Piecework

Two ridiculous problems from Arthur Ford Mackenzie’s 1887 book Chess: Its Poetry and Its Prose:

Left: “White to mate without making a move.”

Right: “White to mate in 1/4 of a move.”

	Select Click for Answer
Left: “White mates without moving simply by removing the Black Pawn from [d5], he having just taken that Pawn en passant with the White Pawn standing on [d6]. The position is impossible unless Black and White had thus played last, for in no other way could the White King be now standing in the check of the Pawn.” Right: “The move of Castling may be divided into four parts. First, lifting up the King: second, placing him on the proper square; third, lifting up the Rook; and fourth, placing the Rook on the proper square. In this position White has made 3/4 of this move: he has taken up the King, and placed him on the Bishop’s square; he has also taken up the Rook, which completes the 3/4 of the move. All that remains, therefore, is to place the Rook on Queen’s square, the remaining fourth of the move, and checkmate is accomplished in a quarter of a move!”

May 6, 2021May 5, 2021 | Puzzles

An Elevated Perspective

tetrahedron example

Consider a triangle ABC and three other triangles (ABD1, BCD2, and ACD3) that share common sides with it, and assume that the sides adjacent to any vertex of ABC are equal, as shown. The altitudes of the three outer triangles, passing through D1, D2, and D3 and orthogonal to the sides of ABC, meet in a point.

This can be made intuitive by imagining the figure in three dimensions. Fold each of the outer triangles “up,” out of the page. Their outer vertices will meet at the apex of a tetrahedron. Now if we imagine looking straight down at that apex and folding the sides down again, each of those vertices will follow the line of an altitude (from our perspective) on the way back to its original position, because each follows an arc that’s orthogonal to the horizontal plane and to one of the sides of ABC. The result is the original figure.

(Alexander Shen, “Three-Dimensional Solutions for Two-Dimensional Problems,” Mathematical Intelligencer 19:3 [June 1997], 44-47.)

May 5, 2021May 4, 2021 | Science & Math

“Appendicitis”

The symptoms of a typical attack
A clearly ordered sequence seldom lack;
The first complaint is epigastric pain
Then vomiting will follow in its train,
After a while the first sharp pain recedes
And in its place right iliac pain succeeds,
With local tenderness which thus supplies
The evidence of where the trouble lies.
Then only — and to this I pray be wise —
Then only will the temperature rise,
And as a rule the fever is but slight,
Hundred and one or some such moderate height.
‘Tis only then you get leucocytosis
Which if you like will clinch the diagnosis,
Though in my own experience I confess
I find this necessary less and less.

From Zachary Cope, The Diagnosis of the Acute Abdomen in Rhyme, 1947.

May 5, 2021May 4, 2021 | Poems · Science & Math

More Loops

Further to my March post “A Lucrative Loop,” reader Snehal Shekatkar of S.P. Pune University notes a similar discovery of iterates leading to strange cycles among natural numbers.

Here is a simple example. Take a natural number and factorize it (12 = 2 * 2 * 3), then add all the prime factors (2 + 2 + 3 = 7). If the answer is prime, add 1 and then factorize again (7 + 1 = 8 = 2 * 2 * 2) and repeat (2 + 2 + 2 = 6). Eventually ALL the natural numbers greater than 4 eventually get trapped in cycle (5 -> 6 -> 5). Instead of adding 1 after hitting a prime, if you add some other natural number A, then depending upon A, numbers may get trapped in a different cycle. For example, for A = 19, they eventually get trapped in cycle (5 -> 24 -> 9 -> 6 -> 5).

For some values of A, several cycles exist. For example, when A = 3, some numbers get trapped in cycle (5 -> 8 -> 6 -> 5) while others get trapped in the cycle (7 -> 10 -> 7).

(Made with Tian An Wong of Michigan University.) (Thanks, Snehal.)

May 4, 2021May 3, 2021 | Science & Math

The Savage Breast

A logic problem from Lewis Carroll: What conclusion follows from these premises?

Nobody who really appreciates Beethoven fails to keep silent while the Moonlight Sonata is being played.
Guinea pigs are hopelessly ignorant of music.
No one who is hopelessly ignorant of music ever keeps silent while the Moonlight Sonata is being played.

	Select Click for Answer
Guinea pigs never really appreciate Beethoven. (From the unpublished manuscript of a second volume of Carroll’s Symbolic Logic, discovered by philosopher William Bartley.)

May 4, 2021May 3, 2021 | Puzzles

“Ode on a Grecian Urn Summarized”

Gods chase
Round vase.
What say?
What play?
Don’t know.
Nice, though.

— Desmond Skirrow

May 3, 2021May 1, 2021 | Poems

Podcast Episode 341: An Overlooked Bacteriologist

https://commons.wikimedia.org/wiki/File:Anti-cholera_inoculation,_Calcutta,_1894_Wellcome_L0037329.jpg — Image: Wikimedia Commons

In the 1890s, Waldemar Haffkine worked valiantly to develop vaccines against both cholera and bubonic plague. Then an unjust accusation derailed his career. In this week’s episode of the Futility Closet podcast we’ll describe Haffkine’s momentous work in India, which has been largely overlooked by history.

We’ll also consider some museum cats and puzzle over an endlessly energetic vehicle.

See full show notes …

May 3, 2021May 2, 2021 | History · Podcast · Science & Math

Expenses

The Massachusetts Archives holds a 1775 bill from Paul Revere for “self and horse.”

It covers the period April 21-May 7, starting three days after the midnight ride. The provisional state government paid it.

