Image: Wikimedia Commons

Set into the corner of Seventh Avenue and Christopher Street in Manhattan’s West Village is a triangular plaque reading PROPERTY OF THE HESS ESTATE WHICH HAS NEVER BEEN DEDICATED FOR PUBLIC PURPOSES. The “Hess triangle” is a remnant from a property dispute that unfolded here 100 years ago: The city was claiming eminent domain in order to demolish hundreds of buildings and expand the subway, but surveyors overlooked this 65-centimeter triangle, owned by Philadelphia landlord David Hess. Hess, outraged at the loss of his five-story apartment building, refused to donate the triangle to the public and added the plaque as a sort of existential revenge. In 1938 it was sold to the adjacent cigar store, and today it’s owned by a local realty corporation, the smallest plot of land in New York City.

Related: In 1973, artist Gordon Matta-Clark bought 13 unused pieces of land that were left over when property lines were redrawn in the borough of Queens. He paid between $25 and $75 for each. The sites are often irregular or isolated, located where other properties meet in a block, some measuring as little as 2×3 feet.

“When I bought those properties at the New York City Auction, the description of them that always excited me the most was ‘inaccessible’,” he said. “They were a group of fifteen micro-panels of land in Queens, leftover properties from an architect’s drawing. One or two of the prize ones were a foot strip down somebody’s driveway and a square foot of sidewalk. And the others were kerbstone and gutterspace. What I basically wanted to do was to designate spaces that wouldn’t be seen and certainly not occupied. Buying them was my own take on the strangeness of existing property demarcation lines. Property is so all-pervasive. Everyone’s notion of ownership is determined by the use factor.”

(Jeffrey Kastner and Brian Wallis, Land & Environmental Art, 2005.)

Who’s Who

John Bevis’ 2010 book Aaaaw to Zzzzzd: The Words of Birds collects the nonsense words that birders have invented to try to convey bird calls and songs:

ag ag ag ag arr: fulmar
beesh: scaled quail
bek bek bek: red-throated loon
chack-weet weet-chack: northern wheater
djadjadja: twite
ee woomp: bittern
hup-hup-a-hwooo: red-billed pigeon
kakakowlp-kowlp: yellow-billed cuckoo
kuk-kuk-cow-cow-cow-cowp-cowp: pied-billed grebe
quickquickquickquick: cuckoo
seedle seedle seedle chup chup: hermit warbler
tiutiu-tiutiutiuk-swee: yellowhammer
trrrrk: wrentit
tzew-zuppity-zuppity-zup: rufous hummingbird
weeta weeta weeta che che che: Lucy’s warbler
wheet-tsack-tsack-tsack: stonechat
zeeda-zeeda-zeeda-sissi-peeso: goldcrest
zoo zee zoo zoo zee: black-throated green warbler

Other interpreters have used actual words — the white-eyed vireo says, “Pick up the beer check quick!”

The Watlington White Mark

In 1764 Oxfordshire squire Edward Horne decided that the parish church of St Leonard on Watlington Hill would look more impressive with a spire. So he gave it one: By cutting a narrow mark 270 feet long into the chalk soil beyond the building, he was able to perch a foreshortened triangle atop the church when it was viewed in perspective from his house. Local residents maintain the mark to this day.

Blind Leading

“Always let your conscience be your guide,” Jiminy Cricket tells Pinocchio. And so we are constantly telling one another: We seem to believe that following our convictions, whatever they are, is better than “giving in to temptation,” regardless of the outcome. Similarly, resisting temptation and doing what I feel is morally right is somehow praiseworthy, even if it can be shown that my convictions were mistaken. We are saying:

If you believe it is wrong for you to do X then it is wrong for you to do X.

And if you do have such a conviction:

You believe it is wrong for you to do X.

Then combining these statements produces the disconcerting conclusion

It is wrong for you to do X.

This seems to mean that all my convictions about my moral conduct are correct, regardless of the facts of the matter, the substance of my convictions, and even whether I’ve considered them.

“Needless to say, this is deeply implausible,” writes University of New Mexico philosopher G.F. Schueler. “For one thing, not only do these ‘proofs’ not depend on the content of our moral convictions; they don’t depend in any way on how we arrived at these convictions. They ‘prove’ the convictions not only of the moral philosopher who has spent her life seriously reflecting on morality, but also those of the most superficial ditz, who has never read or thought about anything more profound than comic books or video games, not to mention the racist bigot who is convinced that it is wrong for her to allow blacks to vote and the religious zealot who thinks all those who don’t accept her religion should be driven out of the country. They ‘prove’ absolutely everybody’s convictions equally.” And that means we must doubt and study even our own convictions … which is much harder than relying on a cricket.

(G.F. Schueler, “Is It Possible to Follow One’s Conscience?”, American Philosophical Quarterly, 44:1 [January 2007], 51-60.)

A Matter of Degree

A clever conundrum from the U.K. Government Communications Headquarters Christmas Puzzle Quiz — it doesn’t even appear to be a question:

42°15′N 72°15′W, 53°52′N 44°50′E, 37°49′N 85°29′W, 39°37′N 75°56′W, 40°57′N 40°17′E, 51°54′N 02°04′W?

The answer is 52°12′N 1°41′W. The listed coordinates identify Ware, Issa, Bardstown, North-East, Of, and Cheltenham. Where is a Bard’s town northeast of Cheltenham? Stratford-upon-Avon! Its own coordinates make up the answer.

