Decisions

A book lover is thinking of buying six books from a group of eight. The price of each book is a whole number of dollars, and none is less than \$2. The prices are such that each possible selection of six books would cost the buyer a different sum. In the end he can’t make up his mind and buys all eight books. What is the smallest amount he must pay?

Each choice of six books from the group of eight leaves two behind. The price of each possible omitted pair must be unique, or else the corresponding sextets would cost the same, which we know is not the case. So we can solve the problem by working out the lowest possible price for each book that ensures that every possible pair has a distinct price.

The three lowest possible prices are \$2, \$3, and \$4. That’s fine so far, but the next book can’t cost \$5, because then we’d have two pairs with the same value (5 + 2 = 3 + 4). So we jump to 6 and see if that works. By looking always for the smallest possible next higher price for each volume, we’ll arrive at 2, 3, 4, 6, 9, 14, 22, 31, which gives a total price of \$91.

But, interestingly, that’s not the answer! After Roland Sprague published this puzzle in his 1963 book Recreation in Mathematics, he found the solution 2, 3, 4, 6, 10, 15, 20, 30, which totals 90. And Fritz Düball later found 2, 3, 4, 6, 10, 16, 21, 26, which totals 88. Is that the lowest sum possible? Sprague doesn’t claim that it is, and I’ve not seen this problem elsewhere than in his book.

06/19/2022 UPDATE: Reader Michael Küll wrote a program to search for all solutions in a reasonable range. There are four solutions with a total amount less than or equal to \$91:

```88 :   2   3   4   6   10   16   21   26
90 :   2   3   4   6   10   15   20   30
91 :   2   3   4   6    9   14   22   31
91 :   2   3   4   6   10   17   22   27
```

So Düball’s solution is indeed the best possible. (Thanks, Michael.)

The Brautigan Library

A reader let me know about this: In Vancouver, Washington, there’s a library for “unwanted” manuscripts — manuscripts that no publisher wanted to publish. The Brautigan Library was inspired by Richard Brautigan’s 1971 novel The Abortion: An Historical Romance 1966, which describes a library for “the unwanted, the lyrical and haunted volumes of American writing.” Authors could place their manuscripts anywhere they liked on the library’s shelves, happy to have them preserved there though no readers could find them.

Inspired by this, in 1990 Todd Lockwood, of Burlington, Vermont, started The Brautigan Library, inviting submissions of unpublished manuscripts and encouraging visitors to read them. Lockwood’s library closed in 2005, but in 2010 its contents were taken from storage and moved to Vancouver, where John Barber, a faculty member at Washington State University, now curates it. It currently contains more than 300 manuscripts, and Barber now accepts electronic submissions. You can browse the catalog here.

The French writer David Foenkinos has written a novel in which a librarian reads Brautigan’s book and decides to create Brautigan’s library as part of the municipal library that he manages in a little town in Brittany. It’s called “Le mystère Henri Pick.”

(Thanks, January.)

Dependent Claws

The first motion picture to feature a live cat is believed to be this 1894 short in which French physiologist Étienne-Jules Marey drops an inverted feline to watch it land on its feet.

When the experiment was published in Nature in 1894, the editors wrote, “The expression of offended dignity shown by the cat at the end of the first series indicates a want of interest in scientific investigation.”

Illumination

In a monastery cloisters on the edge of Venice is a sundial inscribed with the motto Horas non numero nisi serenas.

Literally that means “I don’t count the hours unless they are serene ones” (or “I count only the sunny hours”).

“But it really means, ‘When I come to die, the only moments that matter will have been the moments when I was at ease,'” writes Harry Mount in Amo, Amas, Amat and All That.

Of the motto, William Hazlitt wrote, “There is a softness and a harmony in the words and in the thought unparalleled.”

