Membership

Consider the set (2, 5, 9, 13). Which of these numbers can be tossed out, and for what reason?

We might choose:

  • 2 because it’s the only even number.
  • 9 because it’s the only non-prime.
  • 13 because it doesn’t fit in the sequence An – An-1 = 1 + (An-1 – An-2).

“Hence one could toss out either 2, 9 or 13,” observes Marquette University mathematician George R. Sell. “Therefore one should toss out 5 because it is the only number that cannot be tossed out.”

(George R. Sell, “A Paradox,” Pi Mu Epsilon Journal 2:6 [Spring 1957], 278.)

| Science & Math

“A Matter of Opinion”

https://www.gutenberg.org/cache/epub/52052/pg52052-images.html

A man walks round a pole on the top of which is a monkey. As the man moves, the monkey turns on the top of the pole, so as still to keep face to face with the man. Now, when the man has gone round the pole, has he or has he not gone round the monkey?

— John Scott, The Puzzle King, 1899

| Oddities

A Path-Making Game

Alice and Bob are playing a game. An n×n checkerboard lies between them. Alice begins by marking a corner square, and thereafter the two of them take turns marking squares; each one they choose must be adjacent orthogonally to the last one chosen, so together they’re making a path around the board. When the path can’t continue (because no unmarked adjacent square is available), then the player who moved last wins. For which n can Alice devise a winning strategy? What if she has to start by marking a square adjacent to a corner, rather than the corner itself?

Click for Answer
| Puzzles

Exchange

A Highwayman confronted a Traveler, and covering him with a firearm, shouted: ‘Your money or your life!’

‘My good friend,’ said the Traveler, ‘according to the terms of your demand my money will save my life, my life my money; you imply that you will take one or the other, but not both. If that is what you mean please be good enough to take my life.’

‘That is not what I mean,’ said the Highwayman; ‘you cannot save your money by giving up your life.’

‘Then take it anyhow,’ the Traveler said. ‘If it will not save my money it is good for nothing.’

The Highwayman was so pleased with the Traveler’s philosophy and wit that he took him into partnership and this splendid combination of talent started a newspaper.

— Ambrose Bierce, Fantastic Fables, 1899

| Literature

A Panmagic Geomagic Square

sallows panmagic geomagic square

Another amazing contribution by Lee Sallows:

“The picture above shows a 4×4 geomagic square, which is to say a magic square using geometrical shapes that can be fitted together so as to form an identical target shape, in this case a 4×6 rectangle, rather than numbers adding to a constant sum. In addition, the square is also panmagic, meaning that besides the usual 4 rows, 4 columns, and both main diagonals, the shapes occupying each of the so-called ‘broken’ diagonals, afkn, dejo, cfip, bglm, chin, belo, are also able to tile the rectangle. Lastly, the 4 shapes contained in the corner cells of the four embedded 3×3 sub-squares, acik, bdjl, fhnp, egmo, are also ‘magic’, bringing the total number of target-tiling shape sets to 20, a small improvement over the 16 achieved by a panmagic-only square. With that said, it is worth noting that 4×4 geomagic squares have been found achieving target-tiling scores as high as 48.”

Click the image to expand it. Thanks, Lee!

| Science & Math

The Bebington Puzzle Stones

bebington puzzle stones

Visiting England’s Wirral Peninsula in 1853, Nathaniel Hawthorne came upon a queer battlemented house in the town of Bebington, “quite a novel symbol of decay and neglect,” “probably the whim of some half-crazy person.” “On the wall, close to the street, there were certain eccentric inscriptions cut into slabs of stone, but I could make no sense of them.”

The crazy person was resident Thomas Francis, and the inscriptions had apparently been commissioned to bemuse and entertain passersby. They offer three puzzles. The first presents the image of an inn, The Two Crowns, and the following riddle:

“My name And sign is thirty Shillings just, and he that will tell My Name Shall have a Quart on trust, for why is not Five the Fourth Part of Twenty the Same in All Cases?”

This was easier to guess at the time of its inscription. The landlord of the Two Crowns was Mark Noble, the old English coin known as the noble was worth 6 shillings and eightpence, the mark was worth 13 shillings and fourpence, and two crowns were worth 10 shillings. Together these values total 30 shillings.

The second puzzle is more straightforward: “Subtract 45 From 45 That 45 May Remain.” This seems to refer to the following mathematical curiosity:

  987654321
- 123456789
-----------
  864197532

Each of these figures comprises the digits 1 to 9, so all have the same digit sum: 45.

