# Making the Case

Alexander Shapovalov suggested an unusual coin-weighing problem for the sixth international Kolmogorov math tournament in 2007:

A judge is presented with 80 coins that all look the same, knowing that there are either two or three fake coins among them. All the real coins weigh the same and all the fake coins weigh the same, but the fake coins are lighter than the real ones.

A lawyer knows that there are exactly three fake coins and which ones they are. The lawyer must use a balance scale to convince the judge that there are exactly three fake coins and that it is impossible for there to be only two fake coins. She is bound by her contract not to reveal any information about any particular coin. How should she proceed?

The lawyer might try dividing the 80 coins into three groups of 26, each group containing one fake coin, with two coins left over. With two weighings she could then show that the three groups have the same weight. From this the judge could conclude that either (a) there are 3 fake coins, one in each group, or (b) there are 2 fake coins, both in the leftover group. The lawyer could then weigh one of the leftover coins against a real coin taken from one of the three groups, to show that these balance. This would prove to the judge that there are 3 fake coins (because if there were only 2 then possibility (b) above would be ruled out). However, this strategy is “indiscreet” — it would reveal to the judge the true character of each of the leftover coins, which the lawyer has pledged not to do. How should she proceed instead?

# The Immortal Bridge

This rock formation, formed possibly thousands of years ago, spans a chasm on Mount Tai, one of the Five Great Mountains of China.

It’s said that anyone who crosses the bridge will live forever.

# Endless Variety

In 2022, amateur mathematician David Smith discovered a remarkable tile that will cover an infinite plane but only in a nonperiodic way.

This solves an open problem in mathematics — for years researchers had been seeking an aperiodic monotile, or “einstein,” from the German for “one stone.”

Technically Smith’s tile, known as the “hat,” must be used in combination with its mirror image. But last year his team found another nonperiodic tile, known as the spectre, which is strictly chiral — that is, not only will it tile the plane without its mirror image, but it must be used in that way.

# Nighttide

“Did anyone ever have a boring dream?” — Ralph Hodgson

# Hope Springs Eternal

Ulrich von Cranach of Hamburg devised this perpetual motion machine in 1664, offering it to drive pumps in mines. An Archimedean screw raises a succession of iron balls that then descend by a wheel that turns the screw.

Robert Fludd had proposed a water mill in 1618 that used essentially the same design. The anonymous variant below dates from the Middle Ages.

# Worldly Wise

Proverbs from around the world:

• See that you are wise, but also learn to appear ignorant. (Armenian)
• If men could foresee the future, they would still behave as they do now. (Russian)
• Many have suffered for talking; none ever suffered for keeping silent. (Italian)
• To listen to a lie is harder than to tell it. (Turkish)
• Not your friend but your enemy will tell you who you are. (Greek)
• It is not the fault of the post that a blind man cannot see it. (Sanskrit)
• A good laugh and a long sleep are the best cures in the doctor’s book. (Irish)
• Patience is the door of joy. (German)
• If you are standing upright, do not fear a crooked shadow. (Chinese)
• The thief is sorry that he is to be hanged, not that he is a thief. (English)
• It is the poor who gives alms to the poor. (Japanese)
• Before you marry, think what you are doing. (Spanish)
• Better wisdom than riches. (Swedish)
• Bairns are certain care but no sure joy. (Scottish)
• There are forty kinds of lunacy, but only one kind of common sense. (West African)

And “It is easy to throw a stone into the Danube, but rather difficult to get it out.” (Yugoslavia)

# 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.)

# Resource

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.

# Roll Play

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.)

# In a Word

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.”