“It seems at first blush incongruous, but then again, it’s not,” Massachusetts secretary of state William F. Galvin told the Bangor Daily News. “Even a revolutionary horse needs to be fed, not to mention Paul Revere himself.”

May 2, 2021May 1, 2021 | History

Seating Trouble

The Fall 1978 issue of Pi Mu Epsilon Journal included this problem, submitted by Pier Square. Four men are playing bridge. Their names are Banker, Waiter, Baker, and Farmer, and, as it happens, each man’s name is another man’s job. Mr. Baker’s partner is the baker, Mr. Banker’s partner is the farmer, and the waiter sits at Mr. Farmer’s right. Who is sitting at the banker’s left?

	Select Click for Answer
In his 1994 book Mathematics: A Human Endeavor, Harold R. Jacobs gives a similar problem, by French puzzlist Pierre Berloquin. Around a table sit the four members of the street storekeepers’ committee: a grocer, a baker, a tobacconist, and a butcher, whose name is Andre. If Andre sits on Charmeil’s left, Berton sits at the grocer’s right, and Duclos, who faces Charmeil, is not the baker, then what are the professions of the other members? That’s simple enough that it doesn’t require a solution, I think. But here’s a much harder one, composed by J.B. Parker for the January 1941 issue of Eureka: Seven men, Messrs. Black, Blue, Brown, Gray, Green, Purple and White sat down at a circular table laid for eight. Each man was wearing a tie and a pair of socks — and also had a car, their colours being three of the names of the other six men. There was a tie, a pair of socks and a car of each of the seven colours. Mr. Blue sat next to the man with the green tie; between Mr. Gray and the man with white socks there sat a man wearing a white tie, and opposite him sat Mr. Green. Mr. Brown’s socks were of the same colour as the tie of the man who occupied the chair on his right, and Mr. Green wore a brown tie. The empty chair lay between Mr. Black and the man who wore a green tie. The man with black socks had a gray car, and the man with gray socks had a black car. Mr. Purple’s car was the same colour as Mr. Gray’s tie, and this was of the same colour as Mr. White’s socks. The man with the name of the colour of Mr. White’s car wore socks of the colour of Mr. Black’s car, i.e. blue. Mr. Purple’s tie was of the colour of the car of the man who occupied the chair on his right, and Mr. Brown sat opposite the man with the white car. The colour of the socks of the man whose tie was the colour of Mr. Gray’s car was the same as that of the tie of the man whose socks were the colour of Mr. Black’s car, and this colour was not black. Find the colours of the socks, tie, and car of each man. The answer (but not the solution) appears on page 11 of the May 1941 issue. (The Eureka Digital Archive is published under a Creative Commons license.) (Pier Square, “The Bridge Game,” Pi Mu Epsilon Journal 6:9 [Fall 1978], 530.) 05/24/2021 Reader Dharmesh Jain kindly offers this walkthrough of the Eureka puzzle.

Click for Answer

seating trouble solution

In his 1994 book Mathematics: A Human Endeavor, Harold R. Jacobs gives a similar problem, by French puzzlist Pierre Berloquin. Around a table sit the four members of the street storekeepers’ committee: a grocer, a baker, a tobacconist, and a butcher, whose name is Andre. If Andre sits on Charmeil’s left, Berton sits at the grocer’s right, and Duclos, who faces Charmeil, is not the baker, then what are the professions of the other members?

That’s simple enough that it doesn’t require a solution, I think. But here’s a much harder one, composed by J.B. Parker for the January 1941 issue of Eureka:

Seven men, Messrs. Black, Blue, Brown, Gray, Green, Purple and White sat down at a circular table laid for eight. Each man was wearing a tie and a pair of socks — and also had a car, their colours being three of the names of the other six men. There was a tie, a pair of socks and a car of each of the seven colours.

Mr. Blue sat next to the man with the green tie; between Mr. Gray and the man with white socks there sat a man wearing a white tie, and opposite him sat Mr. Green.

Mr. Brown’s socks were of the same colour as the tie of the man who occupied the chair on his right, and Mr. Green wore a brown tie. The empty chair lay between Mr. Black and the man who wore a green tie. The man with black socks had a gray car, and the man with gray socks had a black car.

Mr. Purple’s car was the same colour as Mr. Gray’s tie, and this was of the same colour as Mr. White’s socks. The man with the name of the colour of Mr. White’s car wore socks of the colour of Mr. Black’s car, i.e. blue. Mr. Purple’s tie was of the colour of the car of the man who occupied the chair on his right, and Mr. Brown sat opposite the man with the white car. The colour of the socks of the man whose tie was the colour of Mr. Gray’s car was the same as that of the tie of the man whose socks were the colour of Mr. Black’s car, and this colour was not black.

Find the colours of the socks, tie, and car of each man.

The answer (but not the solution) appears on page 11 of the May 1941 issue. (The Eureka Digital Archive is published under a Creative Commons license.)

(Pier Square, “The Bridge Game,” Pi Mu Epsilon Journal 6:9 [Fall 1978], 530.)

05/24/2021 Reader Dharmesh Jain kindly offers this walkthrough of the Eureka puzzle.

May 1, 2021May 24, 2021 | Puzzles

Unquote

“To convince any man against his will is hard, but to please him against his will is justly pronounced by Dryden to be above the reach of human abilities.” — Samuel Johnson

“Thou canst not joke an Enemy into a Friend; but thou may’st a Friend into an Enemy.” — Ben Franklin

April 30, 2021April 29, 2021 | Quotations · Society