The Pill Scale (Part 1)

An efficiency-minded pharmacist has just received a shipment of 10 bottles of pills when the manufacturer calls to say that there’s been an error — nine of the bottles contain pills that weigh 5 grams apiece, which is correct, but the pills in the remaining bottle weigh 6 grams apiece. The pharmacist could find the bad batch by simply weighing one pill from each bottle, but he hits on a way to accomplish this with a single weighing. What does he do?

Click for Answer

Dupli-Cat

http://www.flickr.com/photos/cloudsoup/494542407/

Image: Flickr

Two things cannot occupy the same place at the same time. That seems reasonable. But:

A cat called Tibbles loses his tail at time t2. But before t2 somebody had picked out, identified, and distinguished from Tibbles a different and rather peculiar animate entity — namely, Tibbles minus Tibbles’ tail. Let us suppose that he decided to call this entity ‘Tib.’ Suppose Tibbles was on the mat at time t1. Then both Tib and Tibbles were on the mat at t1. … But consider the position from t3 onward when, something the worse for wear, the cat is sitting on the mat without a tail. Is there one cat or are there two cats there? Tib is certainly sitting there. In a way nothing happened to him at all. But so is Tibbles. For Tibbles lost his tail, survived this experience, and then at t3 was sitting on the mat. And we agreed that Tib ≠ Tibbles. We can uphold the transitivity of identity, it seems, only if we stick by that decision at t3 and allow that at t3 there are two cats on the mat in exactly the same place at exactly the same time.

Tibbles and Tib were distinct material objects, but after the amputation they appear to occupy exactly the same space. Were we mistaken?

(From David Wiggins, “On Being in the Same Place at the Same Time,” The Philosophical Review, 77:1 [January 1968], 90-95.)

Further trouble:

No cat has two tails.

Every cat has one tail more than no cat.

Therefore every cat has three tails.

Cardiology

In other words, this is the day on which those charming little missives, ycleped Valentines, cross and intercross each other at every street and turning. The weary and all for-spent twopenny postman sinks beneath a load of delicate embarrassments, not his own. It is scarcely credible to what an extent this ephemeral courtship is carried on in this loving town, to the great enrichment of porters, and detriment of knockers and bell-wires. In these little visual interpretations, no emblem is so common as the heart,–that little three-cornered exponent of all our hopes and fears,–the bestuck and bleeding heart; it is twisted and tortured into more allegories and affectations than an opera-hat. What authority we have in history or mythology for placing the head-quarters and metropolis of god Cupid in this anatomical seat rather than in any other, is not very clear; but we have got it, and it will serve as well as any other thing. Else we might easily imagine, upon some other system which might have prevailed for any thing which our pathology knows to the contrary, a lover addressing his mistress, in perfect simplicity of feeling, ‘Madam, my liver and fortune are entirely at your disposal;’ or putting a delicate question, ‘Amanda, have you a midriff to bestow?’ But custom has settled these things, and awarded the seat of sentiment to the aforesaid triangle, while its less fortunate neighbours wait at animal and anatomical distance.

— Charles Lamb, Essays of Elia, 1823

A mathematical valentine:

http://commons.wikimedia.org/wiki/File:Heart3D.png

Black and White

kuznecov and plaskin chess problem

This week’s puzzle has a twist: Imagine that the board has been rolled into a cylinder so that the a- and h-files are joined and pieces can move across the boundary. How can White mate in two moves?

Click for Answer

Off Duty

A striking story is going the rounds concerning an officer who, being refused leave to go with the Chitral expedition, obtained five days’ leave to go ‘shooting.’ He entrained to a point as near the operations as the railway would carry him, and then, being unable to obtain a horse, set out to march. Equipped with a bottle of gin and a huge sausage as his only rations, he plodded the weary miles over rough ground cheerfully. He reached the head of the column just as the charge was about to be made on the Malakand Pass. He was in time to join the head of the storming column, and was in the first three on the summit. When the battle was over he had to eschew the camp and the rest that awaited the fighting line, and had to make his way back as best he might to a point where the railway would take him up. The London correspondent of the Birmingham Gazette says he heard General Sir Evelyn Wood say that this officer is a full colonel. He went into action as a common soldier, tearing the straps off his Kharka uniform that his rank might not be discovered. For, as Sir Evelyn remarked with a twinkle in his eye, if he had been discovered he would have been put under arrest.

Hawke’s Bay Herald, New Zealand, June 28, 1895

(Thanks, Donald.)

