Futility Closet

Properly speaking, can the top half of an apple exist without the whole?

What is it the top half of?

Any set of 10 positive integers smaller than 100 will always contains two subsets with the same sum.

In any such group, the number of possible subsets (excluding the empty set) is 2¹⁰ – 1, or 1023. And the largest possible sum of any subset is 90 + 91 + … + 99 = 945. Hence, no matter which numbers are chosen, there will always be more subsets than possible sums, and some subsets (dozens, actually) must yield the same sum.

Every road in this little town is a one-way street, and each street is colored either red or blue. This has a helpful effect: If you start at any house in town and follow the sequence blue-red-red three times in a row, you’ll always arrive at the yellow house.

If you follow blue-blue-red three times, you’ll always arrive at the green one.

In 1970 Roy Adler and Benjamin Weiss asked whether it’s always possible to create such a coloring in a given network; in 2009 Avraham Trahtman proved that, within certain constraints, it is.

A rail one mile long is lying on the ground. If you push its ends closer together by a single foot, so that the distance between them is 5279 feet rather than 5280, how high an arc will the rail make?

Click for answer …

If you’re sharing a pizza with another person, there’s no need to cut it into precisely equal slices.

Make four cuts at equal angles through an arbitrary point and take alternate slices. You’ll both get the same amount of pizza.

Also: If a pizza has thickness a and radius z, then its volume is pi z z a.

• Dorothy Parker named Alexander Woollcott’s apartment “Wit’s End.”
• Can you look at something and imagine it at the same time?
• 36850 = (3⁶ + 8) × 50
• AGNOSTIC is an anagram of COASTING.
• “The errors of a man are what make him really lovable.” — Goethe

In June 1978, a Princeton engineering student called structural engineer William LeMessurier with some worrying calculations. LeMessurier’s new Citicorp Tower, which had opened the previous year, was vulnerable to quartering winds — winds that blew from a 45-degree angle. On investigating, LeMessurier found also that the welded joints he had specified had been replaced with weaker bolted joints during construction. This meant that a strong wind could shear the bolts and topple a 59-story building into midtown Manhattan.

With hurricane season approaching, welders worked from 8 p.m. to 4 a.m. every night, reinforcing the building’s joints, and the Red Cross worked out an evacuation plan for the surrounding neighborhood. Because of a press strike at the time, many of these details came to light only 20 years later.

That year’s Hurricane Ella actually bore down on New York as the workers were finishing the job, but the storm veered out to sea before reaching the city. The welding was completed in October, and it’s now estimated that a storm strong enough to rock the tower will occur only once every 700 years.

How fast does time pass? We have no way to measure this. We can reply, helplessly, that it passes at one second per second, but this is not a rate of change — 1 second divided by 1 second is 1. Not 1 of anything, just 1.

“‘One’ can be an answer, right or wrong, to the questions ‘How many children had Lady Macbeth?’, ‘How many Gods are there?’, and ‘How many minutes do sixty seconds make?’,” writes Notre Dame philosopher Peter van Inwagen. “‘One’ can never be an answer, not even a wrong one, to any other sort of question — including those questions that ask ‘how fast?’ or ‘at what rate?’ Therefore, if time is moving, it is not moving at any rate or speed.”

This tiling pattern is sometimes referred to as Pythagorean because it can be construed to prove the Pythagorean theorem.

The red area is a right triangle. The square of its shorter side is equivalent to a green square, and the square of its longer side is equivalent to a yellow square.

One green and one yellow square can be cut up and reassembled to fit into one of the canted white squares, which is equivalent to the square of the red triangle’s hypotenuse. Hence a² + b² = c².

Each year, when the last flight of the summer field season departs the U.S. research station at the South Pole, the remaining staff gather to watch The Thing.

The next flight won’t arrive for eight months.