# Cavalry

This is beautifully simple: The strategy game Jeson Mor, from Mongolia, is essentially a chess variant played on a 9×9 board. Each player gets nine knights, arranged as shown, and the winner is the first player who can occupy the central square and then leave that square on a subsequent turn. (Alternatively you can win by capturing all the opponent’s pieces.)

That’s it. It sounds straightforward, but with good play it becomes a delicate dance in which each side tries to prepare an attack of his own while compromising his opponent’s, and because the knights are short-range pieces, it tends to create a complex scrimmage in which an unexpected move will win the day.

You can play it online here; here’s a video of two new players trying it out.

# The Seven-Spot Problem

How can you arrange seven points in a plane so that if any three are chosen, at least two of them will be a unit distance apart?

The solution is shown here — it’s known as a Moser spindle.

# Lightning Strikes

In 1998, the BBC reported that four card players at a whist club in Bucklesham, Suffolk, had each been dealt one suit from a shuffled deck. Hilda Golding found herself holding 13 clubs, Hazel Ruffles held 13 diamonds, and Alison Chivers held 13 hearts (and won, as this was trumps). The dummy hand, face down on the table, held 13 spades.

Though there were 55 people in the village hall at the time, some of whom claimed to have witnessed the event, it’s vastly more likely that this was some misunderstanding or a false report. In 1939 Horace Norton of University College London calculated the odds of such a deal arising naturally to be 1 in 2,235,197,406,895,366,368,301,599,999.

In The Mathematics of Games (2013), John D. Beasley writes, “A typical evening’s bridge comprises perhaps twenty deals, so a once-a-week player must play for over one hundred million years to have an even chance of receiving a thirteen-card suit. If ten million players are active once a week, a hand containing a thirteen-card suit may be expected about once every fifteen years, but it is still extremely unlikely that a genuine deal will produce four such hands.”

(Martin Gardner points out somewhere that anecdotal reports of four perfect hands are strangely more frequent than reports of two perfect hands, which is more likely — though, I guess, less newsworthy.)

(Via Martin Cohen, 101 Philosophy Problems, 2002.)

# Love Triangle

Male side-blotched lizards compete for mates using a three-sided strategy that resembles a game of rock-paper-scissors. Orange-throated males, the strongest, don’t form strong pair bonds but establish large territories and fight blue-throated males outright for females. The blue-throated males, middle-sized, are less aggressive and tend to pair strongly with individual females. Yellow-throated males, the smallest, have a coloration that resembles that of females; this allows them to approach females in the territories of orange-throated males — though this won’t work with females that have formed strong pair bonds with blue-throated males.

So, broadly speaking, orange beats blue, blue beats yellow, and yellow beats orange, an equilibrium of sorts in which each variety has an advantage over another but not over the third.

# Harmony

The Treaty of Versailles contained an odd provision: It established a standard pitch to which orchestras could tune. For hundreds of years the agreed pitch might vary widely from one region to the next. When it became clear that the average pitch was rising over time, a French commission settled that the standard pitch for the A above middle C would be 435 hertz, and an 1885 convention of European nations adopted that as an international standard. The Treaty of Versailles in 1919 ratified that decision.

It would change again with time and technology. In the 20th century American musicians came to prefer 440 hertz, and that came to be adopted as the new standard. Today the standard tuning frequency is set by the International Organization for Standardization: ISO 16 “specifies the frequency for the note A in the treble stave and shall be 440 hertz.”

# Van Schooten’s Theorem

A pleasing little theorem by Dutch mathematician Frans van Schooten:

Inscribe equilateral triangle ABC in a circle. Now, from a point P on that circle, the length of the longest of segments PA, PB, PC equals the sum of the lengths of the other two segments (in this example, the length of segment PA equals the sum of the lengths of PB and PC).

# The National Razor

Last words at the guillotine, collected by Daniel Gerould in Guillotine: Its Legend and Lore (1992):

• The Comte de Sillery, who was lame, had trouble climbing the steps. When executioner Charles-Henri Sanson told him to hurry, he said, “Can’t you wait a minute? After all, it is I who am going to die. You have plenty of time.”
• As he neared the scaffold, someone suggested to astronomer Jean Sylvain Bailly that he put on a coat. “What’s the matter?” he asked. “Are you afraid I might catch cold?”
• A man named Vigié sang the “Marseillaise” at the top of his lungs as he ascended the steps and continued until the blade fell.
• When an assistant moved to remove his boots, Philippe Égalité suggested, “They’ll be much easier to remove afterward.”
• The Duc de Châtelet attempted suicide by cutting his veins with a piece of broken glass and had to be carried to the tumbril. When Sanson offered to dress his wounds, he said, “Don’t bother, I will be losing the rest of it just now.”
• Journalist Jean-Louis Carra told the executioner, “It annoys me to die. I should have liked to see what follows.”
• General Baron de Biron was executed on the last day of the year. He said, “I will soon arrive in the next world — just in time to wish all my friends there a happy new year!”
• Chrétien Malesherbes asked leave to finish winding his watch before Sanson began his duties.
• When the executioner told Giuseppe Fieschi to put on his coat to keep from shivering, he said, “I shall be a lot colder when they bury me.”
• Georges Danton told the executioner, “Show my head to the people. It’s worth looking at!”

