Reader Alex Freuman passed this along — a simple method of establishing any row in Pascal’s triangle, attributed to Edric Cane. To establish, for example, the seventh row (after the initial solitary 1), create a row of fractions in which the numerators are 7, 6, 5, 4, 3, 2, 1 and the denominators are 1, 2, 3, 4, 5, 6, 7:
Now multiply these in sequence, cumulatively, to get the numbers for the seventh row of the triangle:
These are the coefficients for
.
Cane writes, “It couldn’t be easier to remember or to implement.” Another example — row 10:
obtest
v. to call heaven to witness; to protest against
proditor
n. a traitor; a betrayer
Is a union breaking the law if it posts a giant inflatable rat outside an employer’s facility? No, it’s not, according to a 2011 decision by the National Labor Relations Board. The Sheet Metal Workers’ Union had sought to dissuade a hospital from using non-union workers by stationing a 16-foot rat near the building’s entrance. The NLRB held that the “the use of the stationary Giant Rat (i) constituted peaceful and constitutionally protectable expression, (ii) did not involve confrontational conduct that would qualify as unlawful picketing, and (iii) did not qualify as nonpicketing conduct that was otherwise unlawfully coercive.”
The “rat collosi” are multiplying (gallery). Let’s hope they don’t stage an uprising themselves someday.
That’s a grammatically correct sentence. What does it mean? Most readers have to puzzle over it a bit before seeing the interpretation
The horse [that was] raced past the barn fell.
This is a “garden-path sentence” — the reader naturally assumes one interpretation and is confused to find that another had been intended. Further examples:
The old man the boat.
The government plans to raise taxes were defeated.
The cotton clothing is made of grows in Alabama.
I convinced her children are noisy.
Time flies like an arrow; fruit flies like a banana.
In writing, lexicographer Henry Fowler calls it “an obvious folly — so obvious that no one commits it wittingly except when surprise is designed to amuse. But writers are apt to forget that, if the false scent is there, it is no excuse to say they did not intend to lay it; it is their business to see that it is not there, and this requires more care than might be supposed.”
Here are six new lateral thinking puzzles to test your wits and stump your friends — play along with us as we try to untangle some perplexing situations using yes-or-no questions.
An adventurer wants to explore a desert. He has a jeep that can carry up to 1 unit of fuel at any time and that will travel 1 unit of distance on 1 unit of fuel. As he travels he can leave any amount of the fuel that he’s carrying at any point, as a fuel dump to be picked up later. He starts from a fixed base at the edge of the desert, where there’s an unlimited supply of fuel. How far into the desert can he go if he wants to return safely to the base at the end of each trip?
Surprisingly, with some intelligent planning he can go as far as he likes. The diagram above shows how far he can get with 3 trips:
On the first trip he departs the base with 1 unit of fuel. He drives 1/6 unit into the desert, leaves 2/3 units of fuel at a fuel dump, and returns to the base using the remaining 1/6 unit of fuel.
On the second trip he leaves the base with 1 unit of fuel, drives 1/6 unit to the fuel dump, and draws 1/6 unit from the dump. Now he’s carrying 1 unit of fuel. Then he drives 1/4 unit farther into the desert and leaves 1/2 unit at a new dump. Now he has 1/4 unit of fuel remaining, which is just enough to reach the first fuel dump, where he collects another 1/6 unit of fuel and returns to base.
On the third trip he drives 1/6 unit to reach the first fuel dump, where he tops up with 1/6 unit of fuel (leaving 1/6 unit remaining there). Then he drives 1/4 to the second dump, where he collects 1/4 unit of fuel (again topping up to 1 full unit). (1/4 unit now remains in the second fuel dump.) Now he can drive 1/2 unit distance into the desert before he has to return to the second fuel dump, where he collects the remaining 1/4 unit fuel, which enables him to reach the first fuel dump, where he collects the last 1/6 unit of fuel, which is just enough to get back to base.
So if 3 trips are planned the explorer can travel a round-trip distance of 1 + 1/2 + 1/3 = 11/6 units. You can see the pattern: If the explorer had planned 4 trips, he would set up fuel dumps at distances of 1/8, 1/6, and 1/4 from the base, initially storing 3/4, 2/3, and 1/2 units at each and then drawing 1/8, 1/6, and 1/4 units of fuel from each on each visit. As before, on the final trip he could depart the last fuel dump with a full tank, drive 1/2 unit into the desert, and then return to the base, exhausting each fuel dump on the way. In that case he’d have traveled a round-trip distance of 1 + 1/2 + 1/3 + 1/4 = 25/12 units.
This is just the harmonic series, 1 + 1/2 + 1/3 + 1/4 + 1/5 + …, which is divergent — in principle, at least, the explorer can travel as far as he likes into the desert, provided he plans a large enough series of trips. In practice it would be very difficult, though — both the number of fuel dumps and the total amount of fuel necessary increase exponentially with the distance to be traveled.
At the end of his 1986 book Paradoxes in Probability Theory and Mathematical Statistics, statistician Gábor J. Székely offers a final paradox from his late professor Alfréd Rényi:
Since I started to deal with information theory I have often meditated upon the conciseness of poems; how can a single line of verse contain far more ‘information’ than a highly concise telegram of the same length. The surprising richness of meaning of literary works seems to be in contradiction with the laws of information theory. The key to this paradox is, I think, the notion of ‘resonance.’ The writer does not merely give us information, but also plays on the strings of the language with such virtuosity, that our mind, and even the subconscious self resonate. A poet can recall chains of ideas, emotions and memories with a well-turned word. In this sense, writing is magic.
You’re a venal king who’s considering bribes from two different courtiers.
Courtier A gives you an infinite number of envelopes. The first envelope contains 1 dollar, the second contains 2 dollars, the third contains 3, and so on: The nth envelope contains n dollars.
Courtier B also gives you an infinite number of envelopes. The first envelope contains 2 dollars, the second contains 4 dollars, the third contains 6, and so on: The nth envelope contains 2n dollars.
Now, who’s been more generous? Courtier B argues that he’s given you twice as much as A — after all, for any n, B’s nth envelope contains twice as much money as A’s.
But Courtier A argues that he’s given you twice as much as B — A’s offerings include a gift of every integer size, but the odd dollar amounts are missing from B’s.
Anchormen, chairs, dogs, flowers, and comets are things: If I have one anchorman and add another, I have two anchormen. My chair did not exist until it was assembled into that form. And if a comet hits Paraguay, it is no longer a comet.
Helium, gravy, wood, music, and joy are stuff: If some helium escapes my balloon, it seems wrong to say that I’ve lost a thing. If I divide my gravy into two portions, it’s still gravy. And if I chop my cabin into firewood, the amount of wood in the world does not seem to have changed.
We seem to distinguish between these two classes of existence. We can count things, but stuff forms a sort of cumulative mass. Things are made of stuff (crowns are made of gold), but stuff is made of things (gold is made of molecules). What’s at the bottom? And what leads us to make these distinctions?
(Kristie Miller, “Stuff,” American Philosophical Quarterly 46:1 [January 2009], 1-18.)
Composer Alfred Schnittke’s gravestone bears a musical staff with a semibreve rest under a fermata, indicating that the rest should be held as long as desired. It’s marked fff, or fortississimo, meaning that it should be performed very strongly.
Overall it might be interpreted to mean “a decided rest of indefinite length.”