Turnabout

http://en.wikipedia.org/wiki/File:Mischa_Elman.jpg

In the early 20th century, communications between a concert manager and his artists were typically charged to the musicians. Tired of paying for lengthy telegrams and long-distance calls, violinist Mischa Elman sent this wire to his manager, collect:

AM SITTING IN THE DINING ROOM OF MY HOTEL HAVING FRENCH ONION SOUP, WHOLE WHEAT TOAST, FILET MIGNON MEDIUM RARE, MIXED SALAD WITH THOUSAND ISLAND DRESSING, FRENCH APPLE PIE A LA MODE, COFFEE WITHOUT CREAM AND SUGAR. WEATHER MARVELOUS. HAVE SPLENDID ROOM WITH MAGNIFICENT VIEW. NOW HOW DO YOU LIKE COLLECT TELEGRAMS? YOURS CORDIALLY, MISCHA ELMAN

In a Word

http://commons.wikimedia.org/wiki/File:Hiroshige_Snow_falling_on_a_town.jpg

subnivean
adj. existing, living, or carried out underneath snow

Justice Confused

Suppose that a house is robbed and police find a strand of the burglar’s hair at the scene of the crime. A suspect is in custody, and tests show that the strand matches his hair. A forensic scientist testifies that the chance of a random person producing such a match is 1/2000. Does this mean that there’s a 1999/2000 chance that the suspect is guilty?

No, it doesn’t. In a city of 5 million there will be 1/2000 × 5,000,000 = 2,500 people who produce a match, so on the basis of this evidence alone the probability that the suspect is guilty is only 1/2500.

In a 1987 article, William Thomson and Edward Schumann dubbed this “prosecutor’s fallacy.” Unfortunately, it’s matched by the “defense attorney’s fallacy,” which holds that the hair-match evidence is worthless because it increases the likelihood of the suspect’s guilt by a negligible amount, 1/2500. In fact it drastically narrows the range of possible suspects, from 5 million to 2,500, while failing to exclude the defendant, hardly cause for confidence.

Worryingly, Thompson and Schumann found an experienced prosecutor who insisted that if a defendant and a perpetrator match on a blood type found in 10 percent of the population, then there’s a 10 percent chance that the defendant would have this blood type if he were innocent and hence a 90 percent chance that he’s guilty. “If a prosecutor falls victim to this error,” they write, “it is possible that jurors do as well.”

(William C. Thompson and Edward L. Schumann, “Interpretation of Statistical Evidence in Criminal Trials,” Law and Human Behavior, 11:3 [September 1987], 167-187)

High Hopes

http://books.google.com/books?id=EqwCAAAAYAAJ

American life in 1980, as envisioned by Missouri attorney William McClung Paxton in his 1880 poem “A Century Hence”:

In the midst, at St. Louis, the Capitol loomed,
With lofty and glittering steeple —
The seat of a Nation, where freedom first bloomed,
Containing a billion of people.
“And now,” he exclaimed, “the whole Continent’s ours,
From Panama, North to the pole!
For naught but the ocean can fetter our powers,
Or give to us less than the whole!”

As we walked to the house, my companions reported,
That roads through the land were not found,
That men, on light wings, in the atmosphere sported,
Or walked, as they pleased, on the ground.
With the new motive power, one man could do more
Than fifty, without it, could do;
So people were able to add to their store,
And be generous, noble and true.

An order for supper, by telephone, now,
Had scarcely been made, by my host,
When in sprang a servant, I cannot tell how,
With coffee, ham, biscuit and toast.
He’d come from St. Louis, three hundred miles out,
With dishes delicious and rare;
There were venison, and turkey, and salmon, and trout,
With pine-apple, orange and pear.

When supper was ended, I found it still light;
I looked for a lamp, and found none;
I stepped to the door, and looked forth on the night,
And lo! every house had a sun.
Above me in splendor, surpassing the moon,
A disk, in the heavens gave light;
And neighboring orbs gave the brightness of noon,
And scattered the darkness of night.

By reflectors, the light of these beacons was cast,
On parlor, and chamber, and hall;
And candles and lamps were consigned to the past,
And light, like the air, was for all.
Now worn by the scenes of the day, I need rest,
And find it in slumber elysian;
But rise in the morning, perplexed and distressed;
‘Twas all but a beautiful vision.

More Attachable Mirrors

https://www.google.com/patents/US790537

In 1904 Emmie Alice Thayer lamented that a lady had to hold a hand mirror while attending to her hair or addressing the fit of her garment. The answer, she decided, was to wear the mirror by attaching it to her ears. Kudos.

In the same vein, in 1950 John Kozloff invented a “mirror-attached spectacle frame” (below) to free one’s hands for applying cosmetics or shaving. If you’re nearsighted they can even be fitted with glasses. With a pair of these Narcissus wouldn’t have been tied to that boring pool …

See Self-Regard.

https://www.google.com/patents/US2502224

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