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Albert Einstein is often hailed as the preeminent mind of the 20th century, yet he, like all humans, had missteps and blind spots. Despite his groundbreaking work, he occasionally stumbled over seemingly simple problems—one of which involved a deceptively straightforward math riddle sent by Max Wertheimer, a fellow German refugee and psychologist.
Einstein himself identified the cosmological constant as his greatest error: a term he added to his field equations to reconcile theory with observation. Ironically, this “blunder” later hinted at the universe’s expansion, proving it was no mistake at all. Yet the riddle from Wertheimer highlights a different kind of challenge—one that even a genius struggled to solve.
Wertheimer posed the following question: "An old clattery auto must travel a 2‑mile stretch up and down a hill. Because it is old, it cannot climb the first mile faster than 15 mph. How fast must it go downhill to achieve an average speed of 30 mph for the whole trip?"
At first glance, the problem appears trivial. To average 30 mph over 2 miles, the vehicle must cover the distance in 4 minutes (2 mi ÷ 30 mph = 0.0667 h). However, the ascent alone takes 4 minutes at 15 mph (1 mi ÷ 15 mph = 0.0667 h), leaving no time for the descent. Consequently, no finite speed can satisfy the requirement—Einstein himself noted, "Not until calculating did I notice that there is no time left for the way down!"
In effect, the puzzle is a riddle: the ascent’s fixed speed makes the desired average impossible. Only a minuscule increase above 15 mph on the climb would allow the descent to be extremely fast—hypothetically 600,000 mph—to achieve the 30 mph average, illustrating the absurdity of the constraint.
[Featured image by Los Angeles Times via Wikimedia Commons | Cropped and scaled | CC BY 4.0]
