A visual proof that the sum of all positive odd numbers up to 2n – 1 is a perfect square.
Math Notes
Author: Greg Ross
In a Word
locodescriptive
adj. describing a particular place or places
hippocrepiform
adj. shaped like a horseshoe
elsewhither
adv. in a different direction
mirific
adj. working wonders; wonderful
The Rochester Institute of Technology contains a portal to another dimension. In 2013, student Michael Lacanilao deliberately set out to create a record of an “Escherian stairwell” on campus that forms a perpetual downward loop.
Lacanilao has long since graduated, but the stairwell is still commemorated on the school’s website.
(Thanks, Colin.)
The Blast Shadow
On the morning of Aug. 6, 1945, a customer was sitting on the steps of Sumitomo Bank in Hiroshima, waiting for the branch to open, when an atomic bomb exploded over the city. The bank was only 260 meters from ground zero, and as the intense heat burned its stone face white, the customer’s body shielded one section of the steps, leaving a “shadow” in that place.
The steps are now preserved in the Hiroshima Peace Memorial Museum.
In 1946, the British mission to Hiroshima and Nagasaki noted that the surfaces of asphalt roads “retained the ‘shadows’ of those who had walked there at the instant of the explosion.” It called them “objects of macabre interest and pilgrimage for visitors.”
The Gingerbread Game
Hansel and Gretl have discovered a gingerbread cottage and are wondering whether to eat some of the tiles on its walls. A witch appears and tells them how they must go about it. “Each of you is to name a whole number between 0 and 100. Hansel’s must be odd and Gretl’s even. No conferring. Whoever chooses the lower number can eat twice that number of gingerbread tiles. Whoever chooses the higher number can eat the lower number.” So, for example, if Hansel chooses 57 and Gretl chooses 30, Hansel will get 30 tiles and Gretl will get 60.
This sounds fine, but the children have just had lessons in game theory and regard this as a non-cooperative game between rational utility maximizers. Gretl knows that Hansel will not choose 99, because 97 would leave him better off if she chose 98 and no worse off if she chose any other number. By the same reasoning, she will avoid 98 and choose 96. In her mind she can follow this train all the way to its end: Rationally, it seems, she must choose 2. Hansel, following it also, finds himself indifferent between 3 and 1. In the end he will receive a paltry two tiles and Gretl either one or four.
Is all of this sound? Gretl says, “There is something radically peculiar about trains of thought which proceed in the subjunctive. You are to work out what you would be rational to do, if I were to choose a number which I shall not choose. I am to do likewise, with each train of thought reproduced inside the other. What happens if either player derails a train by choosing in defiance of it? In that case it becomes radically unclear whether either player still has a rational choice.”
(Martin Hollis, “The Gingerbread Game,” Analysis 54:4 [October 1994], 196-200.)
Centers of Attraction
In his 1908 autobiography, Francis Galton described a “beauty map” he’d compiled of the British Isles:
Whenever I have occasion to classify the persons I meet into three classes, ‘good, medium, and bad,’ I use a needle mounted as a pricker, wherewith to prick holes, unseen, in a piece of paper. … I used this plan for my beauty data, classifying the girls I passed in streets or elsewhere as attractive, indifferent, or repellent. … I found London to rank highest for beauty; Aberdeen lowest.
In 2008, psychologists Viren Swami and Eliana Hernandez set out to compile a beauty map of their own, this time focusing on London. They asked 461 residents to rate the physical attractiveness of men and women in the city’s 33 boroughs. For the record, the City of London, the City of Westminster, and Kensington and Chelsea were rated highest — which correlates with the affluence but not the health (life expectancy) of the residents in those boroughs.
The B List
A problem from the Eighth International Mathematical Olympiad, held in Sofia, Bulgaria, in July 1966 (contributed by the Soviet Union):
In a mathematical contest, three problems, A, B, C were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem A, the number who solved B was twice the number who solved C. The number of students who solved only problem A was one more than the number of students who solved A and at least one other problem. Of all students who solved just one problem, half did not solve problem A. How many students solved only problem B?
Limerick
Said a boy to his teacher one day,
“Wright has not written rite right, I say.”
And the teacher replied,
As the error she eyed,
“Right! Wright: Write rite right, right away!”
Elemental Words
In the February 1971 issue of Word Ways, Mary J. Youngquist pointed out that when element symbols are expanded, SATED becomes SULFURATED and FEY becomes IRONY.
That August, she, Philip Cohen, and Murray Pearce extended the list:
FEED = IRONED
AGED = SILVERED
SIC = SULFURIC
SNED = TINED
SNY = TINY
SOUS = SULFUROUS
SET = SULFURET
SING = SULFURING
SIZING = SULFURIZING
CUED = COPPERED
Likewise, BASIS yields BASILICONS, NAZI yields NEONAZI (and then NEONEONEONAZI, and so on forever), and RES can yield either RHENIUMS or RESULFUR.
07/09/2026 UPDATE: A number of readers point out that NAZI doesn’t work, as the symbol for neon is Ne. But JAR = JARGON! (Thanks Paul.)
A Pressing Appointment
Choose a number on this clock face and, starting from 12, spell out that number’s English name as you advance clockwise around the face, one letter per numeral. For example, if you’ve chosen 3, count out T-H-R-E-E and you’ll land on the numeral 5. Adopt this new position as your next chosen number and proceed as before (in this case, counting F-I-V-E and landing on 9). After three or more moves you’ll reliably land on 1.
This works because of a characteristic of Markov chains first observed by Russian mathematician Evgenii Borisovich Dynkin. Here’s a card trick that exploits the same principle.
The Tritone Paradox
This recording presents four pairs of tones, each pair separated by three whole tones, or half an octave. Curiously, some listeners hear the interval as ascending, others as descending. (In fact the tones used are ambiguous as to octave, so there’s no objectively right answer.)
Even more curiously, sometimes a listener’s perception reverses when an interval is transposed, say from C-F# to G#-D, even though nothing else has changed.