GCHQ is a great fount of excellent original puzzles — some are given on their website, they’ve published two collections, and they regularly post brainteasers on Twitter.


Preparing to visit the Dardanelles in July 1915, Winston Churchill sealed a message in an envelope marked “To be sent to Mrs. Churchill in the event of my death”:

Do not grieve for me too much. I am a spirit confident of my rights. Death is only an incident, & not the most important wh happens to us in this state of being. On the whole, especially since I met you my darling one I have been happy, & you have taught me how noble a woman’s heart can be. If there is anywhere else I shall be on the look out for you. Meanwhile look forward, feel free, rejoice in Life, cherish the children, guard my memory. God bless you.

The trip was canceled at the last moment.

(From Geoffrey Best, Churchill: A Study in Greatness, 2001.)

Al Fresco
Image: Wikimedia Commons

In 1848 the French commune of Le Plessis-Piquet distinguished itself with a restaurant built in the boughs of a chestnut tree. Owner Joseph Gueusquin named it Le Grand Robinson, after the treehouse in Swiss Family Robinson.

“Word spread and people started to make the eight-mile pilgrimage from Paris,” writes Pete Nelson in Treehouses of the World. “Soon, other entrepreneurs began opening their own treehouse restaurants. At the height of its popularity, there were ten such restaurants and countless other treehouse attractions.”

The trend persisted even into the 1960s, drawing a steady stream of curious diners to Le Plessis-Piquet — in fact, in 1909 the commune officially changed its name to Le Plessis-Robinson, after Gueusquin’s pioneering idea.

Readouts Revisited

From Lee Sallows:

Earlier this year, Futility Closet featured a puzzle based upon the well-known 7-segment display. Less well known is the 15-cell display shown in Figure 1, in which each decimal digit appears as a pattern of highlighted cells within a 3×5 rectangle. Call these the small rectangles. Observe also that the digit 1 is represented as the vertical column of 5 cells in the centre of its small rectangle rather than as either of the two alternative columns immediately to left and right, a detail that is important in view of what follows.

Figure 1.

Figure 2 shows a pair of readouts using 15-cell displays each arranged in the form of a (large) 3×5 rectangle that mirrors the smaller rectangles just mentioned. The two readouts describe each other. The top left cell in the right-hand readout contains the number 18. A check will show that the number of highlighted top left cells appearing in the left-hand readout is indeed 18. Take for instance the top left-hand cell in the left-hand readout. It contains the number 17, which employs two digits, 1 and 7. None of the 5 cells forming the digit 1 is in top left position within its small rectangle. But the leftmost cell in digit 7 is indeed in top left position. Proceeding next to the left-hand readout’s top centre cell we find two cells in top left position: one in digit 2 and one in digit 4. The score of top left cells so far is thus 1+2 = 3. Continuing in normal reading order, a list of the left-hand readout numbers followed by their top left cell scores in brackets is as follows: 17 (1), 24 (2), 17(1), 13(1), 9(1), 15(1), 17(1), 25(2), 17(1), 8(1), 9(1), 14(1), 15(1), 24(2), 17(1). The sum of the scores is 18, as predicted.

In the same way, a number occupying position x in either of the readouts will be found to identify the total number of cells occurring in position x within the digits of the other readout. That is, the two readouts are co-descriptive, they describe each other.

Figure 2.

Recalling now the solution to the earlier mentioned 7-segment display puzzle, some readers may recall that it involved an iterative process that terminated in a loop of length 4. Likewise, the pair of readouts in Figure 2 are the result of a similar process, but now terminating in a loop of length 2. In that case we were counting segments, here we are counting cells. An obvious question thus prompted is: What kind of a readout would result from a loop of length 1? The answer is simple: a description of the readout resulting from a loop of length 1 would be a copy of the same readout. That is, it will be a self-descriptive readout, the description of which is identical to itself. Such a readout does indeed exist. Can the reader find it?

Click for Answer

Double Talk
Image: Wikimedia Commons

Gottfried Leibniz held that no two distinct objects can have exactly the same properties.

But, Max Black asked, “Isn’t it logically possible that the universe should have contained nothing but two exactly similar spheres? We might suppose that each was made of chemically pure iron, had a diameter of one mile, that they had the same temperature, colour, and so on, and that nothing else existed. Then every quality and relational characteristic of the one would also be a property of the other.”

(We might object that the two spheres are discernible because they occupy different positions in space, but this is true only if we have a third object to use as a reference point in establishing the “objective” location of each sphere. If the only things in the universe are the two spheres, then their positions can be established only in relation to each other, and these would always be identical — for example, each sphere is five miles from another sphere.)

“Now if what I am describing is logically possible,” wrote Black, “it is not impossible for two things to have all their properties in common.”

Sort of vaguely related:

On a visit to Princeton, E.H. Moore began a lecture by saying, “Let a be a point and let b be a point.”

Solomon Lefschetz asked, “But why don’t you just say, ‘Let a and b be points’?”

Moore said, “Because a may equal b.”

In Indiscrete Thoughts, Gian-Carlo Rota writes, “Lefschetz got up and left the lecture room.” Rota calls this “an example of mathematical pedantry.”

(Max Black, “The Identity of Indiscernibles,” Mind 61:242 [April 1952], 153-164.)


A brainteaser by Y. Bogaturov, via Quantum, November-December 1991:

In square ABCD, point L divides diagonal AC in the ratio 3:1 and K is the midpoint of side AB. Prove that angle KLD is a right angle.

Click for Answer