A problem from the February 2006 issue of Crux Mathematicorum:
Prove that if 10a + b is a multiple of 7 then a – 2b must be a multiple of 7 as well.
Author: Greg Ross
Napkin Proof
A sure-fire way to avoid hangover, from philosopher Josh Parsons:
It’s well known that one can alleviate hangover symptoms with a “hair of the dog” — another alcoholic drink. The problem is that this incurs another later hangover.
But think about this. Suppose that a pint of beer produces an hour of drunkenness followed by an hour of hangover, and that smaller quantities produce proportionately shorter periods. Begin, then, from a sober state, and drink half a pint of beer. Wait half an hour, until you’re just about to start feeling the hangover. Then drink a quarter pint of beer and wait a quarter of an hour, then an eighth of a pint, and so on. After an hour, you will have drunk one pint of beer and experienced no hangover, as it’s manifestly true that every incipient hangover you had was prevented by a further drink.
In his paper, cited below, Parsons addresses some objections to this scheme, including the fact that most publicans won’t stock 1/64-pint glasses and that eventually you’ll be swallowing at greater than light speed, and indeed swallowing something that may no longer qualify as beer. He grants that the task might be a “medical impossibility.”
But “by the time you get to the point where you can’t continue, you will only be incurring a very very short hangover with your last sip of beer. You don’t mind suffering a millisecond’s hangover. Besides, isn’t it worth speculating about whether the cure would work, on the counterfactual supposition that you can swallow at any finite speed, and that alcoholic beverages are made out of infinitely divisible gunk? Such speculations can tell us a lot about our concepts of infinity and matter.”
(He thanks “my colleagues and students for many helpful suggestions, and Central Bar for providing a pleasant environment in which to test my theories.”)
(Josh Parsons, “The Eleatic Hangover Cure,” Analysis 64:4 [October 2004], 364-366.)
Two-Step
In this position, devised by Harvard mathematician Noam Elkies, White is in serious trouble. He’s been reduced to a bare king, and his opponent has a knight as well as two pawns on the verge of queening.
It’s tempting to snap up the c2 pawn — this gains material and keeps Black’s king bottled up in the corner, which stops the a-pawn from queening. But now Black can win with 1. … Nd3! This stops the white king from moving to c1, which had been his only way to keep the black king immobilized. Now he’ll have to back off (even though perhaps winning the knight), and Black will play 2. … Kb1 and queen his pawn.
Interestingly, in the original position if White plays 1. Kc1, he’s guaranteed a draw: He can capture the c2 pawn on his next move and then shuffle happily forever between c1 and c2. Because he’s “lost a move,” Black can’t execute the maneuver described above to force the white king away from the corner. That’s because as the white king shuffles between a white and a black square, the knight must move about the board, also alternating between squares of different colors — and always landing on the wrong color to achieve his goal. The knight can wander as far afield as he likes, and execute any complex maneuver; he’ll always arrive at the wrong moment to achieve his aim.
(Noam D. Elkies and Richard P. Stanley, “The Mathematical Knight,” Mathematical Intelligencer 25:1 [December 2003], 22-34.)
“The Pavior”
An Author saw a Laborer hammering stones into the pavement of a street, and approaching him said:
‘My friend, you seem weary. Ambition is a hard taskmaster.’
‘I’m working for Mr. Jones, sir,’ the Laborer replied.
‘Well, cheer up,’ the Author resumed; ‘fame comes at the most unexpected times. To-day you are poor, obscure and disheartened, but to-morrow the world may be ringing with your name.’
‘What are you telling me?’ the Laborer said. ‘Can not an honest pavior perform his work in peace, and get his money for it, and his living by it, without others talking rot about ambition and hopes of fame?’
‘Can not an honest writer?’ said the Author.
— Ambrose Bierce, The Monk and the Hangman’s Daughter, 1911
The Bar Problem
Suppose a local bar has 100 regular patrons. The bar is rather small, and going there is enjoyable on a given night only if fewer than 60 people show up. This is a problem: I want to go to the bar, but I expect to enjoy it only if you don’t come. But I know that you’re thinking the same thing about me. If no one communicates in advance, how many people will tend to turn up at the bar?
In a 1994 computer experiment, Stanford University economist W. Brian Arthur assigned each patron a set of plausible predictive rules on which it might base its decision. One rule might predict that next week’s attendance will be the same as last week’s, while another might take a rounded average of the last four weeks, and so on. Then, in practice, each patron would downgrade its badly performing rules and promote the more successful ones, and revise these ratings continually.
