Once upon a time, a traveling salesman wondered, “With gas getting so expensive, is there a shorter route to all my customers’ cities and back?'”
Rethink, reframe, translate, transform, map … I’m fascinated with how esoteric math – bizarre conjecture – applies to tangible, practical applications. Especially when the context involves an infinite number of arrangements. And a quest for an elusive counterexample.
‘Lonely Runner Problem’
• The ‘Lonely Runner’ Problem Only Appears Simple by Paulina Rowińska (4-18-2026)
[See also: Mathematician Explains Infinity in 5 Levels of Difficulty]
Take a group of runners circling a track at unique, constant paces. Answering the question of how many will always end up running alone, no matter their speed, has vexed mathematicians for decades.
The “lonely runner” problem might seem simple and inconsequential, but it crops up in many guises throughout math. It’s equivalent to questions in number theory, geometry, graph theory, and more—about when it’s possible to get a clear line of sight in a field of obstacles, or where billiard balls might move on a table, or how to organize a network. “It has so many facets. It touches so many different mathematical fields,” said Matthias Beck of San Francisco State University.
For just two or three runners, the conjecture’s proof is elementary. Mathematicians proved it for four runners in the 1970s, and by 2007, they’d gotten as far as seven. But for the past two decades, no one has been able to advance any further.
At first, the lonely runner problem had nothing to do with running.
Instead, mathematicians were interested in a seemingly unrelated problem: how to use fractions to approximate irrational numbers such as pi, a task that has a vast number of applications. In the 1960s, a graduate student named Jörg M. Wills conjectured that a century-old method for doing so is optimal—that there’s no way to improve it.
As versions of the lonely runner problem proliferated throughout mathematics, interest in the question grew. Mathematicians proved different cases of the conjecture using completely different techniques. Sometimes they relied on tools from number theory; at other times they turned to geometry or graph theory.
Derangement
That’s not my grade!
Suppose that a professor gave a test to 4 students – A, B, C, and D – and wants to let them grade each other’s tests. How many ways could the professor hand the tests back to the students for grading, such that no student receives their own test back?
That’s not my hat! (hat-check problem)
If 3 people put their hat in a box, but the hats are mixed up, how likely is it that AT LEAST one person gets their hat back?
The Mad Hatter, March Hare, and Dormouse randomly retrieve their hats. What’s the probability of at least one of them getting their correct hat?
What would you bet? How do the odds compare to betting on an odd/even or red/black spin on a roulette wheel?
What does the mathematical (transcendental) constant e have to do with the general case of N people?
Go down the rabbit hole using a Google search (for an AI Overview); then dive deeper in AI mode, e.g., Explain the math behind the 3-hat shuffling problem in detail.
- Model the problem (start simple)
- Identify all possible outcomes
- Map the matches
- Calculate the probability
- Expand the context of the problem (more hats)
- Look for a pattern
- Find an equation
- Visualize (table, graph)
- Remap the solution as unity minus the probability of zero matches, in the case of infinite hats