12157692622039623539 = 11 + 22 + 13 + 54 + 75 + 66 + 97 + 28 + 69 + 210 + 211 + 012 + 313 + 914 + 615 + 216 + 317 + 518 + 319 + 920
(The largest such number.)
A Second Act
Nathanael West’s 1939 novel The Day of the Locust contains a character named Homer Simpson:
Except for his hands, which belonged on a piece of monumental sculpture, and his small head, he was well proportioned. His muscles were large and round and he had a full, heavy chest. Yet there was something wrong. For all his size and shape, he looked neither strong nor fertile.
In a 2012 interview with Smithsonian, Matt Groening said, “I took that name from a minor character in the novel The Day of the Locust, by Nathanael West. Since Homer was my father’s name, and I thought Simpson was a funny name in that it had the word ‘simp’ in it, which is short for ‘simpleton’ — I just went with it.”
The Petrie Multiplier
This is dismaying. Suppose that men and women are equally sexist, and imagine a group that’s 80 percent men and 20 percent women. If 20 percent of people (represented as squares above) will make sexist comments to people of the opposite gender, we might expect that the women will receive four times as many sexist comments as the men. In fact they receive 16 times as many. Computer scientist Ian Gent explains:
With 20% women the gender ratio is 1:4. So there are 4 times as many men to make sexist remarks, so 4 times as many sexist remarks are made to women as to men. But there are 4 times fewer women to receive sexist remarks, so each individual woman is four times as likely to receive a given remark than an individual man is. These effects multiply, so in this example the mean number of sexist remarks per woman is 16 times the number per man. This holds in general, so with a gender ratio of 1:r, women will receive r2 times as many sexist remarks as men.
Above, after 70 sexist remarks (arrows) are made at random to members of the opposite gender, the women have received a mean of 5.6 remarks each, the men only 0.35. Gent first described the effect in 2013; the mathematical model was devised by computer scientist Karen Petrie.
Circulation
‘I’ll bet you a dollar you won’t give me a dollar to keep,’ Bob says to Sue. She accepts the bet and gives him a dollar. Thus he loses the bet and returns the dollar. But that means he wins the bet, and she has to give him the dollar again. And so Bob and Sue pass the buck back and forth for the rest of their lives.
— Dave Morice, Alphabet Avenue, 1997
Square Diamonds
A puzzle by Henry Dudeney, from 1931:
“Here you have the two to the ten of Diamonds, inclusive. Can you re-arrange them, still retaining the shape of the figure, so that the ‘pips’ total eighteen, across and down, in each line, and also diagonally from the corners?”
The Tullock Paradox
Why is there so little money in politics? If a government subsidy is worth, say, a billion dollars to certain stakeholders, then we might expect them to spend nearly that amount in lobbying and bribes to secure its success.
Generally, this doesn’t happen. The potential beneficiary of a policy will typically spend only a small fraction of its value to bring it about. Economist Gordon Tullock first pointed this out in 1980; the reasons for it are still debated.
Fanciful Creatures
In What Is the Name of This Book? (1986), Raymond Smullyan describes two curious denizens of the Forest of Forgetfulness. The Lion lies on Mondays, Tuesdays, and Wednesdays, and the Unicorn lies on Thursdays, Fridays, and Saturdays. Each tells the truth on the days it doesn’t lie.
One day Alice encounters the two of them resting under a tree. They tell her:
Lion: Yesterday was one of my lying days.
Unicorn: Yesterday was one of my lying days too.
“From these two statements, Alice (who was a very bright girl) was able to deduce the day of the week. What day was it?”
A Gettysburg Puzzle
This 1867 gunnery manual contains a surprising claim:
I don’t know whether it’s accurate. In On Killing (2014), Lt. Col. Dave Grossman writes, “The obvious conclusion is that most soldiers were not trying to kill the enemy,” but that conclusion is disputed. More here.
The Number Checks Puzzle
From Henry Dudeney’s Amusements in Mathematics, 1917. Without removing these checks from their ring, divide them into three groups so that the first group multiplied by the second makes the third. For example, one valid try might be 28, 907, 15463, except that 28 × 907 doesn’t equal 15463.
“Of course, you may have as many of the checks as you like in any group. The puzzle calls for some ingenuity, unless you have the luck to hit on the answer by chance.”
The Disk Covering Problem
The dashed disk here has radius 1. Suppose we want to cover it entirely with n smaller disks. How small can those disks be?
Pleasingly, no one has yet found a general answer to this question. If we have only a single covering disk, then obviously it will need to be fully as large as the target. But if we’re allowed six discs, they can do the job even if each has a radius of only 0.555905…, as shown here.
Similar configurations work up to n = 10. But if we’re allowed 11 disks then some creative thinking again becomes necessary to find the best solution. No one has yet found a general strategy that reliably finds the minimum successful size.