’d try that again!” = [(hours since run ended)/ (Perceived effort, 1-100)]  x (Your IQ).

Named for the diehard Appalachian Trail record-setter Karl Meltzer, this equation calculates the odds that you will try the same soul-sucking challenge … again. Trigger warning: If you’ve ever finished an inhumanly difficult run and found yourself with your ass in the mud, staring numbly up at a friend and muttering, “Wow. That really sucked! When can we do it again?” well, you might find this math hurtful. IQ, you’ll note, is a key variable.

McDermott’s Law of Running Economics

(Miles: Entry fee in dollars) > 1:1

Actually a ratio and not an equation, McDermott’s Law assists you in seeking out the best trail-racing value for your dineros. The higher the ratio, the smugger you’ll feel as you grind out the miles.

The ratio is increasingly useful, given that some entry fees have become notably pricey. My notoriously cheap (he calls it “splendidly frugal”) friend Dave McDermott looks for races with a miles-to-fee ratio of 1:1 or greater. This stricture typically limits him to ultras—arguably karmic payback for his stinginess. Many ultrarunners, in fact, are cheap.  (Does this describe you? Consider the notoriously hard 100-mile Barkley Marathon, in Wartburg, Tennessee. The $1.60 entry fee brings the race at the bargain-basement price of 1.6 cents per mile, with a McDermott ratio of 62.5:1. Bon voyage.)

Trail-Shoe Proliferation Principle

Number of pairs of trail-running shoes = 1 + (number of years running/10) x (trail-running-magazine subscriptions)  x (number of clearly definable seasons).

This one’s simple. The longer you trail run, the more you’re exposed to cool new shoes you covet. Over time, you’ll break down and buy them. Add in variability in your local weather, and you’re looking at a perfect shoe-buying storm. Pro tip: If you’re remodeling your house, design your mudroom with this formula in mind.

Trail-running math, however, can delve into much more complex areas, such as quantum mechanics, that confounding branch of physics where matter can act like a particle and also a wave.

A trail-running physicist I know happens to be trying to quantify what he calls the Kilian Effect.

Have you ever gone to a trail race to cheer on a friend, only to find that he or she is nowhere near the spot you had chosen? That’s quantum mechanics’ Heisenberg Uncertainty Principle at work. The Uncertainty Principle states that the more you know about one quantity, like speed, the less you know about another, like location. Don’t overthink it.

So, how does Kilian Jornet, the world’s greatest trail runner, figure into this? Simple. The more sports scientists study Kilian’s momentum, the harder it becomes to find him. If sports scientists keep it up, we may never see Kilian again.

Doug Mayer lives in Chamonix, France, where he organizes trips for the tour company Run the Alps. He has field-tested each of the above formulas.

This article originally appeared in the 2018 issue of Dirt.