Warnsdorff’s Rule

The knight’s tour is a familiar task in chess: On a bare board, find a path by which a knight visits each of the 64 squares exactly once. There are many solutions, but finding them by hand can be tricky — the knight tends to get stuck in a backwater, surrounding by squares that it’s already visited. In 1823 H.C. von Warnsdorff suggested a simple rule: Always move the knight to a square from which it will have the fewest available subsequent moves.

This turns out to be remarkably effective: It produces a successful tour more than 85% of the time on boards smaller than 50×50, and more than 50% of the time on boards smaller than 100×100. (Strangely, on a 7×7 board its success rate drops to 75%; see this paper.) The video above shows a successful tour on a standard chessboard; here’s another on a 14×14 board:

While we’re at it: British puzzle expert Henry Dudeney once set himself the task of devising a complete knight’s tour of a cube each of whose sides is a chessboard. He came up with this:


If you cut out the figure, fold it into a cube and fasten it using the tabs provided, you’ll have a map of the knight’s path. It can start anywhere and make its way around the whole cube, visiting each of the 364 squares once and returning to its starting point.

Dudeney also came up with this puzzle. The square below contains 36 letters. Exchange each letter once with a letter that’s connected with it by a knight’s move so that you produce a word square — a square whose first row and first column comprise the same six-letter word, as do the second row and second column, and so on.

dudeney knight's tour word square puzzle

So, for example, starting with the top row you might exchange T with E, O with R, A with M, and so on. “A little thought will greatly simplify the task,” Dudeney writes. “Thus, as there is only one O, one L, and one N, these must clearly be transferred to the diagonal from the top left-hand corner to the bottom right-hand corner. Then, as the letters in the first row must be the same as in the first file, in the second row as in the second file, and so on, you are generally limited in your choice of making a pair. The puzzle can therefore be solved in a very few minutes.”

Click for Answer

In a Word

n. one who is ignorant

n. a stupid person

n. a mistake due to ignorance

adj. that does not think

adj. lacking wit or sense

n. gross ignorance or stupidity

adj. knowing little; ignorant

n. an opponent of sloth or stupidity

Different Strokes


G.H. Hardy had a famous distaste for applied mathematics, but he made an exception in 1945 with an observation about golf. Conventional wisdom holds that consistency produces better results in stroke play (where strokes are counted for a full round of 18 holes) than in match play (where each hole is a separate contest). So if two players complete a full round with the same total number of strokes, then the more erratic player should do better if they compete hole by hole.

Hardy argues that the opposite is true. Imagine a course on which every hole is par 4. Player A is so deadly reliable that he shoots par on every hole. Player B has some chance x of hitting a “supershot,” which saves a stroke, and the same chance of hitting a “subshot,” losing a stroke. Otherwise he shoots par. Both players will average par and will be equal over a series of full rounds of golf, but the conventional wisdom says that B’s erratic play should give him an advantage if they play each hole as a separate contest.

Hardy’s insight is that the presence of the hole limits a run of good luck, while there’s no such limit on a run of bad luck. “To do a three, B must produce a supershot at one of his first three strokes, while he will take a five if he makes a subshot at one of his first four. He will thus have a net expectation 4x – 3x of loss on the hole, and should lose the match, contrary to common expectation.”

In general he finds that B’s chance of winning a hole is 3x – 9x2 + 10x3, and his chance of losing is 4x – 18x2 + 40x3 – 35x4, so that there’s a balance f(x) = x – 9x2 + 30x3 – 35x4 against him. If x < 0.37 -- that is, in all realistic cases -- the erratic player should lose.

"If experience points the other way -- and I cannot deny it, since I am no golfer -- what is the explanation? I asked Mr. Bernard Darwin, who should be as good a judge as one could find, and he put his finger at once on a likely flaw in the model. To play a 'subshot' is to give yourself an opportunity of a 'supershot' which a more mechanical player would miss: if you get into a bunker you have an opportunity of recovering without loss, and one which you are naturally keyed up to take. Thus the less mechanical player's chance of a supershot is to some extent automatically increased. How far this may resolve the paradox, if it is one, I cannot say, and changes in the model make it unpleasantly complex."

(G.H. Hardy, "A Mathematical Theorem About Golf," Mathematical Gazette, December 1945.)


