Getting There

I just stumbled into this — in October 1967, IBM published this problem in Eureka, the journal of the Cambridge University Mathematical Society (page 2):

The triplets (whose abilities at walking, cycling, and donkey riding are identical) always leave home together at the last possible minute and arrive at school together on the last stroke of the bell.

They used to walk the 4 1/2 miles, and so had to set out at 8.00; then they acquired a bicycle and found that they did not have to leave home until 8.15 (Charles rode it for the first 1 1/2 miles, left it, and walked on; Donald walked 1 1/2 miles, cycled 1 1/2 miles, and walked again; Edward walked 3 miles and cycled the rest). More recently they have been given a donkey. After experiments to determine the donkey’s speed and to verify that it stood stock still when left, they found that — using the bicycle and the donkey — they did not need to leave home until 8.25. There were several schemes of changing over which they could use to do this, of course; but naturally they chose a scheme which involved the minimum number of changes. Going to school tomorrow Charles will start on foot and Edward will arrive on foot. How far will Donald walk?

In place of an answer they listed the address of their London office, as an invitation to prospective systems analysts. I can’t see that they ever published a solution to the puzzle; I’m posting it here for what it’s worth.

07/01/2024 UPDATE: Reader Catalin Voinescu supplies the solution:

Riding the bicycle for 1.5 miles takes 15 fewer minutes than walking the same distance, so the bicycle saves 10 minutes per mile (compared to walking).

Another way to look at it is that riding the bicycle 4.5 miles saves three people 15 minutes, or an aggregate of 45 person-minutes (see ‘man-month’, another concept popular at IBM at the time; ‘The Mythical Man-Month’, an excellent book by Frederick P. Brooks, Jr, explores this in detail).

Riding the donkey 4.5 miles saves a total of 30 person-minutes, or 6 2/3 minutes per mile (compared to walking).

It’s not possible for any person not to change means of transportation, because they would arrive too late (walking) or too early (other means of transportation). For the same reason, it’s not possible for each person to change only once, because whoever got the bike and the donkey and did not walk at all would also arrive too early. Thus, the minimum number of changes is four, with one person walking and riding the donkey, another person walking and riding the bike (not necessarily in this order), and the third doing all three.

The person who walks and rides the bike needs to save 25 minutes, so they need to ride the bike for 2 1/2 miles and walk 2 miles. Assume they do it in this order (see below). The person who switches twice needs to ride the bike for the final 2 miles.

The person who walks and rides the donkey needs to save the same 25 minutes, which, at a rate of 6 2/3 minutes saved per mile, means 3 3/4 mile of riding the donkey and 3/4 mile of walking. They have to walk first and then ride the donkey, because the person who switches twice needs to ride the donkey for the remaining 3/4 mile, and they can only do this at the beginning (we already have them biking at the end).

The person who switches twice rides the donkey for 3/4 mile at the beginning, and bikes for 2 miles at the end. In between, they walk the remaining 1 3/4 mile.

So: Edward bikes 2 1/2 miles then walks 2 miles. Donald rides the donkey 3/4 mile, walks 1 3/4 mile, then rides the bike 2 miles. Charles walks 3/4 mile, then rides the donkey for the remaining 3 3/4 miles.

If we assume the person who walks and rides the bike does them in the opposite order, we get a mirrored solution: Edward rides the donkey for 3 3/4 miles and walks 3/4 mile; Donald rides the bike for 2 miles, walks 1 3/4 mile, then rides the donkey 3/4 mile; and Charles walks 2 miles, then picks up the bike and rides it for the remaining 2 1/2 miles.

In both cases, Donald walks 1 3/4 mile.

Other solutions exist, but they require more than four changes. It’s even possible for the three people to each walk, ride the donkey and bike equal distances (1 1/2 mile of each, each), but I don’t know what minimum number of changes that would require (more than six, and not all 1 1/2 mile stretches can be contiguous).

(Thanks, Catalin.)

| Puzzles

Resource

https://commons.wikimedia.org/wiki/File:Rudolf_Karel_Drei_Haare_des_Gro%C3%9Fvaters_Allwissend_Skizze.jpg

Imprisoned at Theresienstadt during World War II, Czech composer Rudolf Karel wrote a five-act opera, Three Hairs of the Wise Old Man, on toilet paper using the medicinal charcoal he’d been prescribed for his dysentery.

The illness eventually claimed his life, but a sketch of the opera was preserved by a friendly warden, and the orchestrations were finished by Karel’s pupil Zbynek Vostrák.

https://commons.wikimedia.org/wiki/File:Rudolf_Karel_Drei_Haare_des_Gro%C3%9Fvaters_Allwissend_Skizze.jpg

| Art · History

Roll Play

https://archive.org/details/LangePhysicalParadoxesAndSophismsScienceForEveryoneMir1987/page/n29/mode/2up

These two handcarts have the same mass. Newton tells us that equal forces applied to equal masses impart equal accelerations. So why does the second handcart pick up speed more quickly than the first? (This is a Soviet problem; Н is the Russian abbreviation for newtons.)

