Easy Street

The following was rather widely quoted a few years ago. It bothered one banker so much that he made a hasty trip to consult his neighbor, a college professor of mathematics. Assume we make a deposit of $50 in a bank.

    withdraw $20.00 leaving $30.00
now withdraw  15.00 leaving  15.00
now withdraw   9.00 leaving   6.00
now withdraw   6.00 leaving   0.00
             ------         ------
             $50.00         $51.00

We now present our figures to the bank, showing the discrepancy, and demand the extra dollar. Repeat ten thousand times, and retire for a while.

— Cecil B. Read, “Mathematical Fallacies,” School Science and Mathematics, June 1933

| Science & Math

On the Wing

https://en.wikipedia.org/wiki/File:Phoebe_snetsinger.jpg

Diagnosed with terminal melanoma at 50, Phoebe Snetsinger resolved to devote her remaining time to watching birds. Between 1981 and 1999, as her cancer went periodically into remission, she visited every continent several times over, traversing jungles, swamps, deserts, and mountains and surviving malaria, a boat accident, abduction in Ethiopia, and rape in Papua New Guinea. In 1995 she became the first person to see 8,000 species of bird, and in time she extended the list to 8,398, nearly 85 percent of the world’s known species. She died in 1999 when her van overturned during a birding trip in Madagascar. The last bird she’d observed was a red-shouldered vanga, a species that had been described as new to science only two years previously.

In her memoir, Birding on Borrowed Time, she wrote, “When I was given my death sentence by the doctors, one of my immediate reactions that I clearly remember was ‘Oh no — there are all those things I haven’t yet done, and now will never have a chance to do.’ … The preparation and primarily the birding itself, plus the record keeping afterwards, all enabled me to forget the threat to my life (or at least push it aside) and to immerse myself totally in what I was doing.”

| Death · Science & Math

Also-Ran

Arthur Conan Doyle tells us little about James Moriarty, the criminal mastermind in the Sherlock Holmes stories. But he does mention one intriguing accomplishment in The Valley of Fear:

Is he not the celebrated author of The Dynamics of an Asteroid, a book which ascends to such rarefied heights of pure mathematics that it is said that there was no man in the scientific press capable of criticizing it?

Mathematicians Alain Goriely and Simon P. Norton have both pointed out that in 1887 King Oscar II of Sweden offered a bounty for the solution to the n-body problem in celestial mechanics. Doyle’s story was set in 1888, so it’s possible that Moriarty had intended his book as his entry in this contest.

If he did, he was disappointed — the prize went to Henri Poincaré.

| Literature · Science & Math

Fitting

From reader Snehal Shekatkar:

There exist exactly 17 numbers the sum of whose distinct prime factors is exactly 17:

17 = 17
52 = 2 × 2 × 13
88 = 2 × 2 × 2 × 11
99 = 3 × 3 × 11
147 = 3 × 7 × 7
175 = 5 × 5 × 7
210 = 2 × 3 × 5 × 7
224 = 2 × 2 × 2 × 2 × 2 × 7
250 = 2 × 5 × 5 × 5
252 = 2 × 2 × 3 × 3 × 7
300 = 2 × 2 × 3 × 5 × 5
320 = 2 × 2 × 2 × 2 ×2 × 2 × 5
360 = 2 × 2 × 2 × 3 × 3 × 5
384 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3
405 = 3 × 3 × 3 × 3 × 5
432 = 2 × 2 × 2 × 2 × 3 × 3 × 3
486 = 2 × 3 × 3 × 3 × 3 × 3

(Thanks, Snehal.)

| Science & Math

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

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

Action at a Distance

https://commons.wikimedia.org/wiki/File:Jupiter_-_Flickr_-_upsidedown_astronomer.png

On April 1, 1976, English astronomer Patrick Moore announced a unique opportunity on BBC Radio 2. At 9:47 a.m., he said, Pluto would pass behind Jupiter, and the combined gravitational forces of the two bodies would noticeably reduce gravity on Earth. If listeners jumped in the air during the conjunction, they’d feel a floating sensation.

Shortly after the appointed moment, the BBC began to receive hundreds of phone calls from listeners who’d confirmed the effect, including “a woman who said that she and 11 friends had been wafted from their chairs and orbited gently around the room.”

None of them had noted the date.

| Hoaxes · Science & Math

Dunbar’s Number

In the 1990s, after studying the relation between primate brain size and social groups, British anthropologist Robin Dunbar proposed that human beings can comfortably maintain about 150 stable relationships — relationships in which one knows all the other members and how they relate to one another. Informally, he said, this is “the number of people you would not feel embarrassed about joining uninvited for a drink if you happened to bump into them in a bar.”

Notably, in a 2018 article for the Financial Times, Dunbar added that we maintain an inner core of about five people with whom we spend about 40 percent of our social time and 10 more with whom we spend another 20 percent. “In other words, about two-thirds of our total social effort is devoted to just 15 people.”

| Science & Math · Society

Smell Test

In an 1899 lecture, University of Wyoming chemistry professor Emory Edmund Slosson announced that he wanted to see how rapidly a certain odor diffused through the air and asked students to raise their hands as soon as they could smell it. Then he poured distilled water over cotton, explaining that it was a chemical with a distinctly unusual odor. Within 15 seconds most of the students in the front row had raised their hands, and within a minute three quarters of the class had.

Similarly, University of California food scientist Michael O’Mahoney told a British TV and radio audience in 1978 that a certain sound frequency would induce the perception of an outdoorsy smell. This suggestion and some audiovisual prompts led hundreds of people to write in claiming they’d smelled something when the frequency was broadcast.

In The Scent of Desire (2009), Brown University psychologist Rachel Herz writes, “This media stunt shows how simply being told that an odor is ‘there’ can convince you that you are smelling something with all of its full-blown consequences.”

12/13/2023 UPDATE: It seems likely that Slosson is Edwin Emery Slosson — possibly Herz mistook his name. (Thanks, Charlotte.)

| Science & Math