“The monuments of wit survive the monuments of power.” — Francis Bacon

Bug Hunt

terletzky patent

Frustrated in catching insects in 1904, Max Terletzky hit on this rather alarming solution. A basket with an open mouth is attached to the business end of a feathered arrow; the prospective bug hunter props open the basket’s mouth, stalks his prey, and fires at it using a bow. The arrow is attached to a cord in the archer’s hand, which closes the basket doors when the arrow has intercepted the bug and reached the limit of its flight. At that point the arrow drops to the ground and the archer can draw in the cord and claim his prize.

Terletzky writes, “This particular construction of the automatic device for closing the doors of the basket is extremely strong, simple, and durable in construction, as well as thoroughly efficient in operation.” For all I know he’s right.

A Puzzling Exit


Canadian doctor Samuel Bean created a curious tombstone for his first two wives, Henrietta and Susanna, who died in succession in the 1860s and are buried side by side in Rushes Cemetery near Crosshill, Wellesley Township, Ontario. The original stone weathered badly and was replaced with this durable granite replica in 1982. What does it say?

Click for Answer

Asking Directions

You’re a logician who wants to know which of two roads leads to a village. Standing nearby, inevitably, are three natives: one always lies, one always tells the truth, and one answers randomly. You don’t know which is which, and you can ask only two yes-or-no questions, each directed to a single native. How can you get the information you need?

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The Thunder Stone

In Saint Petersburg, an equestrian statue of Peter the Great stands atop an enormous pedestal of granite. The statue was conceived by French sculptor Étienne Maurice Falconet, who envisioned the horse rearing at the edge of a great cliff under Peter’s restraining hand.

Casting the horse and rider was relatively easy; harder was finding a portable cliff. In September 1768 a peasant led authorities to an enormous boulder half-buried near the village of Konnaia, four miles north of the Gulf of Finland and about 13 miles from the center of Saint Petersburg. Falconet proposed cutting it into pieces, but Catherine the Great, who wanted to show off Russia’s technological potential, ordered it moved whole, “first by land and then by water.”

Incredibly, she got her wish. The unearthed boulder measured 42 feet long, 27 feet wide, and 21 feet high; even when trimmed by a third it weighed an estimated 3 million pounds. But it was mounted on a chassis and rolled along atop large copper ball bearings, a “mountain on eggs,” as stonecutters worked continuously to shape it. When they reached the Gulf of Finland it was transferred precariously to a barge mounted between two cutters of the imperial navy, which carried it carefully to the pier at Senate Square, where it was installed in 1770, after two years of work. The finished pedestal stands 21 feet tall.

“The daring of this enterprise has no parallel among the Egyptians and the Romans,” marveled the Journal Encyclopédique; the English traveler John Carr said that the feat astonished “every beholder with a stupendous evidence of toil and enterprise, unparalleled since the subversion of the Roman empire.” It remains the largest stone ever moved by man.


A Dark Day


On Oct. 21, 1966, an avalanche of mining debris descended into the Welsh mining village of Aberfan, filling the classrooms of a local junior school with mud and killing 144 people, 116 of them children. In response to a subsequent newspaper appeal, Shrewsbury psychiatrist J.C. Barker received 76 letters from people who claimed to have had precognition of the event. Of these, 22 were supported by witnesses. This account, by the parents of 10-year-old Eryl Mai Jones, was compiled by a local minister and signed by them as correct:

She was an attractive dependable child, not given to imagination. A fortnight before the disaster she said to her mother, who at the time was putting some money aside for her, ‘Mummy, I’m not afraid to die.’ Her mother replied, ‘Why do you talk of dying, and you so young; do you want a lollipop?’ ‘No,’ she said, ‘But I shall be with Peter and June’ (schoolmates). The day before the disaster she said to her mother, ‘Mummy, let me tell you about my dream last night.’ Her mother answered gently, ‘Darling, I’ve no time now. Tell me again later.’ The child replied, ‘No, Mummy, you must listen. I dreamt I went to school and there was no school there. Something black had come down all over it.’ Her mother answered, ‘You mustn’t have chips for supper for a bit.’ The next day off to school went her daughter as happy as ever. In the communal grave she was buried with Peter on one side and June on the other.

“This last point may not, however, be significant, since the order of burial was apparently influenced by parents’ requests.”

(From the Oxford Book of the Supernatural.)

Five of a Kind

kordemsky multigrade

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