Jupiter gave to every man a sack,
To hold his faults and carry on his back;
Another, Jupiter gave, which from his breast
Hung heavy with his neighbour’s faults oppressed:
On this account man never can behold
His own, but can his neighbour’s faults unfold.
— Phaedrus
Painter Giulio Romano decorated the Palazzo del Te outside Mantua with a series of illusionistic spaces and special effects, culminating in a bewildering room in which giants that have rebelled against Zeus are crushed for their transgression — Giulio “paints the walls away,” leaving the viewer in a crumbling city into which Zeus flings lightning from the heavens. Poet Gregorio Comanini praised Giulio’s fantastic imagination:
In Mantua, in a room in the Palazzo del Te, Giulio Romano has painted giants struck by lightning at Flegra. They are crushed under the rubble of rock and mountain, in positions so strange and horrible that anyone who saw such a spectacle in reality would surely be horrified and feel great distress. None the less, since this is an imitation and a painting, anyone would welcome a chance to see it and would be highly pleased with it, as can be attested to by the frequency with which visitors flock to view it.
Giorgio Vasari wrote, “Let no one think ever to see any work of the brush more horrifying, or more realistic, than this.”
In the Japanese logic puzzle Akari, you’re presented with a grid of black and white squares. The goal is to place “light bulbs” into white cells until the whole grid is illuminated. Each bulb sends out rays of light horizontally and vertically, illuminating its row and column unless a black cell blocks the rays.
There are two constraints: The bulbs must not shine on one another, and each numbered black cell must bear that many bulbs (orthogonally adjacent to it) in the finished diagram. An unnumbered black cell can bear any number of bulbs.
Here’s a moderately difficult puzzle. Can you solve it?
In 1967 German filmmaker Arno Peters promoted a new map of the world in which areas of equal size on the globe appear of equal sizes on the map, so that poor, less powerful nations near the equator are restored to their rightful proportions.
Peters promoted the map by comparing it the popular Mercator projection, which is useful to navigators but makes Europe appear larger than South America and Greenland larger than China.
Peters’ goal was to empower underdeveloped nations, which he felt had suffered from “cartographic imperialism.” But his own map badly distorts the polar regions — cartographic educator Arthur Robinson wrote that its “land masses are somewhat reminiscent of wet, ragged, long winter underwear hung out to dry on the Arctic Circle” — and observers noted that Peters’ native Germany suffered less distortion than the underdeveloped nations he was trying to help.
To quell what they felt was an ill-founded controversy, in 1990 seven North American geographic organizations adopted a resolution urging media and government to stop using all rectangular world maps “for general purposes or artistic displays,” as they necessarily distort the planet’s features. That included both Mercator’s and Peters’ projections.
Peters’ map wasn’t even new. It had first been proposed by Scottish clergyman James Gall — who had noted in 1885 that “we may obtain comparative area with mathematical accuracy” by using this projection, but “we must sacrifice everything else.”
The most common coins in U.S. circulation are worth 1¢, 5¢, 10¢, and 25¢. University of Waterloo computer scientist Jeffrey Shallit found that with this system the average cost of making change is 4.7; that is, if every amount of change between 0¢ and 99¢ is equally likely to be needed, then on average a change-maker must return 4.7 coins with each transaction.
Can we do better? Shallit found two four-coin sets that reduce the average cost to a minimum: (1¢, 5¢, 18¢, 25¢) and (1¢, 5¢, 18¢, 29¢). Either reduces the average cost to 3.89.
“We would therefore gain about 17% efficiency in change-making by switching to either of these four-coin systems,” he writes. And “the first system, (1, 5, 18, 25), possesses the notable advantage that we only need make one small alteration in the current system: replace the current 10¢ coin with a new 18¢ coin.”
(Jeffrey Shallit, “What This Country Needs Is an 18¢ Piece,” Mathematical Intelligencer 25:2 [June 2003], 20-23.)
A three-digit number is evenly divisible by 7 if and only if twice its first digit added to the number formed by its two last digits gives a result that’s divisible by 7. So, for example, 938 is divisible by 7 because 2 × 9 + 38 = 56 = 7 × 8.
In fact this can be extended to numbers of any length: 229187 → 2 × 2291 + 87 = 4669 → 2 × 46 + 69 = 161 → 2 × 1 + 61 = 63 = 7 × 9.
(J. Kashangaki, “A Test for Divisibility by Seven,” Mathematical Gazette 80:487 [March 1996], 226.)
Does there, I wonder, exist a being who has read all, or approximately all, that the person of average culture is supposed to have read, and that not to have read is a social sin? If such a being does exist, surely he is an old, a very old man, who has read steadily that which he ought to have read sixteen hours a day, from early infancy. … My leisure has been moderate, my desire strong and steady, my taste in selection certainly above average, and yet in ten years I seem scarcely to have made an impression upon the intolerable multitude in volumes which ‘everyone is supposed to have read.’
— Arnold Bennett, Journal, Oct. 15, 1896
When Toronto attorney Charles Vance Millar died in 1926, he left behind a mischievous will that promised a fortune to the woman who gave birth to the most children in the next 10 years. In this week’s episode of the Futility Closet podcast we’ll follow the Great Stork Derby and the hope and controversy it brought to Toronto’s largest families during the Great Depression.
We’ll also visit some Portuguese bats and puzzle over a suspicious work crew.
The London borough of Camden enshrined disapproval in 2012 with a concrete bench designed to deter sleeping, skateboarding, drug dealing, graffiti, and theft. Its surface discourages any activity but sitting, it contains no crevices or hiding places, its surface repels paint, and it weighs two tons.
The result has been called a “masterpiece of unpleasant design,” a “perfect anti-object” “defined far more by what it is not than what it is,” and an example of “hostile architecture” oppressive to the homeless. The designers, Factory Furniture, responded by saying, “Homelessness should never be tolerated in any society and if we start designing in to accommodate homeless then we have totally failed as a society. Close proximity to homelessness unfortunately makes us uncomfortable so perhaps it is good that we feel that and recognise homelessness as a problem rather than design to accommodate it.”
Whether it discourages skateboarders is debatable.
From the Kanja Otogi Zoshi (“Collection of Interesting Results”) of Nakane Genjun, 1743:
Your friend has 30 go stones. He lines them up out of your sight, placing down either one or two stones with each deposit and calling “here” so you’ll know this has been done. When all 30 stones have been placed, you have heard him say “here” 18 times. How many deposits contained one stone, and how many two?