Catching sight of the statue of liberty opposite the scaffold, Madame Roland cried, “Oh, Liberty, what crimes are committed in thy name!”

# Extra-ordinary Magic

From Lee Sallows:

A recent contribution to Futility Closet showed an atypical type of 3×3 geometric magic square in which the 4 pieces occupying each of its nine 2×2 subsquares are able to tile the same rectangle. A different square with the same property is seen in the figure here shown, where the nine tiled rectangles appear at right.

As in the earlier example, the square is to be interpreted as if drawn on a torus, the relations among its peripheral cells then being the same as those that result if the square is surrounded with copies of itself, as seen in the following figure showing four such copies, one in each quadrant:

The figure makes it easier to identify the different 2 × 2 subsquares, exactly nine distinct examples of which can be identified. A brief commentary on the square pointed out that the number of ‘magic’ conditions it satisfies is one greater than the eight conditions demanded by a conventional 3 × 3 magic square. Hence the title of the piece, ‘Extra Magic.’

It was while perusing this diagram that an alternative division of the cells into sets of 4 suggested itself. Instead of 2 × 2 subsquares, consider the four cells defined by a cross that can be centered on any chosen cell. The above figure shows a yellow-shaded example, along with a rectangle tiled by its four associated shapes. It is interesting to note that, as before, there are just nine distinct crosses of this kind to be found in a 3 × 3 square. An obvious question thereby prompted was whether or not a new 3 × 3 magic square could be found based upon such crosses rather than 2×2 subsquares? The answer turned out to be yes, but in the process of scrutinizing an initial specimen I noticed that although it embodied nine cross-based sets of 4 rectangle-tiling pieces, as required, it also included a couple of additional rectangle-tiling sets contained within 2 × 2 subsquares. Clearly the maximum number of such surplus sets would be nine, one for each cross, but could a specimen showing nine cross-based and nine subsquare-based rectangle-tiling sets really exist? I lost no time in seeking an answer.

Regrettably, I was unable to find one. However, the figure below shows a close approach to perfection. It is the same 3 × 3 square with which we started, but now shown alongside no less than eight additional rectangles, each of them tiled with a set of 4 pieces belonging to a cross. Note that the missing rectangle is the one belonging to the non-magic central cross, a show of symmetry that seems appropriate.

So whereas a 3 × 3 magic square, numerical or geometric, satisfies at least 8 separate conditions (3 rows + 3 columns + 2 diagonals), the square here shown satisfies no less than eight more.

(Thanks, Lee.)

# Q&A

In 2011, journalist Alex Renton’s 6-year-old daughter Lulu passed him a letter and asked him to see that it reached the addressee:

To God how did you get invented?

From Lulu

He sent the letter to family members, Christian friends, the Scottish Episcopal Church, the Church of Scotland, and the Scottish Catholic Church. None sent a satisfactory reply. Then he sent it to the Anglican Communion and received this response from Rowan Williams, then Archbishop of Canterbury:

Dear Lulu,

‘Dear Lulu – Nobody invented me – but lots of people discovered me and were quite surprised. They discovered me when they looked round at the world and thought it was really beautiful or really mysterious and wondered where it came from. They discovered me when they were very very quiet on their own and felt a sort of peace and love they hadn’t expected.

Then they invented ideas about me – some of them sensible and some of them not very sensible. From time to time I sent them some hints – specially in the life of Jesus – to help them get closer to what I’m really like.

But there was nothing and nobody around before me to invent me. Rather like somebody who writes a story in a book, I started making up the story of the world and eventually invented human beings like you who could ask me awkward questions!’

And then he’d send you lots of love and sign off. I know he doesn’t usually write letters, so I have to do the best I can on his behalf. Lots of love from me too.

Archbishop Rowan

Renton read it to Lulu. “It went down well,” he wrote later. “What worked particularly was the idea of ‘God’s story.’

“‘Well?’ I asked when we reached the end. ‘What do you think?’ She thought a little. ‘Well, I have very different ideas. But he has a good one.'”

# The Beard Tax

In his efforts to reform Russian society, Peter the Great once resorted to banning beards. To bring Russian society more in line with Western Europe, in 1698 he began to charge a fee for the privilege of wearing whiskers, ranging from 100 rubles a year for wealthy merchants down to 1 kopek for a peasant entering a city. Police were empowered to shave scofflaws forcibly.

If you paid your tax you were given a “beard token” with a Russian eagle on one side and a beard on the other. One coin bore the legend THE BEARD IS A SUPERFLUOUS BURDEN.

Because Russians generally resented the law, the tokens are quite valuable now. As early as 1845 collector Walter Hawkins wrote, “The national aversion to the origin of this token probably caused their destruction or dispersion, after they had served their purpose for the year, as they are now very rarely to be met with even in Russia.”