Back Matter

More entries from unusual indexes:

From Robert Burton’s Anatomy of Melancholy:

Cabbage brings heaviness to the soul, 192
Calis, who would wash in no common water, 397
Fish discommended, 192; defended, 398
Kisses, honest and otherwise, 701 et seq.
Pork, naught for quasy stomachs, 190
Roman courtesans, their elegancy of speech, 699
Spider in a nutshell, medicine for ague, 596
Statues, love in, 649
Venison, a melancholy meat, 190
Verjuice and oatmeal is good for a parrot, 80 note

And from Gilbert White’s 1789 The Natural History and Antiquities of Selborne:

ANNE, Queen, came to Wolmer-forest to see the red deer
August, the most mute month respecting the singing of birds
Castration, its strange effects
Cats, house, strange that they should be so fond of fish
Daws breed in unlikely places
Dispersion of birds, pretty equal, why
Fishes, gold and silver, why very amusing in a glass bowl
Fly, bacon, injurious to the housewife
Hogs, would live, if suffered, to a considerable age
Slugs, very injurious to wheat just come out of the ground, by eating
Worms, earth, no inconsiderable link in the chain of nature, some account of

There’s much more in Hazel K. Bell’s wonderful Indexers and Indexes in Fact & Fiction.

A Peruvian Puzzle

In the Pisco Valley on Peru’s Nazca Plateau is a band of 5,000 to 6,000 holes, each about 1 meter in diameter and 100 centimeters deep. The band, which averages about 20 meters in width, starts at the edge of a valley and extends about 1.5 kilometers up a hill understandably known as Cerro Viruela, or Smallpox Hill. (You can see its extent in Google Earth.)

No one knows who dug the holes or why. They’re circular and lined with stone, resembling pits found elsewhere that serve as resting places for mummies. But these are empty. Possibly they were designed for storage by users of the Inca road system, or possibly they were used to measure quantities of produce owed as tribute to the Inca state. Archaeologists are still investigating.

Two Trains

Two trains set out at 7 a.m., one headed from A to B and the other from B to A. The first reaches its destination in 8 hours, the second in 12. At what hour will the two trains pass one another?

A Little Decorating

This is a miniature, a tiny replica of an English dining room of the late 18th century. Between 1932 and 1940, American artist Narcissa Niblack Thorne made about 100 such diminutive rooms, often enlisting architects, designers, and craftsmen whose talents became available during the Depression.

Of the miniatures still in existence, most are on display at the Art Institute of Chicago, where they serve as documents of the decorating styles of periods past. Generally designed on a scale of one inch per foot, many of them include authentic materials: bowls of real silver, chandeliers of real crystal, even original paintings and sculpture contributed by admired artists.

In 2010, the institute began to decorate them for the holidays each year, after researchers discovered Thorne’s affection for Christmas. But “Some of the rooms will never have a holiday theme,” the museum’s Lindsey Mican Morgan told the Chicago Tribune. “That is because many of them depict a room from a time when holidays were simply not celebrated as they are now.”

Square Meal

In the game Chomp, two players begin with a rectangular grid. The first player chooses any square and removes it from the grid, together with all the squares above and to the right of it. The second player chooses one of the remaining squares and removes that, together with all the squares above it and to its right. The two take turns in this way until one of them is forced to remove the last square. That player loses.

If the starting grid is square, can either player force a win?

The Backward Club

A self-working card curiosity by Shippensburg University mathematician Douglas E. Ensley:

I give you the four aces from a deck of cards and turn my back. Then I ask you to stack the four cards face up with the heart at the bottom, then the club, the diamond, and the spade. Now turn the uppermost card, the spade, face down.

Now you’re invited to perform any of these operations as many times and in any order that you wish:

• Cut any number of cards from the top of the stack to the bottom.
• Turn the top two cards over as one.
• Turn the entire stack over.

When you’ve finished, I ask you to turn the topmost card over, then turn the top two cards over as one, then turn the top three cards over as one. I predict that the club is the only card facing the opposite way from the others, and as long as you’ve followed the directions above, it always will be.

The answer is explained by group theory — see the article below for the details.

(Douglas E. Ensley, “Invariants Under Group Actions to Amaze Your Friends,” Mathematics Magazine 72:5 [December 1999], 383-387.)