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Bootstrap Percolation

grid infection puzzle

On a 12 × 12 grid, some squares are infected and some are healthy. On each turn, a healthy square becomes infected if it has two or more infected orthogonal neighbors. (In the example above, the black squares are infected, the white squares are healthy, and the gray squares will be infected next turn.) What’s the smallest number of initially infected squares that can spread an infection over the whole board?

Click for Answer

“When a Man’s Busy”

When a man’s busy, why, leisure
Strikes him as wonderful pleasure:
‘Faith, and at leisure once is he?
Straightway he wants to be busy.

– Robert Browning


Steven Bartlett and Peter Suber’s Self-Reference: Reflections on Reflexivity contains a bibliography of works on reflexivity.

It includes an entry for Steven Bartlett and Peter Suber’s Self-Reference: Reflections on Reflexivity.



“Whether you can observe a thing or not depends on the theory which you use. It is the theory which decides what can be observed.” — Albert Einstein

“When we observe nature, and especially the ordering of nature, it is always ourselves alone we are observing.” — G.C. Lichtenberg

Counterfeit Redux

A more challenging version of the Counterfeit Coin puzzle from 2011:

You have 12 coins, one of which has been replaced with a counterfeit. The false coin differs in weight from the true ones, but you don’t know whether it’s heavier or lighter. How can find it using three weighings in a pan balance?

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Moving Images

By turning the traditional rules of perspective inside out, the paintings of British artist Patrick Hughes induce a compelling illusion in three dimensions.

“Reverspectives are three-dimensional paintings that when viewed from the front initially give the impression of viewing a painted flat surface that shows a perspective view,” he writes. “However as soon as the viewer moves their head even slightly the three dimensional surface that supports the perspective view accentuates the depth of the image and accelerates the shifting perspective far more than the brain normally allows.

“This provides a powerful and often disorienting impression of depth and movement. The illusion is made possible by painting the view in reverse to the relief of the surface, that is, the bits that stick farthest out from the painting are painted with the most distant part of the scene.”

See Termespheres.

“The Anatomy of Humor”

“What is funny?” you ask, my child,
Crinkling your bright-blue eye.
“Ah, that is a curious question indeed,”
Musing, I make reply.

“Contusions are funny, not open wounds,
And automobiles that go
Crash into trees by the highwayside;
Industrial incidents, no.

“The habit of drink is a hundred per cent,
But drug addiction is nil.
A nervous breakdown will get no laughs;
Insanity surely will.

“Humor, aloof from the cigarette,
Inhabits the droll cigar;
The middle-aged are not very funny;
The young and the old, they are.

“So the funniest thing in the world should be
A grandsire, drunk, insane,
Maimed in a motor accident,
And enduring moderate pain.

“But why do you scream and yell, my child?
Here comes your mother, my honey,
To comfort you and to lecture me
For trying, she’ll say, to be funny.”

– Morris Bishop

Well Bounded

In the Indian state of West Bengal lies a district known as Cooch Behar which is curiously merged with its neighbor, Bangladesh. The Indian land contains 92 Bangladeshi enclaves, and the Bangladeshi land contains 106 Indian enclaves.

The largest Indian enclave itself contains a Bangladeshi enclave, and that Bangladeshi enclave contains a bare hectare of Indian farmland known as Dahala Khagrabari. That makes Dahala Khagrabari the world’s only instance of an enclave in an enclave in an enclave.

See Concentric Landmarks.

“A Financial Puzzle”

This, now, is straightforward and business-like: A. applied to B. for a loan of $100. B. replied, ‘My dear A., nothing would please me more than to oblige you, and I’ll do it. I haven’t $100 by me, but make a note and I’ll indorse it, and you can get the money from the bank.’ A. proceeded to write the note. ‘Stay,’ said B., ‘make it $200. I want $100 myself.’ A. did so, B. indorsed the paper, the bank discounted it, and the money was divided. When the note became due, B. was in California, and A. had to meet the payment. What he is unable to cipher out is whether he borrowed $100 of B., or B. borrowed $100 of him.

– Henry C. Percy, Our Cashier’s Scrap-Book, 1879

The Comma Strike

In September 1905, printers in Ivan Sytin’s Moscow publishing house went on strike, demanding pay for punctuation marks. Discontented workers in other trades and other cities soon joined them in sympathy: bakers, railroad workers, lawyers, bankers, even the Imperial Ballet. Without the railroad, steel and textile mills were forced to shut down; soon nearly the entire adult population of Petrograd had ceased work. The general strike led Tsar Nicholas II to issue the October Manifesto, granting a constitution to Russia for the first time in its history. Thus, wrote Trotsky, “a strike which started over punctuation marks ended by felling absolutism.”

A somewhat related story, from David Kahn’s The Codebreakers: In June 1887 Philadelphia wool dealer Frank J. Primrose sent his agent William B. Toland west, ordering him to buy 50,000 pounds of wool in Kansas and Colorado and await further instructions. The two corresponded by telegram using phrase codes like these to shorten the messages.

On June 16 Primrose planned to send the message Yours of the 15th received; am exceedingly busy; I have bought all kinds, 500,000 pounds; perhaps we have sold half of it; wire when you do anything; send samples immediately, promptly of purchases. Shortened with phrase codes this read DESPOT AM EXCEEDINGLY BUSY BAY ALL KINDS QUO PERHAPS BRACKEN HALF OF IT MINCE MOMENT PROMPTLY OF PURCHASES.

Unfortunately, somewhere between Brookville and Ellis, Kansas, someone added a dot, converting BAY into BUY. Consequently Toland bought 300,000 pounds of wool. Primrose lost more than $20,000 in settling with the sellers and sued Western Union, but the Supreme Court ruled against him on a technicality (he had declined to have his message read back to him). He collected only the cost of the telegram, $1.15.

(Thanks, Folkard.)

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