Some “Odd” Theorems

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Draw any triangle and divide each leg into three equal segments. Connect each vertex to one of the trisection points on the opposite leg, as shown, and the triangle formed in the center will have 1/7 the area of the original triangle.

2/5 semicircle theorem

A square inscribed in a semicircle has 2/5 the area of a square inscribed in a circle of the same radius.

1/5 square theorem

Draw a square and connect each vertex to the midpoint of an opposite side, as shown. The square formed in the center will have 1/5 the area of the original square.

A “proof without words”:

1/5 square theorem - proof

Trisect each side of a triangle and join each vertex to the opposite trisection points. Then write a hexagram in the hexagon in the center. The area of the hexagram is 7/100 the area of the original triangle.

| Science & Math

Ice Water

A boat floats in a swimming pool. In the boat is a block of ice. If the block is dumped into the water and melts, does the water level rise or fall?

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| Puzzles

Haunting Attire

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Why do ghosts wear clothes? If a ghost is the spirit of a living creature, how can it carry its inanimate garments into the afterlife?

“How do you account for the ghosts’ clothes — are they ghosts, too?” asked the Saturday Review in 1856. “What an idea, indeed! All the socks that never came home from the wash, all the boots and shoes which we left behind us worn out at watering-places, all the old hats which we gave to crossing-sweepers … What a notion of heaven — an illimitable old clothes-shop, peopled by bores, and not a little infested with knaves!”

In 1906 psychic researcher Andrew Lang argued that, far from confusing the notion of an afterlife, ghosts’ clothing might even help to corroborate its existence. “A pretty instance occurs, I think, in a biography of Warren Hastings. The anecdote, as I remember it, avers that at a meeting of the Council of the East India Company in Calcutta one of the members (I think several shared the experience) saw his own father, wearing a hat of a peculiar shape, hitherto strange to the observers. In due time came a ship from London bearing news of the father’s death, and a large and well-selected assortment of the new hat fashionable in England. It was the hat worn by the paternal appearance! If the circumstances are recorded in the minutes of the proceedings of the Council, which I have not consulted, then the hat of that spook becomes important as evidence.”

Even if we grant that a dead person can convey his most personal belongings into the afterlife, how are we to account for phantom ships, coaches, and railway trains? In his 1879 book The Spirit World, American spiritualist Eugene Crowell decided that, rather than being the spirits of “dead” earthly conveyances, these are constructed in the afterlife by the ghosts of mariners and railwaymen who want to ply their trades again. Spectral ships “glide over the waves without sinking,” Crowell explained, “and earthly winds propel them at rates of speed which our ships cannot attain.” If that’s true, then perhaps some ghostly tailor is simply manufacturing clothes for the naked spirits of the newly dead. Decent of him.

| Oddities

The Broken Deck

When eight certain cards are removed from a standard poker deck, it becomes impossible to deal a straight flush. What are the cards? (Assume the deck contains no jokers.)

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| Puzzles

Presidential Wordplay

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In 1936, as Franklin Roosevelt campaigned for re-election against Republican Alf Landon, a group from Wall Street’s financial district sponsored a competition to find the best anagram for FRANKLIN DELANO ROOSEVELT. The winner was VOTE FOR LANDON ERE ALL SINK.

During the 1980 campaign, Jimmy Carter was accused of reversing his position on several issues to maintain his popularity. Edward Scher of New York University coined the palindrome TO LAST, CARTER RETRACTS A LOT.

Carter lost that election to Ronald Reagan, who inspired Howard Bergerson to compose a “press conference” in 1982 using only the letters in RONALD WILSON REAGAN. Here the Gipper describes his foreign policy:

We are sworn nonaggressors; we need law and order, we disallow war as lawless and senseless, and in a larger sense we also regard war as, now and again, needed. A needed war is no dead end or swan song, nor need we ride in war as no-good sinners on genderless geldings! We need androgens and derring-do! We need Old Glories, and seasoned soldiers garrisoned worldwide, generals in golden regalia, and raised dander! We need all-seeing, world-girdling radar, seagoing sonar and liaison ensigns, newer DEW lines and earlier NORAD warnings, larger arsenals and deadlier arrows in silos, R-and-D on lasers, and goodlier anger! We need no ring-a-ding dissensions and wild-goose rallies, nor do we need addled ding-a-ling diagnoses on wielding dread winged swords and daggers — or on wielding God’s own grenades! Ordained grenadiers alone assess, and ordained godlings alone will wield Gold’s sidereal grenades riding on Odin’s arrows. Godless Leningrad warlords and roodless, religionless Red warriors sold on Red-engendered Warsaw agreeings are as sidling sidewinders in loose sand! In nine innings (I disdain gridiron analogies) we will win — no one is dawdling! We are leaning on oars! We and God will engage all Red raiders, and, God willing, we will win odds-on! No one dragoons or goads God!

Reagan’s name can also be rearranged to spell INSANE ANGLO WARLORD.

| History · Language

Black and White

erlinger chess problem

By Konrad Erlinger. White to play and mate in two moves.

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| Puzzles

“An Hellespont of Cream”

If there were, O! an Hellespont of cream
Between us, milk-white Mistress, I would swim
To you, to show to both my love’s extreme,
Leander-like, — yea, dive from brim to brim.
But met I with a butter’d pippin-pie
Floating upon’t, that would I make my boat,
To waft me to you without jeopardy:
Though sea-sick I might be while it did float.
Yet if a storm should rise, by night or day,
Of sugar snows or hail of care-aways,
Then if I found a pancake in my way,
It like a plank should bear me to your quays,
Which having found, if they tobacco kept,
The smoke should dry me well before I slept.

— John Davies of Hereford, 1598

| Poems

Going Down

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One last, almost unbelievable, anecdote from the San Francisco earthquake:

When the quake struck at 5:12 a.m., William Stehr was in his room on the top floor of the Nevada House at 132 Sixth Street. At first he considered jumping from his window onto a roof below, but “as I was waiting to make up my mind the house I was looking at collapsed with a deafening roar.” He was dressing himself hurriedly when he heard another crash and saw that the Brunswick House had also collapsed “and was tumbling into a heap of ruins in a smother of dust.” He leapt to his door but found that the quake had jammed it shut.

As I was tugging at it I felt the floor tilting and sinking under me, and I knew that the house was going down like the others. So I hung on instinctively to the door handle while the whole floor dropped. As it sank I felt three distinct bumps as the lower floors collapsed in turn under the weight of the roof and the top story. With each bump came a frightful crash and cracking of timbers and glass and the cries of other people in the house who were being destroyed. …

The cries of these people who were being killed, especially the women, were dreadful to hear; even to me, in my own peril, thinking every instant that I would be crushed, they were the most dreadful part of the experience.

Then came another bump, very sudden and very severe. The place fell in on top of me, the breath seemed to be knocked out of my body and I went unconscious.

He had survived. Of the 50 people in the Nevada, only seven had escaped. “On all sides, where the Nevada, the Ohio, the Brunswick and other lodging houses had been, there was nothing but a big pile of debris.”

(From Malcolm E. Barker’s Three Fearful Days, 1998.)

| History