Click for Answer
| Puzzles

In a Word

https://commons.wikimedia.org/wiki/File:The_Chronicle_of_Ioannis_Skylitzis_Bulagar_Defeat.jpg

manqueller
n. a man killer; an executioner

In 1014, after a decisive victory over the Bulgarian Empire at the Battle of Kleidion, Byzantine emperor Basil II followed up with a singularly cruel stroke. He ordered that his 14,000 prisoners be divided into groups of 100; that 99 of each group be blinded; and that the hundredth retain one eye so that he could lead the others home. The columns were then released into the mountains, each man holding on to the belt of the man in front. It’s not known how many were lost on the journey, but when the survivors reached the Bulgar capital, their tsar collapsed at the sight and died of a stroke two days later. Basil is remembered as “the Bulgar slayer.”

| Death · History · Language

Fogbound

From 1995 to 1998, the journal Philosophy and Literature ran a bad writing contest to mark “the most stylistically lamentable passages found in scholarly books and articles.” The final year’s winners:

First prize:

The move from a structuralist account in which capital is understood to structure social relations in relatively homologous ways to a view of hegemony in which power relations are subject to repetition, convergence, and rearticulation brought the question of temporality into the thinking of structure, and marked a shift from a form of Althusserian theory that takes structural totalities as theoretical objects to one in which the insights into the contingent possibility of structure inaugurate a renewed conception of hegemony as bound up with the contingent sites and strategies of the rearticulation of power.

— Judith Butler, “Further Reflections on the Conversations of Our Time,” in the journal Diacritics

Second prize:

If, for a while, the ruse of desire is calculable for the uses of discipline soon the repetition of guilt, justification, pseudo-scientific theories, superstition, spurious authorities, and classifications can be seen as the desperate effort to ‘normalise’ formally the disturbance of a discourse of splitting that violates the rational, enlightened claims of its enunciatory modality.

— Homi K. Bhabha in his book The Location of Culture

Third prize:

As my story is an august tale of fathers and sons, real and imagined, the biography here will fitfully attend to the putative traces in Manet’s work of ‘les noms du père’, a Lacanian romance of the errant paternal phallus (‘Les Non-dupes errent’), a revised Freudian novella of the inferential dynamic of paternity which annihilates (and hence enculturates) through the deferred introduction of the third term of insemination the phenomenologically irreducible dyad of the mother and child.

— Steven Z. Levine in Bradford Collins, ed., Twelve Views of Manet’s Bar

The contest’s only condition was that entries not be ironic: “Deliberate parody cannot be allowed in a field where unintended self-parody is so widespread.” The full list of winners is here.

| Language

Hazard

From an April Fools’ feature in LIFE, April 4, 1938:

https://books.google.com/books?id=40oEAAAAMBAJ&pg=PA6

The photo appeared in a German paper, which claimed the story was American. “To Germans, actual American goings-on are fantastic enough to be April Fools’ tricks.”

| Hoaxes

Andy

https://en.wikipedia.org/wiki/File:Andy_the_goose_and_Gene_Fleming.jpg

When a goose without feet was hatched in Harvard, Nebraska, in 1987, local inventor Gene Fleming adopted the bird, fitted him with baby-sized shoes, and taught him to walk. In time “Andy” learned to swim and fly as well; he gave inspiration to disabled children and received a lifetime supply of shoes from Nike, but it wasn’t to last — in 1991 he was killed by an unnamed perpetrator.

| Oddities

Strain’d

If mercy modifies the demands of justice, then to be merciful is perhaps to be unjust. But manifesting injustice is a vice, not a virtue. This seems to mean that mercy is a vice. A sentencing judge has been hired to enforce the rule of law that society has agreed upon. If he tempers this, even through love or compassion, then arguably he’s departing from his sworn obligation. Angelo says in Measure for Measure:

I show [pity] most of all when I show justice,
For then I pity those I do not know,
Which a dismissed offense would after gall,
And do him right that, answering one foul wrong,
Lives not to act another.

(If we try to claim that mercy is a form of justice, so that every act of mercy is just, then we’re saying that one has a right to mercy, that it’s not a gift. That seems wrong too.)

(Jeffrie G. Murphy and Jean Hampton, Forgiveness and Mercy, 1988, via George W. Rainbolt, “Mercy: An Independent, Imperfect Virtue,” American Philosophical Quarterly 27:2 [April 1990], 169-173.)

| Society

Roommates

A conference is attended by 1,000 delegates from various countries. It’s known that any three delegates can speak together without help, though one of the three may have to serve as interpreter for the other two. Prove that all the attendees can be accommodated in double rooms so that the two occupants of each room can speak to each other.

Click for Answer
| Puzzles