A Bimagic Queen’s Tour

walkington semi-bimagic queen's tour
Image: William Walkington (CC BY-NC-SA 4.0)

A queen’s tour is the record of a chess queen’s journey around an empty board in which she visits each of the squares once. If the squares are numbered by the order in which she visits them, then the resulting square is magic if the numbers in each rank and file sum to the same total. It’s bimagic if the squares of these numbers also produce a consistent total.

William Walkington has just found the first bimagic queen’s tour, which also appears to be the first bimagic tour of any chess piece. (William Roxby Beverley published the first magic knight’s tour in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science in 1848.)

Note that here the long diagonals don’t produce the magic sum, as they would in a magic square. This constraint is normally dropped in a magic tour — in fact, of the 140 magic knight tours possible on an 8×8 board, none have two long magic diagonals, and no bimagic queen’s tour with qualifying diagonals is possible on such a board either.

More details, and an interesting description of the search, are on William’s blog. He has been told that a complete list of such bimagic queen’s tours is within reach of a computer search, though the field is daunting — there are more than 1.7 billion essentially different semi-bimagic squares possible on an 8×8 board, and each allows more than 400 million permutations.

For What It’s Worth

In 1882 Anton Chekhov published eight “Questions Posed by a Mad Mathematician.” Here are the first three:

  1. I was chased by 30 dogs, 7 of which were white, 8 gray, and the rest black. Which of my legs was bitten, the right or the left?
  2. Ptolemy was born in the year 223 A.D. and died after reaching the age of eighty-four. Half his life he spent traveling, and a third, having fun. What is the price of a pound of nails, and was Ptolemy married?
  3. On New Year’s Eve, 200 people were thrown out of the Bolshoi Theater’s costume ball for brawling. If the brawlers numbered 200, then what was the number of guests who were drunk, slightly drunk, swearing, and those trying but not managing to brawl?

The full list appears in The Undiscovered Chekhov, translated by Peter Constantine (1998). No answers are provided.


At the 1961 Solvay conference on physics, Abdus Salam overheard this conversation between Richard Feynman and Paul Dirac:

Feynman extended his hand towards Dirac and said: ‘I am Feynman.’ It was clear from his tone that it was the first time they were meeting. Dirac extended his hand and said: ‘I am Dirac.’ There was silence, which from Feynman was rather remarkable. Then Feynman, like a schoolboy in the presence of a Master, said to Dirac: ‘It must have felt good to have invented that equation.’ And Dirac said: ‘But that was a long time ago.’ Silence again. To break this, Dirac asked Feynman: ‘What are you yourself working on?’ Feynman said: ‘Meson theories’ and Dirac said: ‘Are you trying to invent a similar equation?’ Feynman said: ‘That would be very difficult.’ And Dirac, in an anxious voice, said: ‘But one must try.’

“At that point the conversation finished because the meeting had started.”

(Abdus Salam, “Physics and the Excellences of the Life It Brings,” in Ideals and Realities: Selected Essays of Abdus Salam, 1987.)

All Aboard

Image: Chemistry Blog

This is brilliant: University of Hull chemist Mark Lorch has combined the periodic table with London’s classic Tube map to create an Underground Map of the Elements.

“My son loves trains. So I came up with a train related twist to an inspection of the periodic table. We sat and cut up a copy of the table and then rearranged each element as a ‘station’ on an underground rail system. Each line represents a characteristic shared by the elements on that line.”

More details at the Guardian and at Chemistry Blog.

In a Nutshell


“The difference between the amoeba and Einstein is that, although both make use of the method of trial and error elimination, the amoeba dislikes erring while Einstein is intrigued by it.” — Karl Popper, Objective Knowledge: An Evolutionary Approach, 1972

Neuberg’s Theorem

Image: Wikimedia Commons

Construct squares outwardly on the sides of triangle ABC, and make a triangle of their centers. Now the centers of squares constructed inwardly on the sides of that triangle will fall on the midpoints of the sides of ABC.

(Due to Luxembourger mathematician Joseph Neuberg, 1840-1926.)

The Last Banana

A thought experiment in probability by Leonardo Barichello: Two people are stranded on an island with only one banana to eat. To decide who gets it, they agree to play a game. Each of them will roll a fair 6-sided die. If the largest number rolled is a 1, 2, 3, or 4, then Player 1 gets the banana. If the largest number rolled is a 5 or 6, then Player 2 gets it. Which player has the better chance?

Click for Answer

Counting Sheep

Image: Flickr

Shepherds in Northern England used to tally their flocks using a base-20 numbering system. They’d count a score of sheep using the words:

Yan, tan, tether, mether, pip,
Azer, sayzer, acka, konta, dick,
Yanna-dick, tanna-dick, tethera-dick, methera-dick, bumfit,
Yanna-bum, tanna-bum, tethera-bum, methera-bum, jigget

… and then denote the completion of a group by taking up a stone or marking the ground before commencing the next count.

These systems vary by region — Wikipedia has them laid out in pleasing tables.

(Thanks, Brieuc.)

Finger Numerals


Writing in the north of England in the early 8th century, the Venerable Bede described a Roman system of finger counting:

1 = the little finger bent at the middle joint
2 = the ring and little fingers bent at the middle joints
3 = the middle, ring, and little fingers bent at the middle joints
4 = the middle and ring fingers bent at the middle joints
5 = the middle finger only bent at the middle joint
6 = the ring finger bent at the middle joint
7 = the little finger closed on the palm
8 = the ring and little fingers closed on the palm
9 = the middle, ring, and little fingers closed on the palm
10 = the tip of the index finger touching the middle joint of the thumb
11 to 19 = the actions denoting each numeral from 1 to 9 plus that of 10
20 = the thumb tucked between the index and middle fingers, so that the thumbnail touches the middle joint of the index finger
21 to 29 = the actions denoting each numeral from 1 to 9 plus that of 20
30 = the tips of the thumb and index finger touching and forming a circle or ring
40 = the thumb and index finger standing erect and close together
50 = the thumb bent at both joints and held against the palm
60 = the index finger closed over the thumb
70 = the first joint of the index finger resting over the first joint of the thumb, which is held nearly straight
80 = the tip of the index finger resting on the first joint of the thumb
90 = the thumb bent over the first joint of the index finger

The signs for 100, 200, 300, and so on are the same as 10, 20, 30, but made by the right hand; and the signs for 1,000, 2,000, 3,000 and so on are the same as 1, 2, 3 but made by the right hand. “To add two numbers, one simply signed the first, then made the mental arithmetical calculation and reproduced the gesture corresponding with the correct sum,” writes Angus Trumble in The Finger: A Handbook (2010). “The process was cumulative; to add a further number to the sum of the first two, you proceeded to represent the gesture corresponding with the new total, and so on. Likewise, the task of subtraction merely threw the whole system into reverse. It was perfectly clear to anyone observing you carry out these separate procedures whether the job in hand was one of addition or subtraction.”

Trumble says that at the end of the 19th century Wallachian peasants were discovered to have preserved a few methods of digital multiplication and division that had been preserved throughout the Roman empire. Here’s one.



Choose one of these cards and fix it clearly in your mind. Then open the answer box.

Click for Answer