Cameo

Western Illinois University mathematician Iraj Kalantari published an unusual puzzle in Math Horizons in February 2019. A sphere B of radius 150 is centered at (150, 150, 0). A sphere M of radius 144, centered on the z-axis, lies entirely below the (x, y)-plane so that the volume of its intersection with B is 1/2. “Can we find a sphere S of radius 73 that has its center on the circle (x – 73)2 + (y – 73)2 = 1502 in the plane z = 73 so that the volume of B minus its intersections with M and S equals the volume of M minus its intersection with B plus the volume of S minus its intersection with B?”

The answer is no, because Vol(B – (M ∪ S)) = Vol((M – B) ∪ (S – B)) if and only if Vol(B) = Vol(M) + Vol(S), and that’s the case if and only if r_{B}^{3} = r_{M}^{3} + r_{S}^{3} , where rB, rM, and rS are the radii of the three spheres. “[A]nd because the radii are integers, this equality is impossible by Fermat’s last theorem!”

The placement of the spheres and the fact that the values differ by 1 are red herrings.

(Iraj Kalantari, “The Three Spheres,” Math Horizons 26:3 [February 2019]: 13, 25.)

02/28/2026 UPDATE: In my original statement of the problem I left out a vital phrase in Kalantari’s presentation: The sphere S should have its center on the circle (x – 73)2 + (y – 73)2 = 1502 in the plane z = 73.

I’d omitted the last phrase, a condition that guarantees that S lies above the xy plane and so does not intersect sphere M, which is required to deduce the equation involving the volumes. Many thanks to reader Francesco Veneziano for pointing this out.

| Science & Math

Devotion

A “prayer to the local deities” offered by Socrates in Plato’s Phaedrus:

Beloved Pan, and all ye other gods who haunt this place, give me beauty in the inward soul; and may the outward and inward man be at one. May I reckon the wise to be the wealthy, and may I have such a quantity of gold as a temperate man and he only can bear and carry.

“Anything more? The prayer, I think, is enough for me.”

| Society

Down Under

https://commons.wikimedia.org/wiki/File:La_rotondit%C3%A9_de_la_Terre,_Image_du_monde_(cropped).jpg

How is it with those who imagine that there are antipodes opposite to our footsteps? Do they say anything to the purpose? Or is there any one so senseless as to believe that there are men whose footsteps are higher than their heads? Or that the things which with us are in a recumbent position, with them hang in an inverted direction? That the crops and trees grow downwards? That the rains, and snow, and hail fall upwards to the earth? And does any one wonder that hanging gardens are mentioned among the seven wonders of the world, when philosophers make hanging fields, and seas, and cities, and mountains?

— Lactantius, Institutiones Divinae, 303

| History · Science & Math

Shapes of Things

https://commons.wikimedia.org/wiki/File:Sierra_Leone-Mappa.gif

In 2016, University of Buenos Aires computer science student Gonzalo Ciruelos worked out that the roundest country in the world is Sierra Leone, with a roundness index of 0.934 on a scale of 0 to 1.

He’d been inspired by David Barry, who’d found that the world’s most rectangular country is Egypt (0.955 on the same scale).

Metropolitan France is known as the Hexagon. I suppose each country has its claim to fame.

(Gonzalo Ciruelos, “What Is the Roundest Country?”, Math Horizons 26:3 [February 2019], 26-27.)

| Science & Math · Trivia

The Art of Living

https://pixabay.com/photos/old-stove-hot-fire-flame-2568425/

When I am going out for an evening I arrange the fire in my stove so that I do not fail to find a good one when I return, though it would have engaged my frequent attention present. So that, when I know I am to be at home, I sometimes make believe that I may go out, to save trouble. And this is the art of living, too, — to leave our life in a condition to go alone, and not to require a constant supervision. We will then sit down serenely to live, as by the side of a stove.

— Thoreau, Journal, Feb. 20, 1841

| Literature

In a Word

https://www.pinterest.com/pin/the-chevening-maze-in-kent-is-the-first-multiplyconnected-puzzle-maze-this-means-it-cannot-be-solved-by-the-put-your-left-hand--48624870947902374/

anfractuous
adj. having many windings and turnings

loof
n. the palm of the hand

penetralia
n. the innermost recesses of a building

swither
n. a state of perplexity

It’s commonly said that you can defeat a hedge maze by placing one hand on a wall and carefully maintaining that contact as you advance. If the hedges are all connected, this method will reliably lead you to the center of the maze (and, indeed, to every other part of it before you return to the entrance).

The Chevening maze, in Kent, was designed deliberately to thwart this technique. Its center is concealed in an “island” of hedges distinct from the outer wall, so following either a left- or a right-hand rule will return you to the entrance without ever passing the goal.

| Puzzles · Science & Math