Some “Odd” Theorems

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.


“People who count their chickens before they are hatched act very wisely because chickens run about so absurdly that it’s impossible to count them accurately.” — Oscar Wilde

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?

Click for Answer

Haunting Attire

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.

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.)

Click for Answer

Presidential Wordplay

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.


Grandpapa fell down a drain;
Couldn’t scramble out again.
Now he’s floating down the sewer
There’s one grandpapa the fewer.

— Harry Graham


book cover

At last, here’s a book! Futility Closet: An Idler’s Miscellany of Compendious Amusements collects my favorite finds in nine years of dedicated curiosity-seeking: lawyers struck by lightning, wills in chili recipes, a lost manuscript by Jules Verne, dreams predicting horse race winners, softball at the North Pole, physicist pussycats, 5-year-olds in the mail, camels in Texas, balloons in the arctic, a lawsuit against Satan, starlings amok, backward shoes, revolving squirrels, Dutch Schultz’s last words, Alaskan mirages, armored baby carriages, pig trials, rivergoing pussycats, a scheme to steal the Mona Lisa, and hundreds more, plus a selection of the curious words, odd inventions, and quotations that are regular features on the site, as well as 24 favorite puzzles and a preface explaining how Futility Closet came to be and how I come up with this stuff. A million thanks to Greg Mortimer for making it look so good.

The paperback is available on Amazon and Amazon Europe now, and at other online retailers imminently. An ebook is also forthcoming, which I’ll announce here as soon as it’s ready. Thanks for your support, and thanks, as always, for reading!

Black and White

erlinger chess problem

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

Click for Answer

“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