Double and Half

A “curious paradox” presented by Raymond Smullyan at the first Gathering for Gardner: Consider two positive integers, x and y. One is twice as great as the other, but we’re not told which is which.

  • If x is greater than y, then x = 2y and the excess of x over y is equal to y. On the other hand, if y is greater than x, then x = 0.5y and the excess of y over x is y – 0.5y = 0.5y. Since y is greater than 0.5y, then we can say generally that the excess of x over y, if x is greater than y, is greater than the excess of y over x, if y is greater than x.
  • Let d be the difference between x and y. This is the same as saying that it’s equal to the lesser of the two. Generally, then, the excess of x over y, if x is greater than y, is equal to the excess of y over x, if y is greater than x.

The two conclusions contradict one another, so something is amiss. But what?

| Science & Math

Close to Home

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

On Sept. 9, 1942, a lookout on Mount Emily in Oregon’s Siskiyou National Forest reported a plume of smoke near the town of Brookings. The Forest Service contained the fire easily, but investigators turned up something odd at the site: fragments of an incendiary bomb of Japanese origin.

It turned out that a Japanese submarine had surfaced off the Oregon/California border and 31-year-old navy officer Nobuo Fujita had piloted a seaplane into the forest, hoping to start a fire that would divert U.S. military resources from the Pacific. Recent rains had wet the forest, so the plan failed, but it marked the first time the continental United States had been bombed by enemy aircraft.

Fujita returned safely to Japan, where he opened a hardware store after the war, and he became an agent of amity with the United States. In 1962 he accepted an invitation to return to Oregon, where he donated his family’s samurai sword to Brookings, and he invited three local students to visit Japan in 1985. The city made him an honorary citizen shortly before his death in 1997, and his daughter spread his ashes at the site of the bombing.

| History · Oddities

Sand Reckoning

Using only a 4-minute hourglass and a 7-minute hourglass, how can you measure 9 minutes?

Click for Answer
| Puzzles

Boor Laws

http://commons.wikimedia.org/wiki/File:Chamberlin_-_Benjamin_Franklin_(1762).jpg

Ben Franklin’s “rules for making oneself a disagreeable companion,” 1750:

  1. If possible engross the whole Discourse; and when other Matter fails, talk much of your-self, your Education, your Knowledge, your Circumstances, your Successes in Business, your Victories in Disputes, your own wise Sayings and Observations on particular Occasions, &c. &c. &c.;
  2. If when you are out of Breath, one of the Company should seize the Opportunity of saying something; watch his Words, and, if possible, find somewhat either in his Sentiment or Expression, immediately to contradict and raise a Dispute upon. Rather than fail, criticise even his Grammar.
  3. If another should be saying an indisputably good Thing; either give no Attention to it; or interrupt him; or draw away the Attention of others; or, if you can guess what he would be at, be quick and say it before him; or, if he gets it said, and you perceive the Company pleas’d with it, own it to be a good Thing, and withal remark that it had been said by Bacon, Locke, Bayle, or some other eminent Writer; thus you deprive him of the Reputation he might have gain’d by it, and gain some yourself, as you hereby show your great Reading and Memory.
  4. When modest Men have been thus treated by you a few times, they will chuse ever after to be silent in your Company; then you may shine on without Fear of a Rival; rallying them at the same time for their Dullness, which will be to you a new Fund of Wit.

“Thus you will be sure to please yourself,” he concluded. “The polite Man aims at pleasing others, but you shall go beyond him even in that. A Man can be present only in one Company, but may at the same time be absent in twenty. He can please only where he is, you where-ever you are not.”

| Society

Spin Control

http://www.sxc.hu/photo/443691

We’re playing Russian roulette. The revolver has six chambers, all empty. I put bullets in two adjacent chambers, spin the cylinder, hold the gun to my head, and pull the trigger. It clicks. Now it’s your turn. Before pulling the trigger, you can choose to spin the cylinder again or leave it as it is. Which is better?

Click for Answer
| Puzzles

“A Magic Circle of Cubes”

kaprekar circle of cubes

Reading this circle clockwise produces the numbers 04, 20, 34, 12, 50, 42, 03, 41, 53, 15, 31, 25.

Reading it counterclockwise gives 05, 21, 35, 13, 51, 43, 02, 40, 52, 14, 30, 24.

