Suppose there are an infinite number of Grim Reapers. Each has an appointed time to kill Fred if it finds him alive.

The last Grim Reaper (call it #1) is appointed to do this exactly one minute after noon. The next-to-last (#2) is appointed to do it one half minute after noon. And so on: If it finds him alive, Reaper n will kill Fred exactly 1/2^(n-1) minutes after noon.

Thus there is no first Reaper. For any given Reaper, there are infinitely many others who precede it by moments.

Whatever happens, we know that Fred can’t survive this ordeal — to go on with his life he must still be alive at 12:01, and we know for certain that if he lives that long then Reaper #1 will kill him. But in order to survive to 12:01 he must still be alive at 30 seconds after 12 — and at that time Reaper #2 will kill him. And so on. It appears that no Reaper will ever get the chance to kill Fred, because each is preceded by another who will rob him of the opportunity.

So it’s impossible that Fred survives, but it’s also impossible that any Reaper kills him. Must we say that he dies for certain but of no cause?

(From José Benardete’s Infinity: An Essay in Metaphysics, 1964.)

I think if there’s one thing that you could truly say it the most beautiful sight you can possibly see as a human, it is watching sunrise over the Earth, because imagine, you’re looking at blackness out the window, black Earth, black space, and then as the Sun comes up, the atmosphere acts as a prism, and it splits the light into the component colors. It splits the white light of the Sun into the component colors, so you get this rainbow effect, and it starts with this deep indigo eyelash, just defining the horizon, and then as the Sun rises higher, you get these reds and oranges and blues in this rainbow. … You never got tired of looking at those.

— Astronaut Mike Mullane, quoted in Ariel Waldman, What’s It Like in Space?, 2016

Astronomer Clyde Tombaugh assembled his first telescope from spare parts on his family’s Kansas farm — the crankshaft of a 1910 Buick, a cream-separator base, and mechanical components from a straw spreader. He used this to make sketches of Jupiter and Mars that so impressed the astronomers at Lowell Observatory that they gave him a job there.

Years later, after he had made his name by discovering Pluto, the Smithsonian Institution asked if it could exhibit this early instrument. He told them he was still using it — he was making observations from his backyard near Las Cruces, N.M., until shortly before his death in 1997.

“Its mirror was hand-ground and tested in a storm cellar,” wrote Peter Manly in Unusual Telescopes in 1991. “It’s not the most elegant-looking optical instrument I’ve ever used, but it is one of the better planetary telescopes around. … Because of its role in the history of astronomy, I would classify this as one of the more important telescopes in the world.”

From Chapter 12 of Ken Follett’s novel The Pillars of the Earth:

‘My stepfather, the builder, taught me how to perform certain operations in geometry: how to divide a line exactly in half, how to draw a right angle, and how to draw one square inside another so that the smaller is half the area of the larger.’

‘What is the purpose of such skills?’ Josef interrupted.

‘Those operations are essential in planning buildings,’ Jack replied pleasantly, pretending not to notice Josef’s tone. ‘Take a look at this courtyard. The area of the covered arcades around the edges is exactly the same as the open area in the middle. Most small courtyards are built like that, including the cloisters of monasteries. It’s because these proportions are most pleasing. If the middle is bigger, it looks like a marketplace, and if it’s smaller, it just looks as if there’s a hole in the roof. But to get it exactly right, the builder has to be able to draw the open part in the middle so that it’s precisely half the area of the whole thing.’

How is this done? Inscribe a diamond within a square and then rotate it 45 degrees:

Edward Mann Langley, founder of the Mathematical Gazette, posed this problem in its pages in 1922:

ABC is an isosceles triangle. B = C = 80 degrees. CF at 30 degrees to AC cuts AB in F. BE at 20 degrees to AB cuts AC in E. Prove angle BEF = 30 degrees.

(Langley’s description makes no mention of D; perhaps this is at the intersection of BE and CF.)

A number of solutions appeared. One, offered by J.W. Mercer in 1923, proposes drawing BG at 20 degrees to BC, cutting CA in G. Now angle GBF is 60 degrees, and angles BGC and BCG are both 80 degrees, so BC = BG. Also, angles BCF and BFC are both 50 degrees, so BF = BG and triangle BFG is equilateral. But angles GBE and BEG are both 40 degrees, so BG = GE = GF. And angle FGE is 40 degrees, so GEF is 70 degrees and BEF is 30 degrees.

Suppose we cover a chessboard with 32 dominoes so that each domino covers two squares. What is the likelihood that there will be an even number of dominoes in each of the two orientations (horizontal and vertical)?

In fact this will always be the case. Consider the 32 squares in the odd-numbered horizontal rows. Each horizontal domino on the board covers either two of these squares or none of them. And each vertical domino covers exactly one of these squares. So the horizontal dominoes cover an even number of these squares (call it n), and the number of squares remaining in this group (32 – n) must also be even. This latter number is also equal to the number of vertical dominoes, so both quantities are even.

(By Vyacheslav Proizvolov.)