The last puzzle is the easiest:

  AR
    UBB
I
  NGS
TONEF
  ORAS
 SE
S

Read this straight through and you get A RUBBING STONE FOR ASSES — possibly a comment by Francis on the loiterers who would gather outside his home.

The house was demolished in the 1960s, but the stones can be seen today in the foyer of the library at the Bebington civic center.

| Oddities · Puzzles

“Fifty-Seven to Nothing”

https://books.google.com/books?id=FS8PAAAAQAAJ&pg=PA98

A puzzle by Henry Dudeney:

“It will be seen that we have arranged six cigarettes so as to represent the number 57. The puzzle is to remove any two of them you like (without disturbing any of the others) and so replace them as to represent 0, or nothing.”

Click for Answer
| Puzzles

A Fateful Choice

https://commons.wikimedia.org/wiki/File:Dying_Valentine.jpg

A disease is spreading rapidly across the country. Half the people who have contracted it have died, and half have recovered on their own. A crash program to ward off the epidemic has produced two serums, A and B, but there’s been little time to test them. All three of the patients who were given serum A recovered, and so did 7 of the 8 patients who were given serum B. Unfortunately, you’ve just learned that you have the disease. If you get no treatment, your chances of surviving are 50-50. Both serums have a better record than that, but which one should you take?

“There doesn’t seem to be anything we can do other than appeal to our intuitive feelings on the matter,” writes University of Waterloo mathematician Ross Honsberger. “However, a very ingenious notion, the so-called ‘null hypothesis,’ permits a measure of analysis which, in this case, yields a definite preference.”

The key is to ask how likely it is that 3 out of 3 patients would have recovered if serum A were neither helping nor hindering them. An untreated patient has a 50-50 chance of recovery, so the answer is

\displaystyle \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}.

On the other hand, if serum B had no effect, then the chance of 7 recoveries out of 8 is

\displaystyle 8 \left ( \frac{1}{2} \times \frac{1}{2} \cdots \frac{1}{2} \right ) = 8 \left ( \frac{1}{2} \right )^{8} = \frac{1}{32}.

(Here the factor 8 reflects the fact that there are 8 different possible victims, and again the probability of dying is 1/2.)

So the available evidence suggests that it’s 4 times as likely that serum A has no effect as that serum B has no effect. Your best course is to take serum B.

(Ross Honsberger, “Some Surprises in Probability,” in his Mathematical Plums, 1979.)

| Science & Math

The Ta Prohm Stegosaur

https://commons.wikimedia.org/wiki/File:Dinosaur_carving_at_Ta_Prohm_temple,_Siem_Reap,_Cambodia_(5534467622).jpg
Image: Wikimedia Commons

At Cambodia’s Ta Prohm temple, near Angkor Wat, visitors have noticed an unusual carving on a crumbling wall — it appears to resemble a stegosaur. Well, viewed from the right angle it does, and allowing for the large head, large ears, and horn. In fact, it better resembles a boar, a rhino, or a chameleon viewed against a leafy background.

Because the temple was built in the 12th century, some creationists have cited the carving as evidence that humans and dinosaurs once coexisted in the region. But it’s not even clear when the carving was made — the temple is a favorite spot for filmmakers, one of whom may have added it, or it may even be a deliberate hoax.

In any case, writes Smithsonian‘s Riley Black, “the temple carving can in no way be used as evidence that humans and non-avian dinosaurs coexisted. Fossils have inspired some myths, but close scrutiny of geological layers, reliable radiometric dating techniques, the lack of dinosaur fossils in strata younger than the Cretaceous, and other lines of evidence all confirm that non-avian dinosaurs became extinct tens of millions of years before there was any type of culture that could have recorded what they looked like.”

| Oddities

Simple

The 1968 Putnam Competition included a beautiful one-line proof that π is less than 22/7, its common Diophantine approximation:

\displaystyle 0 \enspace \textless \int_{0}^{1}\frac{x^{4}\left ( 1 - x \right )^{4}}{1 + x^{2}} \: dx = \frac{22}{7} - \pi .

The integral must be positive, because the integrand’s denominator is positive and its numerator is the product of two non-negative numbers. But it evaluates to 22/7 – π — and if that expression is positive, then 22/7 must be greater than π.

University of St Andrews mathematician G.M. Phillips wrote, “Who will say that mathematics is devoid of humour?”

| Science & Math