Self-Tiling Tile Sets

These tiles have a remarkable property — by working together, the four can impersonate any one of their number (click to enlarge):

sallows rep-tiles 1

The larger versions could then perform the same trick, and so on. Here’s another set:

sallows rep-tiles 2

In this set, each of the six pieces is paved by some four of them:

sallows rep-tiles 3

By the fathomlessly imaginative Lee Sallows. There’s more in his article “More on Self-Tiling Tile Sets” in last month’s issue of Mathematics Magazine.

Fallout

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

When George’s Grandmamma was told
That George had been as good as Gold,
She Promised in the Afternoon
To buy him an Immense BALLOON.

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

And so she did; but when it came,
It got into the candle flame,
And being of a dangerous sort
Exploded with a loud report!

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

The Lights went out! The Windows broke!
The Room was filled with reeking smoke.
And in the darkness shrieks and yells
Were mingled with Electric Bells,
And falling masonry and groans,
And crunching, as of broken bones,
And dreadful shrieks, when, worst of all,
The House itself began to fall!
It tottered, shuddering to and fro,
Then crashed into the street below —
Which happened to be Savile Row.

When Help arrived, among the Dead

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

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

The Still-Room Maid.
And I am dreadfully afraid
That Monsieur Champignon, the Chef,
Will now be

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

While George, who was in part to blame,
Received, you will regret to hear,
A nasty lump

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

MORAL.
The moral is that little Boys
Should not be given dangerous Toys.

— Hilaire Belloc, Cautionary Tales for Children, 1922

Two Olive Problems

1. A friend gives you a bottle that contains seven olives. Two of them are green and five are black. He bets that if you remove three olives at random from the bottle, they’ll include a green one. Should you take the bet?

2. Agnes has a tin of olives. It originally contained both black and green ones, but someone has been eating them, so she’s not sure of the colors of the 14 olives that remain. She removes 7 at random and finds that they’re all green. If the odds of this happening were exactly 50-50, what are the colors of the remaining 7?

Click for Answer

Railway Mazes

http://puzzlemuseum.com/luppitt/lmb02.htm

In 2000, the residents of Luppitt, East Devon, installed a granite bench decorated with a variety of puzzles and curiosities that “it is hoped will be practical and entertaining for most of the next millennium.”

Among the puzzles is this “railway maze,” contributed by Roger Penrose. Make your way from Start in the upper left to Finish in the lower right. The catch is that your train has no reverse gear — you must move continually forward, following the natural curve of the track and making no sharp turns.

http://demonstrations.wolfram.com/PenrosesRailwayMazes/

Click for Answer

Unquote

http://books.google.co.uk/books?id=lmcMAAAAIAAJ

“Foolish man, what do you bemoan, and what do you fear? Wherever you look there is an end of evils. You see that yawning precipice? It leads to liberty. You see that flood, that river, that well? Liberty houses within them. You see that stunted, parched, and sorry tree? From each branch liberty hangs. Your neck, your throat, your heart are all so many ways of escape from slavery … Do you enquire the road to freedom? You shall find it in every vein in your body.” — Seneca

But:

In the West Indies, according to the Spanish historian Girolamo Benzoni, four thousand men and countless women and children died by jumping from cliffs or by killing each other. He adds that, out of the two million original inhabitants of Haiti, fewer than 150 survived as a result of the suicides and slaughter. In the end the Spaniards, faced with an embarrassing labor shortage, put a stop to the epidemic of suicides by persuading the Indians that they, too, would kill themselves in order to pursue them in the next world with even harsher cruelties.

— Alfred Alvarez, The Savage God: A Study of Suicide, 1971

Podcast Episode 10: A Baboon Soldier, Lighthouse Rescues, and a Parliament of Owls

http://www.samvoa.org/jackie.html

When Albert Marr joined the South African army in 1915, he received permission to bring along his pet baboon, Jackie. In this week’s episode of the Futility Closet podcast we’ll follow Jackie’s adventures in England, Egypt, and Belgium, his work for the Red Cross after the war, and his triumphant return to Pretoria in 1919.

We’ll also meet a Rhode Island lighthouse keeper’s daughter who saved the lives of 18 people over a period of 48 years, and present the next Futility Closet Challenge.

Above is Jackie in Johannesburg in 1919, on his way home. Note the knife and fork. More photos, including one of Jackie saluting, can be found at the website of the South African Military Veterans Organisation of Australasia (see the gallery at the bottom of the page).