The sum of the first group equals that of the second, and this holds true if the numbers are squared or cubed. Further, if the numbers in the first group are arranged in ascending order and those in the second in descending, then:

kaprekar circle sums

(Devised by D.R. Kaprekar in 1956.)

| Science & Math

Art and Commerce

In 1896 the U.S. Treasury introduced some beautifully high-minded currency — instead of American presidents, the “educational series” of silver certificates bear neoclassical allegories:

http://commons.wikimedia.org/wiki/File:1896$1obv.jpg

On the $1 note, the Goddess of History instructs a youth, pointing to the U.S. Constitution, a panorama of Washington D.C., and a roster of famous Americans, including Franklin, Longfellow, Hawthorne, and Emerson.

http://commons.wikimedia.org/wiki/File:1896$2obv.jpg

On the $2 note, Science presents Steam and Electricity (as children) to Commerce and Manufacture. The back bears portraits of Robert Fulton and Samuel Morse.

http://commons.wikimedia.org/wiki/File:1896$5obv.jpg

The almost impossibly glorious $5 note depicts Electricity Presenting Light to the World. She is flanked by Strength, Fame, and Peace. The New York Times wrote, “The arrangement of this composition, the grace of pose in each figure, and the idea connected with the designs of this artist entitle it to a place beside the finest allegorical designs in the world.”

Unfortunately, the Treasury got a new secretary the following year, one who favored simple, clear designs, and he canceled more than $54 million in certificates as they came into the Treasury. “It can be said authoritatively … that no more of the so-called ‘new certificates’ will be printed,” the Times reported sadly in May 1897. “Neither will fresco painters be called in to make designs for the substitutes.”

(Thanks, Stephenson.)

| Art · History

No Smoking

http://www.google.com/patents/US156187

In 1874 John Thomas suggested a novel way to keep a locomotive’s smoke out of the passenger cars: a jointed pipe would carry the smoke the length of the train and release it harmlessly in its wake.

Smoke and sparks were a real danger on early railroads — in 1916 one Edson Hains filed a claim against the Wheeling & Lake Erie Railroad Company in Cuyahoga County, Ohio, for burning down his outhouse. The case is doubly memorable because Hains filed his pleading in verse:

Now comes the plaintiff, Edson Hains, and prays this august Court
To heed the very earnest plea, enshrined in this report.
The Wheeling & Lake Erie Co., a heartless corporation,
But licensed under certain laws of this fair State and Nation,
Did, by a spark through carelessness, from a locomotive owned
By the defendant in this Court, and by it not atoned,
Destroy and render Null and Void, a building situate
Upon the rear of premises, known as E. Hains Estate,
Which in the town of Bedford lies, a placid rural spot,
Until the conflagration, which spoiled the plaintiff’s lot.
The second of November last was the most woeful date
On which the said defendant did this outrage perpetrate.
Said building being plain but good, and open to all callers,
Was worth in money of the realm, the sum of fifty dollars.
Now, the aforesaid Hains will not cite Blackstone, Coke or Livy,
The incinerated building was, in vulgar terms, a privy.
The wealthy have from two to eight, but this case is more sad,
For like the poor man’s one ewe lamb, ’twas the only one he had.
And now in frigid winter’s time, before beginning labor,
He eke perforce must use the can of an obliging neighbor.

The plaintiff does still further state that he is an inventor,
And since the last catastrophe he has no place to center
His great inventive genius which, before in chaste seclusion
Of the before in mentioned can, did blossom in profusion.
True genius cannot be appraised, but plaintiff’s was so nifty
He thinks he should receive therefor at least one hundred fifty.
He therefore prays the Court to grant two hundred Iron Men,
That he may once more take his place midst Bedford’s upper ten,
And walk abroad midst citizens, and meet them man to man,
Which now he cannot rightly do, he being shy a can.
And so he prays that he may go from hence with compensation,
Commensurate with what he’s lost, through this his degradation.
And so he windeth up his prayer, and hopes the Court will grant
One fifty for his peace of mind, and fifty for the plant.

There’s a photostat of the pleading in John Allison Duncan’s Strangest Cases on Record (1940). Hains won — Judge William H. McGannon assessed the railroad $25 and costs.

| Technology

Red Letters

‘All the world loves a lover.’ But in Venezuela, they do something about it. T.R. Lahey, in the Catholic periodical, Ave Maria, is responsible for the statement that the postal authorities there allow love letters to go through the mails at half price! But there is a condition. The letters must be mailed in bright-colored envelopes (pansy-blue for loving thoughts, and pink cloud effects; there would be a place for yellow and green to express the feelings of envious suitors and jealous lovers). These bright tints are intended to help the postal clerks and postmen to recognize the nature of the missives; but what a temptation to the carriers to open the letters and cull precious thoughts and phrases!

— The Lutheran, Nov. 6, 1940

| Oddities