“The Staircase Race”

https://books.google.com/books?id=FS8PAAAAQAAJ&pg=PA16

A puzzle from Henry Dudeney’s Modern Puzzles and How to Solve Them, 1926:

This is a rough sketch of the finish of a race up a staircase in which three men took part. Ackworth, who is leading, went up three risers at a time, as arranged; Barnden, the second man, went four risers at a time, and Croft, who is last, went five at a time.

Undoubtedly Ackworth wins. But the point is, How many risers are there in the stairs, counting the top landing as a riser?

I have only shown the top of the stairs. There may be scores, or hundreds, of risers below the line. It was not necessary to draw them, as I only wanted to show the finish. But it is possible to tell from the evidence the fewest possible risers in that staircase. Can you do it?

Click for Answer
| Puzzles

Hitchcock’s Bomb Theory

https://commons.wikimedia.org/wiki/File:Hitchcock,_Alfred_02.jpg

There is a distinct difference between ‘suspense’ and ‘surprise,’ and yet many pictures continually confuse the two. I’ll explain what I mean. We are now having a very innocent little chat. Let’s suppose that there is a bomb underneath this table between us. Nothing happens, and then all of a sudden, ‘Boom!’ There is an explosion. The public is surprised, but prior to this surprise, it has seen an absolutely ordinary scene, of no special consequence. Now, let us take a suspense situation. The bomb is underneath the table and the public knows it, probably because they have seen the anarchists place it there. The public is aware the bomb is going to explode at one o’clock and there is a clock in the decor. The public can see that it is a quarter to one. In these conditions, the same innocuous conversation becomes fascinating because the public is participating in the scene. The audience is longing to warn the characters on the screen: ‘You shouldn’t be talking about such trivial matters. There is a bomb beneath you and it is about to explode!’ In the first case, we have given the public fifteen seconds of surprise at the moment of the explosion. In the second we have provided them with fifteen minutes of suspense. The conclusion is that whenever possible the public must be informed. Except when the surprise is a twist, that is, when the unexpected ending is, in itself, the highlight of the story.

(Via François Truffaut’s Hitchcock, 1967.)

| Entertainment

Short Work

A simple proof that \sqrt{2} is irrational:

Assume that it’s rational. Then \sqrt{2} = p/q and p2 = 2q2. But in the latter equation, the left side must have an even number of prime factors and the right an odd number. That’s a contradiction, so our assumption must be wrong.

In the paper below, Manchester Polytechnic mathematician T.J. Randall credits this “marvellous” proof to Philip J. Davis and Reuben Hersh in their 1982 book The Mathematical Experience, but I can’t find it there.

(T.J. Randall, “67.45 \sqrt{2} Revisited,” Mathematical Gazette 67:442 [December 1983], 302-303.)

12/20/2023 UPDATE: Reader Hans Havermann finds the proof mentioned in Stuart Hollingdale’s Makers of Mathematics (1989), after the more familiar proof based on parity of p and q. Hollingdale writes that the alternative proof “can be traced back to the Classical period.”

| Science & Math

Long Haul

In Penn & Teller’s 1994 game Desert Bus, you must pilot a bus along an arrow-straight road through 360 miles of featureless desert from Tucson, Arizona to Las Vegas. There are no passengers, no other vehicles, and no scenery but dead trees and bushes. The bus tends to drift, so you have to work continuously to keep it on the road, but apart from that there is nothing to do. At a top speed of 45 mph, it will take at least 8 hours of continuous attention to reach your destination.

Absolute Entertainment ceased operations before it could release the game, but a press copy came to light in 2000, and since then versions have been released on Android, iOS, and virtual reality. A recurring livestream had raised $6 million for charity by 2019.

(When you reach Vegas, you’re given one point and the option to drive back to Tucson.)

| Entertainment

Elementary

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

A logic exercise by Lewis Carroll: What conclusion can be drawn from these premises?