Our main source for the segment about Ida Lewis is Lenore Skomal’s 2002 biography The Keeper of Lime Rock. Some images:

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

http://commons.wikimedia.org/wiki/File:Ida_Lewis_001.jpg

One of the soldiers she saved during her fifth rescue, on March 29, 1869, remembered, “When I saw the boat approaching and a woman rowing, I thought, She’s only a woman and she will never reach us. But I soon changed my mind.” Her brother Thomas said, “Ida knows how to handle a boat. She can hold one to wind’ard in a gale better than any man I ever saw, wet an oar, and yes, do it too, when the sea is breaking over her.”

Here’s the lighthouse in 1869, the first year of her fame:

http://commons.wikimedia.org/wiki/Category:Ida_Lewis_Lighthouse#mediaviewer/File:Lime_Rock_Island_in_1869_Harper%27s_Weekly.jpg

At 14 Ida was accounted the best swimmer in Newport, and at 15 she had finished her formal schooling but rowed her siblings to Newport and back each day. Her father said: “Again and again, have I seen the children from the window as they were returning from school in some heavy blow, when Ida alone was with them, and old sailor that I am, I felt that I would not give a penny for their lives, so furious was the storm — yes sir. I have watched them ’til I could not bear to look any longer, expecting every moment to see them swamped and the crew at the mercy of the waves, and then I have turned away and said to my wife — let me know if they get safe in, for I could not endure to see them perish and realize that we were powerless to save them. And oh you cannot tell the relief when she cried out: they have got safe to the rock, Father. It was a mighty weight off my mind, I can assure you. I have seen Ida in the bitter winter weather obliged to cut off her frozen stockings at the knee.”

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at http://feedpress.me/futilitycloset. The show notes are on the blog, where you can also enter your submissions in this week’s Challenge. Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

Intersections

Here’s a way to visualize multiplication that reduces it to simple counting:

multiplication lattice

Express the digits in each factor with rows of parallel lines, as shown, and then count the intersections to derive the product. This is more cumbersome than the traditional method, but its visual nature is appealing, and it permits anyone who can count to reach the right answer even if he doesn’t know the multiplication table.

The example above uses small digits, so no “carrying” is required, but the method does accommodate more complex sums — it’s explained well in this video:

See Two by Two.

(Thanks, Dieter.)

Foursquare

foursquare puzzle

Print out two copies of this pattern, cut them out, and fold each along the dotted lines, making two identical solids. Then fit these two pieces together to make a regular tetrahedron.

This sounds dead simple, but it stumped me for a while. See if you can do it. (There’s no trick — the task is just what it seems.)

Presto

A card trick by Mark Wilson:

wilson card trick

Put your finger on any red card. Move it left or right to the nearest black card. Move it up or down to the nearest red card. Move it diagonally to the nearest black card. Now move it down or to the right to the nearest red card.

You’ll always land on the ace of diamonds.

(From Harold R. Jacobs, Mathematics: A Human Endeavor, 1970.)

Words and Music

The German comedian known as Loriot (Vicco von Bülow) used to perform a narrative version of Camille Saint-Saëns’ The Carnival of the Animals with members of the Berlin Philharmonic Orchestra, using words to convey music. “His style enters the fairy-tale world the composer has portrayed musically,” writes Siglind Bruhn in Musical Ekphrasis (2000). “He sees and hears the orchestra’s depictions from the inside. Here, the verbal medium happily supplements the little details that might otherwise escape the music listener.” Here’s part of Bruhn’s translation:

A wood-ant, no longer in her prime, taps the giant ant-eater in front of her on the shoulder. ‘Excuse me, I cannot see anything if you keep your hat on,’ Grumpily the ant-eater takes off her headdress, an unwieldy contraption braided from wild asparagus and chicken feathers. ‘Thank you!’ says the ant. Then she lets her eyes wander across the jungle clearing. On the arena seats alone she counts 4791 strangely costumed animals, not to mention the innumerable monkeys and birds that are crowding the overburdened treetops.

Just now there is a stir of anticipation, for the moon is ascending from behind the branches of a mango tree to signal the beginning of the festivity. ‘I think I hear something,’ says a pigeon and she isn’t altogether wrong, for over there near the entrance, in the twigs of a bare oak, sixty-four horned owls take up their instruments. And now the marabou raises his baton, the two squirrels at the pianos lower their paws into the keyboards … and then he enters, with all the members of the royal family: His Majesty, the Lion.

Accompanied by moderate applause the lion has ambled twice around the arena, looking rather bored as he waved to the crowd. Together with his spouse, his three sons, one daughter, five cousins, and an imperfectly colored aunt, he has then taken the seats of honor and closed his eyes. …

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