To earn some money during college, Raymond Smullyan applied for a job as a salesman. He had to take an examination, and one of the questions asked whether he had any objection to telling a small lie now and then. Smullyan did object, but he was afraid that he wouldn’t get the job if he said so. So he lied and said no.

“Later on, I realized I was in a kind of paradox!” Smullyan wrote later. “Did I object to the lie I told the sales company? I realized that I did not! Then since I didn’t object to that particular lie, it therefore followed that I don’t object to all lies, hence my answer ‘No’ was not a lie, but the truth! So was I lying or not?”

(From his book A Mixed Bag, 2016.)

The vortex tube is a bit of a magic trick: When a stream of compressed gas is injected into the chamber, it accelerates to a high rate of rotation and moves toward the nozzle on the right. Because of the nozzle’s shape, though, only the quickly rotating outer shell of this gas can escape; the rest moves back through the center of the vortex and escapes through the opening on the left.

The result, perplexingly, is that even though the tube has no moving parts, it emits hot air (up to 200°C) on the right and cold air (down to -50° C) on the left.

Could this principle be used to air-condition a home or vehicle? “That’s what everyone thinks when they first hear about it,” engineer Leslie Inglis told Popular Science in 1976. “I always tell them that they wouldn’t buy a toaster for the kitchen if they had to buy the generator to produce the electricity. You’ve got to think of this as a compressed-air appliance.”

On Thursday, Numberphile published this video, which features a startling wall hanging in the Senior Combination Room at Trinity Hall, Cambridge: Junior research fellow James McKee devised a 1350-digit prime number whose image forms a likeness of the college’s coat of arms. (The number of digits is significant, as it’s the year that Bishop William Bateman founded the college.)

It turns out that finding such “prime” images is easier than one might think. In the video description, McKee explains: “Most of the digits of p were fixed so that: (i) the top two thirds made the desired pattern; (ii) the bottom third ensured that p-1 had a nice large (composite) factor F with the factorisation of F known. Numbers of this shape can easily be checked for primality. A small number of digits (you can see which!) were looped over until p was found that was prime.'”

Indeed, on the following day, Cambridge math student Jack Hodkinson published his own prime number, this one presenting an image of Corpus Christi College and including his initials and date of birth:

Hodkinson explains that he knew he wanted a 2688-digit prime, and the prime number theorem tells us that approximately one in every 6200 2688-digit numbers is prime. And he wasn’t considering even numbers, which reduces the search time by half: He expected to find a candidate in 100 minutes, and in fact found eight overnight.

(Thanks, Danesh.)

In 2014 I described the Peaucellier–Lipkin linkage, a mechanism that transforms a rotary motion into a perfect straight-line motion:

That linkage was invented in 1864 by French army engineer Charles-Nicolas Peaucellier. A decade later, Harry Hart invented two more. “Hart’s inversor” is a six-bar linkage — links of the same color are the same length. The fixed point on the left is at the midpoint of the red link, and the “input” and “output” are at the midpoints of the two blue links:

In “Hart’s A-frame,” the short links are half the length of the long ones, and the center link is a quarter of the way down the long links:

Pleasingly, the motion perpendicularly bisects a fixed link across the bottom, which is the same length as the long links.

A thrifty space traveler can explore the solar system by following the Interplanetary Transport Network, a series of pathways determined by gravitation among the various bodies. By plotting the course carefully, a navigator can choose a route among the Lagrange points that exist between large masses, where it’s possible to change trajectory using very little energy.

In the NASA image above, the “tube” represents the highway along which it’s mathematically possible to travel, and the green ribbon is one such route.

The good news is that these paths lead to some interesting destinations, such as Earth’s moon and the Galilean moons of Jupiter. The bad news is that such a trip would take many generations. Virginia Tech’s Shane Ross writes, “Due to the long time needed to achieve the low energy transfers between planets, the Interplanetary Superhighway is impractical for transfers such as from Earth to Mars at present.”

The sum of any two of these numbers is a perfect square:

7442 + 28658 = 190²

7442 + 148583 = 395²

7442 + 177458 = 430²

7442 + 763442 = 878²

28658 + 148583 = 421²

28658 + 177458 = 454²

28658 + 763442 = 890²

148583 + 177458 = 571²

148583 + 763442 = 955²

177458 + 763442 = 970²

Two other such sets:

{-15863902, 17798783, 21126338, 49064546, 82221218, 447422978}

{30823058, 63849842, 150187058, 352514183, 1727301842}

Whether there’s a set of six positive integers with this property is an open question.

(A.R. Thatcher, “Five Integers Which Sum in Pairs to Squares,” Mathematical Gazette 62:419 [March 1978], 25-29.)