Paradoxical Puzzle Pair

Two interesting puzzles, one of which looks hard and is in fact easy, while the other looks easy and is in fact hard.

1. Three Cards

The values attached to a deck of bridge cards start with the Two of Clubs as lowest, with Diamonds, Hearts and Ace of Spades as highest.

If you draw three cards at random from the deck, what is the probability that they will be drawn in order of increasing value? (Answer 1)

2. The Hungry Hunter

A hunter, having run out of food, met two shepherds. One of the shepherd had three loaves of bread and the other had five loaves. When the hunter asked for food, the shepherds agreed to divide the eight identical loaves equally between the three of them. The hunter thanked them and gave them $8. How should the shepherds divide the money? (Answer 2)

• Puzzles and answers from Erwin Brecher’s How Do You Survive a Duel? And Other Mathematical Diversions, Puzzles and Brainteasers (Carlton Books 2018)










Answer #1: The puzzle sounds far more complicated than it is. The deck of cards is a red herring. The question reduces to this: Take three cards, say 2, 3 and 4 of clubs, facedown. What is the probability of turning them over in the order 2, 3, 4? There are six possible ways of arranging three cards. Therefore the probability is one-sixth.










Answer #2: It would be wrong to split the money into $3 and $5. Each of the three ended up with 2⅔ loaves. In other words, the first shepherd parted with ⅓ of a loaf and the other shepherd with 2⅓ or 7/3 loaves. The first shepherd should therefore get $1 and the second shepherd $7.

Agnathous Analysis

In Mandibular Metamorphosis, I looked at two distinct fractals and how you could turn one into the other in one smooth sweep. The Sierpiński triangle was one of the fractals:

Sierpiński triangle

The T-square fractal was the other:

T-square fractal (or part thereof)

And here they are turning into each other:

Sierpiński ↔ T-square (anim)
(Open in new window if distorted)

But what exactly is going on? To answer that, you need to see how the two fractals are created. Here are the stages for one way of constructing the Sierpiński triangle:

Sierpiński triangle #1

Sierpiński triangle #2

Sierpiński triangle #3

Sierpiński triangle #4

Sierpiński triangle #5

Sierpiński triangle #6

Sierpiński triangle #7

Sierpiński triangle #8

Sierpiński triangle #9

When you take away all the construction lines, you’re left with a simple Sierpiński triangle:

Constructing a Sierpiński triangle (anim)

Now here’s the construction of a T-square fractal:

T-square fractal #1

T-square fractal #2

T-square fractal #3

T-square fractal #4

T-square fractal #5

T-square fractal #6

T-square fractal #7

T-square fractal #8

T-square fractal #9

Take away the construction lines and you’re left with a simple T-square fractal:

T-square fractal

Constructing a T-square fractal (anim)

And now it’s easy to see how one turns into the other:

Sierpiński → T-square #1

Sierpiński → T-square #2

Sierpiński → T-square #3

Sierpiński → T-square #4

Sierpiński → T-square #5

Sierpiński → T-square #6

Sierpiński → T-square #7

Sierpiński → T-square #8

Sierpiński → T-square #9

Sierpiński → T-square #10

Sierpiński → T-square #11

Sierpiński → T-square #12

Sierpiński → T-square #13

Sierpiński ↔ T-square (anim)
(Open in new window if distorted)

Post-Performative Post-Scriptum

Mandibular Metamorphosis also looked at a third fractal, the mandibles or jaws fractal. Because I haven’t included the jaws fractal in this analysis, the analysis is therefore agnathous, from Ancient Greek ἀ-, a-, “without”, + γνάθ-, gnath-, “jaw”.

Sins of the Sesh


Dî magni, salaputium disertum* — CAT[ullus]. Lib. LIII.

AT the Session of Poets held lately in London,
   The Bard of Freshwater was voted the chair:
With his tresses unbrush’d, and his shirt-collar undone,
   He loll’d at his ease like a good-humour’d Bear;
“Come, boys,” he exclaimed, “we’ll be merry together!”
   And lit up his pipe with a smile on his cheek;
While with eye like a skipper’s cock’d up at the weather,
   Sat the Vice-Chairman Browning, thinking in Greek.

The company gather’d embraced great and small bards,
   Both strong bards and weak bards, funny and grave,
Fat bards and lean bards, little and tall bards,
   Bards who wear whiskers, and others who shave.
Of books, men, and things, was the bards’ conversation
   Some praised Ecce Homo, some deemed it so-so —
And then there was talk of the state of the nation,
   And when the unwash’d would devour Mr. Lowe.

