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.)
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.
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?
The figure below shows one of two solutions to the puzzle. As the reader can verify, the position of each number in the readout is the position of those yellow cells within their smaller rectangles that the number counts.
Consider now the digit 6 that occurs in the leftmost cell of the fourth row of the readout. If we excise the yellow cell in that same position within the digit it will be turned from a 6 into a 5. Which will still be an accurate count of the yellow cells occupying that position. So changing the 6 into a 5 does not affect the readout’s self-descriptive property, and thus provides us with a genuine, if trivially distinct, second solution.
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.)
If we divide the square into 16 smaller squares, as shown, K and L fall at nodes in the resulting grid. Now rotating the diagram through 90° around L carries triangle LMD into triangle LNK. So angle DLK must equal the angle of rotation, or 90°.
CUNY philosopher Noël Carroll notes, “It is a remarkable fact about the creatures of horror that very often they do not seem to be of sufficient strength to make a grown man cower. A tottering zombie or a severed hand would appear incapable of mustering enough force to overpower a co-ordinated six-year-old. Nevertheless, they are presented as unstoppable, and this seems psychologically acceptable to audiences.” Why is this?
As children Maurice Baring and his brother Hugo invented a gibberish language in which the word for yes was Sheepartee and the word for no was Quiliquinino. This grew so tiresome to the adults around them that they were eventually threatened with a whipping:
The language stopped, but a game grew out of it, which was most complicated, and lasted for years even after we went to school. The game was called ‘Spankaboo.’ It consisted of telling and acting the story of an imaginary continent in which we knew the countries, the towns, the government, and the leading people. These countries were generally at war with one another. Lady Spankaboo was a prominent lady at the Court of Doodahn. She was a charming character, not beautiful nor clever, and sometimes a little bit foolish, but most good-natured and easily taken in. Her husband, Lord Spankaboo, was a country gentleman, and they had no children. She wore red velvet in the evening, and she was bien vue at Court.
There were hundreds of characters in the game. They increased as the story grew. It could be played out of doors, where all the larger trees in the garden were forts belonging to the various countries, or indoors, but it was chiefly played in the garden, or after we went to bed. Then Hugo would say: ‘Let’s play Spankaboo,’ and I would go straight on with the latest events, interrupting the narrative every now and then by saying: ‘Now, you be Lady Spankaboo,’ or whoever the character on the stage might be for the moment, ‘and I’ll be So-and-so.’
“Everything that happened to us and everything we read was brought into the game — history, geography, the ancient Romans, the Greeks, the French; but it was a realistic game, and there were no fairies in it and nothing in the least frightening. As it was a night game, this was just as well.”
Gertrude Stein’s 1935 lecture “Poetry and Grammar” includes a section on punctuation, for which she had a peculiar disdain:
There are some punctuations that are interesting and there are some punctuations that are not. Let us begin with the punctuations that are not. Of these the one but the first and the most the completely most uninteresting is the question mark. The question mark is alright when it is all alone when it is used as a brand on cattle or when it could be used in decoration but connected with writing it is completely entirely completely uninteresting.
In 2000, Kenneth Goldsmith rather archly removed the words from this passage and offered the bare punctuation as a poem titled “Gertrude Stein’s Punctuation from ‘Gertrude Stein on Punctuation'” (the full passage and the poem are both here). Goldsmith did the same thing with the punctuation chapter from Strunk & White’s Elements of Style and with Molly Bloom’s soliloquy at the end of Ulysses — a few hyphens and a period.
Carl Reuterswärd’s 1960 novel Prix Nobel consists entirely of punctuation marks. Reuterswärd felt that ordinary writing robs punctuation of its meaning; the surrounding words convey concepts and the commas, colons, and periods simply help to mark it. Removing the words, though, revealed an “interesting alternative: not to ignore syntax but certainly to forgo ‘the preserved meaning of others.’ The ‘absence’ that occurs is not mute. For want of ‘governing concepts’ punctuation marks lose their neutral value. They begin to speak an unuttered language out of that already expressed. This cannot help producing a ‘colon concept’ in you, a need of exclamation, of pauses, of periods, of parentheses.”
In 2005, Chinese novelist Hu Wenliang offered 140,000 yuan ($16,900 U.S.) to the reader who could decipher his novel «?», which consists entirely of punctuation marks.
The autobiography of the American eccentric “Lord” Timothy Dexter (1748-1806) contains 8,847 words and no punctuation. When readers complained, he added a page of punctuation marks to the second edition, inviting them to “peper and solt it as they plese.”
This solution is by Geneviève Lalonde. Any integer must fall into one of three groups:
It’s a multiple of 3.
It’s one less than a multiple of 3.
It’s one more than a multiple of 3.
In our set of five numbers, either it’s the case that all three of these groups are represented or it’s not.
If all three are represented, then the set includes numbers of the form 3a – 1, 3b, and 3c + 1. The sum of these is 3a – 1 + 3b + 3c + 1 = 3(a + b + c), which is a multiple of 3.
If not all three groups are represented, then one of the groups must contain at least three numbers from our set of five.
If there are three numbers in the first group, they’ll have the form 3a, 3b, and 3c.
If there are three numbers in the second group, they’ll have the form 3a – 1, 3b – 1, and 3c – 1.
If there are three numbers in the third group, they’ll have the form 3a + 1, 3b + 1, and 3c + 1.
In each case, the sum of the three numbers is a multiple of 3.
The bird known as the red phalarope in North America is the grey phalarope in England — it bears red plumage during its breeding season, but the British see only its drab winter dress.
A poem by Lord Kennet, from my notes:
I live in hope some day to see
The crimson-necked phalarope;
(Or do I, rather, live in hope
To see the red-necked phalarope?)
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