In 2014, Princeton mathematician John Conway proposed a simple conjecture:

Let n be a positive integer. Write the prime factorization in the usual way, e.g. 60 = 2² × 3 × 5, in which the primes are written in increasing order, and exponents of 1 are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number f(n). Now repeat.

So, for example, f(60) = f(2² × 3 × 5) = 2235. Next, because 2235 = 3 × 5 × 149, it maps, under f, to 35149, and since 35149 is prime, we stop there forever.

The conjecture, in which I seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that 20 → 225 → 3252 → 223271 → …, eventually getting to more than one hundred digits without reaching a prime!

Conway offered $1,000 for a counterexample, a number that doesn’t “climb to a prime.” The challenge stood for three years before James Davis, who calls himself “not a mathematician by any stretch,” found the pretty 13532385396179 = 13 × 53² × 3853 × 96179, a composite number that loops onto itself and thus never reaches a prime — a find worth $1,000.

It’s been established that a volume of fluid can be suspended stably above a layer of air by vibrating the whole system vertically. Now researchers in Paris have shown that lightweight objects can “float” on the underside of this suspended slab of liquid — thus inverting an entire seaside scene.

A YouTube commenter writes, “We’ve been sailing like that for years in Australia.”

(Benjamin Apffel et al., “Floating Under a Levitating Liquid,” Nature 585:7823 [2020], 48-52.)

I am nearly driven wild with the Dorcas accounts, and by Mrs. Wakefield’s orders they are to be done now. I do hate sums. There is no greater mistake than to call arithmetic an exact science. There are Permutations and Aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of Number which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then again from the top down, the result is always different. Again if you multiply a number by another number before you have had your tea, and then again after, the product will be different. It is also remarkable that the Post-tea product is more likely to agree with other people’s calculations than the Pre-tea result.

Try the experiment, and if you do not find it as I say, you are a mere sciolist, a poor mechanical thinker, and not gifted as I am, with subtle perceptions.

— Maria Price La Touche, The Letters of a Noble Woman, 1908

When you’re driving and see an upcoming traffic light turn yellow, you face an urgent choice: stop quickly or try to run through the intersection before the light turns red. In 1962, Stanford aeronautics professor Howard Seifert worked out that you can choose either alternative if

where your car’s initial speed is v₀ ft/sec, its maximum acceleration is a⁺ ft/sec², its maximum deceleration is a^– ft/sec², the duration of the yellow light is T seconds, and the intersection is s feet wide and d₀ feet away.

“[I]f the left or right inequality is reversed, you will not be able to run through or to stop, respectively,” he concluded. “It can be shown that there are situations where neither alternative will work and hazard and law violation are inevitable, as some palpitating drivers will testify.”

(Howard S. Seifert, “The Stop-Light Dilemma,” American Journal of Physics 30:3 [1962], 216-218.)