Right stately sat Arnold — his black gown adjusted
   Genteelly, his Rhine wine deliciously iced, —
With puddingish England serenely disgusted,
   And looking in vain (in the mirror) for “Geist.”
He heark’d to the Chairman, with “Surely!” and “Really?”
   Aghast at both collar and cutty of clay, —
Then felt in his pocket, and breath’d again freely,
   On touching the leaves of his own classic play.

Close at hand lingered Lytton, whose Icarus-winglets
   Had often betrayed him in regions of rhyme —
How glitter’d the eye underneath his grey ringlets,
   A hunger within it unlessened by time!
Remoter sat Bailey — satirical, surly —
   Who studied the language of Goethe too soon,
Who sang himself hoarse to the stars very early,
   And crack’d a weak voice with too lofty a tune.

How name all that wonderful company over —
   Prim Patmore, mild Alford — and Kingsley also?
Among the small sparks who was realler than Lover?
   Among misses, who sweeter than Miss Ingelow?
There sat, looking moony, conceited, and narrow,
   Buchanan, — who, finding when foolish and young,
Apollo asleep on a coster-girl’s barrow,
   Straight dragged him away to see somebody hung.

What was said? what was done? was there prosing or rhyming?
   Was nothing noteworthy in deed or in word?
Why, just as the hour for the supper was chiming,
   The only event of the evening occurred.
Up jumped, with his neck stretching out like a gander,
   Master Swinburne, and squeal’d, glaring out through his hair,
“All Virtue is bosh! Hallelujah for Landor!
   I disbelieve wholly in everything! — there!”

With language so awful he dared then to treat ’em, —
   Miss Ingelow fainted in Tennyson’s arms,
Poor Arnold rush’d out, crying “Sæcl’ inficetum!”
   And great bards and small bards were full of alarms;
Till Tennyson, flaming and red as a gipsy,
   Struck his fist on the table and uttered a shout:
“To the door with the boy! Call a cab! He is tipsy!”
   And they carried the naughty young gentleman out.

After that, all the pleasanter talking was done there
   Whoever had known such an insult before?
The Chairman tried hard to re-kindle the fun there,
   But the Muses were shocked, and the pleasure was o’er.
Then “Ah!” cried the Chairman, “this teaches me knowledge,
   The future shall find me more wise, by the powers!
This comes of assigning to younkers from college
   Too early a place in such meetings as ours!”

CALIBAN, The Spectator, September 15, 1866

*Dî magni, salaputium disertum = “Great gods, an eloquent mannikin!”
†”The Bard of Freshwater” is Tennyson, who lived at Freshwater on the Isle of Wight
‡”Sæcl’ inficetum!” = “Uncouth age!”

Caliban was Robert Buchanan (1841-1901), later the author of “The Fleshly School of Poetry”, an attack on immorality and sensuality in the poetry of Swinburne and Rossetti.

Mandibular Metamorphosis

Here’s the famous Sierpiński triangle:

Sierpiński triangle

And here’s the less famous T-square fractal:

T-square fractal (or part of it, at least)

How do you get from one to the other? Very easily, as it happens:

From Sierpiński triangle to T-square (and back again) (animated)
(Open in new window if distorted)

Now, here are the Sierpiński triangle, the T-square fractal and what I call the mandibles or jaws fractal:

Sierpiński triangle

T-square fractal

Mandibles / Jaws fractal

How do you cycle between them? Again, very easily:

From Sierpiński triangle to T-square to Mandibles (and back again) (animated)
(Open in new window if distorted)

Toxic Turntable #19

Currently listening…

• Robin Westerhouse Quintet, Tripolis (2009)
• Lux Harmonica, Siever of the River (1988)
• Rudyard Kirby, Pudd’nhead Rash (2011)
• Symimi, Speculum Deæ (2007)
• Oil Koir, Bordergland (1983)
• Shiggiv, Shiggiv Lives (2014)
• Edenic Leap, Toilers in the Heat of the Day (2008)
• Joe Keef, Photophilic Ossifrage (2018)
• Olga Mjuko, Cocotte (Remixes) (2019)
• Ahmrikkimlua / Uusuut-Iagoi, Unitary / Half-Torus (split EP) (2009)
• Nzvoh, Whispers from the Willows (1964)
• 97-Jeulo, Hast Ðu? (1996)
• Aiden Snail, Trafalgar (1982)
• Os Dolocos, Coronemus Rosis (2014)
• Gottzunge, Musikkimus (1968)
• Papa Rupu, Ibunun Vavolos (1974)
• Newth, We • Are • Newth (1997)
• Johann Baptist Cramer, Piano Concerti (2002)
• Shake the Snake, Hakúr Mā Qú (1980)

Previously pre-posted:

Toxic Turntable #1#2#3#4#5#6#7#8#9#10#11#12#13#14#15#16#17#18