  1. All the human race, except my footmen, have a certain amount of common sense.
  2. No one who lives on barley sugar can be anything but a mere baby.
  3. None but a hopscotch player knows what real happiness is.
  4. No mere baby has a grain of common sense.
  5. No engine driver ever plays hopscotch.
  6. No footman of mine is ignorant of what true happiness is.
Click for Answer
| Puzzles

Signifying Nothing

https://commons.wikimedia.org/wiki/File:Shakespeare_Droeshout_1623_From_the_First_Folio_Edition.jpg

If … we were asked to select one monument of human civilization that should survive to some future age … we should probably choose the works of Shakespeare. In them we recognize the truest portrait and best memorial of man. Yet the archæologists of that future age … would misconceive our life in one important respect. They would hardly understand that man had had a religion. …

Shakespeare could be idealistic when he dreamed, as he could be spiritual when he reflected. … It is all the more remarkable, therefore, that we should have to search through all the works of Shakespeare to find half a dozen passages that have so much as a religious sound, and that even these passages, upon examination, should prove not to be the expression of any deep religious conception. If Shakespeare had been without metaphysical capacity, or without moral maturity, we could have explained his strange insensibility to religion; but as it is, we must marvel at his indifference and ask ourselves what can be the causes of it.

— George Santayana, “The Absence of Religion in Shakespeare,” in Interpretations of Poetry and Religion, 1900

| Literature · Religion

Changes

In 1967, psychiatrists Thomas Holmes and Richard Rahe asked 5,000 medical patients about recent difficult events in their lives. They found a positive correlation with illness. The sum of the “life change units” you’ve amassed in the past year, they said, gives a rough estimate of the effect on your health:

Death of a spouse: 100
Divorce: 73
Marital separation: 65
Imprisonment: 63
Death of a close family member: 63
Personal injury or illness: 53
Marriage: 50
Dismissal from work: 47
Marital reconciliation: 45
Retirement: 45
Change in health of family member: 44
Pregnancy: 40
Sexual difficulties: 39
Gain a new family member: 39
Business readjustment: 39
Change in financial state: 38
Death of a close friend: 37
Change to different line of work: 36
Change in frequency of arguments: 35
Major mortgage: 32
Foreclosure of mortgage or loan: 30
Change in responsibilities at work: 29
Child leaving home: 29
Trouble with in-laws: 29
Outstanding personal achievement: 28
Spouse starts or stops work: 26
Beginning or end of school: 26
Change in living conditions: 25
Revision of personal habits: 24
Trouble with boss: 23
Change in working hours or conditions: 20
Change in residence: 20
Change in schools: 20
Change in recreation: 19
Change in church activities: 19
Change in social activities: 18
Minor mortgage or loan: 17
Change in sleeping habits: 16
Change in number of family reunions: 15
Change in eating habits: 15
Vacation: 13
Major holiday: 12
Minor violation of law: 11

A score of 300+ means you’re at risk of illness.

A score of 150-299 means your risk of illness is moderate (but 30 percent lower than the foregoing group).

A score of less than 150 means your risk of stress-related illness is slight.

| Science & Math

Kempe’s Jig

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

In 1600, actor William Kempe traveled from London to Norwich, a distance of 177 kilometers. Why is that notable? He morris-danced the whole way, a feat that took nine days spread over several weeks. He published an account of the journey to convince doubters:

Vpon Fryday morning I set on towardes Thetford, dauncing that tenne mile in three houres; for I left Bury somewhat after seauen in the morning, and was at Thetford somewhat after ten that same forenoone. But, indeed, considering how I had been booted the other iourneys before, and that all this way, or the most of it, was ouer a heath, it was no great wonder; for I far’d like one that had escaped the stockes, and tride the vse of his legs to out-run the Constable: so light was my heeles, that I counted the ten mile no better than a leape.

He’s remembered for an even more distinctive accomplishment — he was one of the original stage actors in Shakespeare’s early dramas.

(Thanks, Tyler.)

| Oddities

To the Point

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

In 1755 Samuel Richardson published an index of the “moral and instructive sentiments, maxims, cautions and reflexions” in his novels Pamela, Clarissa, and The History of Sir Charles Grandison, so you can receive all the edification without actually reading them:

Courage.

One of the characteristics of a good man, is, to be studious to avoid danger, and to be unappalled in it, iv. 104. [276].
In a case of inevitable danger, the way to avoid it, is not to appear to be intimidated. One man’s fear gives another man Courage, v. 289. [vi. 267].
Courage is a glorious quality when it is divested of rashness, and founded on integrity of life and manners, v. 296. [vi. 274].
But otherwise founded, it is rather to be called savageness and brutality, than Courage, ibid.
See Challenges. Duelling. Good Man. Magnanimity.

The whole thing is here. See Asides.

| Literature