Moessner’s Theorem

moessner's theorem

Write out the positive integers in a row and underline every fifth number. Now ignore the underlined numbers and record the partial sums of the other numbers in a second row, placing each sum directly beneath the last entry that it contains.

Now, in this second row, underline and ignore every fourth number, and record the partial sums in a third row. Keep this up and the entries in the fifth row will turn out to be the perfect fifth powers 15, 25, 35, 45, 55

If we’d started by ignoring every fourth number in the original row, we’d have ended up with perfect fourth powers. In fact,

For every positive integer k > 1, if every kth number is ignored in row 1, every (k – 1)th number in row 2, and, in general, every (k + 1 – i)th number in row i, then the kth row of partial sums will turn out to be just the perfect kth powers 1k, 2k, 3k

This was discovered in 1951 by Alfred Moessner, a giant of recreational mathematics who published many such curiosa in Scripta Mathematica between 1932 and 1957.

(Ross Honsberger, More Mathematical Morsels, 1991.)


In March 1945 the Japanese painted the giant image of an American B-29 on the Tien Ho airfield in China. They gave it a burning engine and a 300-foot wingspan, so that when viewed from a great altitude it would look like a stricken bomber flying at several thousand feet. Their hope was that this would induce high-flying Allied planes to drop down to investigate, bringing them within range of their anti-aircraft guns. I don’t know whether it worked.

The Atlantic has a collection of similar deceptive exploits from World War II.

Black and White

dawson chess puzzle

A “maximummer-selfmate” by T.R. Dawson, from 1934. White wants to force Black to checkmate him, and Black always makes the geometrically longest move available to him. How can White accomplish his goal in three moves?

Click for Answer

All Roads

Another puzzle from Kendall and Thomas’ Mathematical Puzzles for the Connoisseur (1971):

Take three consecutive positive integers and cube them. Add up the digits in each of the three results, and add again until you’ve reached a single digit for each of the three numbers. For example:

463 = 97336; 9 + 7 + 3 + 3 + 6 = 28; 2 + 8 = 10; 1 + 0 = 1
473 = 103823; 1 + 0 + 3 + 8 + 2 + 3 = 17; 1 + 7 = 8
483 = 110592; 1 + 1 + 0 + 5 + 9 + 2 = 18; 1 + 8 = 9

Putting the three digits in ascending order will always give the result 189. Why?

Click for Answer

In a Word

adj. high-flying

adj. capable of being seen, visible

n. a source of fear

John Lithgow’s eyes pop out of his head momentarily at the climax of “Nightmare at 20,000 Feet,” the final segment in Twilight Zone: The Movie (1983). In the segment, a remake of the famous television episode from 1963, Lithgow plays a nervous air passenger who discovers a gremlin on the wing of his plane. At the moment when he lifts the shade, the edit shows the monster for 17 frames, then Lithgow’s face for 10 frames, then the monster for 42 frames, and then a 5-frame shot of Lithgow’s head incorporating the eye-popping effect.

Of these 5 frames, the first three show a wild-eyed Lithgow, the fourth shows bulging eyes, and the fifth is shown below. “This 5-frame sequence is on the screen for 1/5 second, but the most distorted image is only visible for 1/24 second,” writes William Poundstone in Bigger Secrets. “Blink at the wrong time, and you miss it. But if you watch the shot carefully at normal speed, the sequence is detectable. Lithgow’s eyes seem to inflate with an accelerated, cartoon-like quality.”

Here’s the frame:

twilight zone movie

The Test
Image: Wikimedia Commons

“If you think that you can think about a thing, inextricably attached to something else, without thinking of the thing it is attached to, then you have a legal mind.” — Thomas Reed Powell

A lawyer advertised for a clerk. The next morning his office was crowded with applicants — all bright, many suitable. He bade them wait until all should arrive, and then ranged them in a row and said he would tell them a story, note their comments, and judge from that whom he would choose.

‘A certain farmer,’ began the lawyer, ‘was troubled with a red squirrel that got in through a hole in his barn and stole his seed corn. He resolved to kill the squirrel at the first opportunity. Seeing him go in at the hole one noon, he took his shot gun and fired away; the first shot set the barn on fire.’

‘Did the barn burn?’ said one of the boys.

The lawyer without answer continued: ‘And seeing the barn on fire, the farmer seized a pail of water and ran to put it out.’

‘Did he put it out?’ said another.

‘As he passed inside, the door shut to and the barn was soon in flames. When the hired girl rushed out with more water’ —

‘Did they all burn up?’ said another boy.

The lawyer went on without answer:–

‘Then the old lady came out, and all was noise and confusion, and everybody was trying to put out the fire.’

‘Did any one burn up?’ said another.

The lawyer said: ‘There that will do; you have all shown great interest in the story.’ But observing one little bright-eyed fellow in deep silence, he said: ‘Now, my little man, what have you to say?’

The little fellow blushed, grew uneasy, and stammered out:–

‘I want to know what became of that squirrel; that’s what I want to know.’

‘You’ll do,’ said the lawyer; ‘you are my man; you have not been switched off by a confusion and a barn burning, and the hired girls and water pails. You have kept your eye on the squirrel.’

Ballou’s Monthly Magazine, February 1892

Image: Wikimedia Commons

Poet Brendan Behan began his career as a housepainter. While in Paris, he was asked to paint a sign on the window of a café to attract English-speaking tourists. He painted:

Come in, you Anglo-Saxon swine
And drink of my Algerian wine.
‘Twill turn your eyeballs black and blue
And damn well good enough for you.

“At least I got paid for it,” he said later. “But I ran out of the place before the patron could get my handiwork translated.”

(From his wife Beatrice’s My Life With Brendan, 1973.)

Mind’s Eye #3

mind's eye #3

Here’s this month’s piece from designer Von Glitschka — each month he composes an engraved artwork to accompany a Futility Closet quotation.

He’s provided a high-resolution PDF (2 MB) suitable to collect, print, and hang.

We’re releasing a new image each month, here and on social media — as with the wallpaper, I’ll put the links on the About page as well. Thanks for reading, and thanks again to Von!

Safety First

In 1936, J.E. Torbert patented a taillight for horses:

When a person is riding a horse along a road at night and an automobile approaches the horse from the rear, the signals will be illuminated by reflecting light from the headlights of the automobile and thus permit the driver of the automobile to see that there is a horse ahead of him and eliminate danger of the automobile striking and injuring the horse.

Simple enough. By that time we already had headlights for horses. What’s next?

Too Late

For 500 years it was thought that Geoffrey Chaucer had written The Testament of Love, a medieval dialogue between a prisoner and a lady.

But in the late 1800s, British philologists Walter Skeat and Henry Bradshaw discovered that the initial letters of the poem’s sections form an acrostic, spelling “MARGARET OF VIRTU HAVE MERCI ON THINUSK” [“thine Usk”].

It’s now thought that the poem’s true author was Thomas Usk, a contemporary of Chaucer who was accused of conspiring against the duke of Gloucester. Apparently he had written the Testament in prison in an attempt to seek aid — Margaret may have been Margaret Berkeley, wife of Thomas Berkeley, a literary patron of the time.

If it’s aid that Usk was seeking, he never found it: He was hanged at Tyburn in March 1388.