What he found is that the mean attendance converges to about 60, forming an “emergent ecology” that Arthur said is “almost organic in nature.” The population of active predictors splits into a 60/40 ratio, even though it keeps changing in membership forever. “To get some understanding of how this happens, suppose that 70 percent of their predictors forecasted above 60 for a longish time. Then on average only 30 people would show up; but this would validate predictors that forecasted close to 30 and invalidate the above-60 predictors, restoring the ‘ecological’ balance among predictions, so to speak.”
This is heartening to see, because life is full of such murky decisions. “There is no deductively rational solution — no ‘correct’ expectational model,” Arthur writes. “From the agents’ viewpoint, the problem is ill-defined, and they are propelled into a world of induction.”
(W. Brian Arthur, “Inductive Reasoning and Bounded Rationality,” American Economic Review 84:2 [May 1994], 406-411.)
Unquote
“Foreign nations are a contemporaneous posterity.”
— “A forgotten American essayist,” quoted by Brander Matthews in his introduction to The Oxford Book of American Essays, 1914
Readouts
A puzzle by Lee Sallows. In this readout from a computer-driven electronic display, the digits in the fifth row have been obscured. What are they?
In a Word
https://www.youtube.com/watch?v=9udNrOh5DyA
res angusta domi
n. straitened financial circumstances
appaumé
adj. having the hand opened out so as to display the palm
mammering
n. a state of hesitation or doubt
manuduction
n. careful guidance
dactylonomy
n. the art of counting on the fingers
belve
v. to roar or bellow
Short-Timer
Soliloquy of a mayfly, imagined by Benjamin Franklin in a 1778 letter to Madame Brillon:
It was the opinion of learned philosophers of our race, who lived and flourished long before my time, that this vast world, the Moulin Joly, could not itself subsist more than eighteen hours; and I think there was some foundation for that opinion, since, by the apparent motion of the great luminary that gives life to all nature, and which in my time has evidently declined considerably towards the ocean at the end of our earth, it must then finish its course, be extinguished in the waters that surround us, and leave the world in cold and darkness, necessarily producing universal death and destruction. I have lived seven of those hours, a great age, being no less than four hundred and twenty minutes of time. How very few of us continue so long! I have seen generations born, flourish, and expire. My present friends are the children and grandchildren of the friends of my youth, who are now, alas, no more! And I must soon follow them; for, by the course of nature, though still in health, I cannot expect to live above seven or eight minutes longer. What now avails all my toil and labor in amassing honey-dew on this leaf, which I cannot live to enjoy! What the political struggles I have been engaged in for the good of my compatriot inhabitants of this bush, or my philosophical studies for the benefit of our race in general! for in politics what can laws do without morals? Our present race of ephemeræ will in a course of minutes become corrupt, like those of other and older bushes, and consequently as wretched. And in philosophy how small our progress! Alas! art is long, and life is short! My friends would comfort me with the idea of a name they say I shall leave behind me; and they tell me I have lived long enough to nature and to glory. But what will fame be to an ephemera who no longer exists? And what will become of all history in the eighteenth hour, when the world itself, even the whole Moulin Joly, shall come to its end and be buried in universal ruin?
Franklin added, “To me, after all my eager pursuits, no solid pleasures now remain, but the reflection of a long life spent in meaning well, the sensible conversation of a few good lady ephemeræ, and now and then a kind smile and a tune from the ever amiable Brillante.”
All Covered
Can any 10 points on a plane always be covered with some number of nonoverlapping unit discs?
It’s not immediately clear that the answer is yes. The most efficient way to pack circles together on the plane is the hexagonal packing shown above; it covers about 90.69 percent of the surface. But if our 10 points are inconveniently arranged, it’s not clear that we’ll always be able to shift the array of circles around in order to get them all covered.
In this case there’s a neat proof that takes advantage of a technique called the probabilistic method — if, for a group of objects, the probability is less than 1 that a randomly chosen object does not have a certain property, then there must exist an object that has this property.
Take a hexagonal packing randomly. Then, for any point on the plane, the probability that it’s not covered by the chosen packing is about 1 – 0.9069 = 0.0931. This means that for any 10 points, the chance that one or more points are not covered is approximately 0.0931 × 10 = 0.931. And that’s less than 1.
“Therefore, we obtain from the principle that there exists some closest packing that covers all the 10 points,” writes mathematician Hirokazu Iwasawa of the Institute of Actuaries of Japan. “And, in such a packing, we actually need at most 10 discs to cover the 10 points.”
(Hirokazu Iwasawa, “Using Probability to Prove Existence,” Mathematical Intelligencer 34:3 [September 2012], 11-14. The puzzle was created by Naoki Inaba.)