Falling Currency

A problem from the October 1964 issue of Eureka, the journal of the Cambridge University Mathematical Society:

My friend tosses two coins and covers them with his hand. ‘Is there at least one “tail”?’ I ask. He affirms this (a).

Just then he accidentally knocks one of them to the floor (b). On finding the dropped coin under the table, we discover it to be a ‘tail’ (c).

‘That is all right,’ he says, ‘because it was a “tail” to start with.’ (d).

At each point (a), (b), (c) and (d) of this episode I calculated what, to the best of my knowledge, was the probability that both coins showed ‘tails’ at the time. What were these probabilities?

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| Puzzles

Modern Times

https://commons.wikimedia.org/wiki/File:Coelopleurus_exquisitus.jpg
Image: Wikimedia Commons

The sea urchin Coelopleurus exquisitus was discovered on eBay. Marine biologist Simon Coppard was directed to a listing on the site in 2004 and realized that the species had not previously been described. When it was properly named and introduced in Zootaxa two years later, the value of specimens on eBay shot up from $8 to $138.

In 2008 a fossilized aphid on eBay was similarly found to be unidentified. Eventually it was named Mindarus harringtoni, after the buyer.

| Science & Math · Technology

Heesch’s Problem

https://commons.wikimedia.org/wiki/File:Heesch_number_2_minimal_polyomino.svg
Image: Wikimedia Commons

The dark polyomino at the center of this figure, devised by Craig S. Kaplan, has an unusual property: It can be surrounded snugly with copies of itself, leaving no overlaps or gaps. In this case, the “corona” (red) can be surrounded with a second corona (amber), itself also composed of copies of the initial shape. But that’s as far as we can get — there’s no way to create a third corona using the same shape.

That gives the initial shape a “Heesch number” of 2 — the designation is named for German geometer Heinrich Heesch, who had proposed this line of study in 1968.

Shapes needn’t be polyominos: Heesch himself devised the example below, the union of a square, an equilateral triangle, and a 30-60-90 triangle:

https://commons.wikimedia.org/wiki/File:Heesch_1.svg
Image: Wikimedia Commons

It earns a Heesch number of 1, as it can bear only the single corona shown.

Can all positive integers be Heesch numbers? That’s unknown. The Heesch number of the square is infinite, and that of the circle is zero. The highest finite number reached so far is 6.

| Science & Math

Memories

Excerpts from Mark Twain’s boyhood journal:

Monday — Got up, washed, went to bed.
Tuesday — Got up, washed, went to bed.
Wednesday — Got up, washed, went to bed.
Thursday — Got up, washed, went to bed.
Friday — Got up, washed, went to bed.
Next Friday — Got up, washed, went to bed.
Friday fortnight — Got up, washed, went to bed.
Following month — Got up, washed, went to bed.

“I stopped, then, discouraged. Startling events appeared to be too rare, in my career, to render a diary necessary.”

(From The Innocents Abroad.)

| Literature · Society

Summing Up

Much of the success of the administrator in carrying out a program depends upon how far it is his sole object overshadowing everything else, or how far he is thinking of himself; for this last is an obstruction that has caused many a good man to stumble and a good cause to fall. The two aims are inconsistent, often enough for us to state as a general rule that one cannot both do things and get the credit for them.

— A. Lawrence Lowell, What a University President Has Learned, 1938

| Society

Black and White

wills chess problem

By W.F. Wills. White to mate in two moves.

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| Puzzles

Solo

https://commons.wikimedia.org/wiki/File:Kangchenjunga_PangPema.JPG
Image: Wikimedia Commons

A poignant little detail from my podcast research on Maurice Wilson, who in 1934 set out to climb Everest alone:

There was only one precedent in mountaineering history for such an impossible lone assault. In May of 1929 a young American climber, E.F. Farmer of New Rochelle, N.Y., had set off from Darjeeling on a suicidal attack on 28,146-foot Kangchenjunga. He disappeared into the clouds and was never seen again.

(John Cottrell, “The Madman of Everest,” Sports Illustrated, April 30, 1973.)

| Death

One Solution

[Jerome Sankey] challenged Sir William [Petty] to fight with him. Sir William is extremely short-sighted, and being the challengee it belonged to him to nominate place and weapon. He nominates for the place a dark cellar, and the weapon to be a great carpenter’s axe. This turned the knight’s challenge into ridicule, and so it came to nought.

— John Aubrey, Brief Lives, 1697

| Society