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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.
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.”
Why do we cultivate friendships? What reason do I have to be a friend to another person, that is, to care about him for his own sake? In order to make the friendship worthwhile, such a reason would have to explain how doing it makes my own life better. But that’s a problem: If I pursue the friendship in order to improve my own life, then I’m not really being a true friend, caring about my friend for his own sake.
University of Newcastle philosopher Joe Mintoff writes, “The problem is that, even though many of us think that being a true friend makes our lives better, paradoxically this thought had better not guide our pursuit of friendship, lest this mean that we are not true friends and that our lives are not made better.” Why, then, do we seek to befriend others?
(Joe Mintoff, “Could an Egoist Be a Friend?,” American Philosophical Quarterly 43:2 [April 2006], 101-118.)
Another conundrum from Henry Dudeney’s Canterbury Puzzles:
A country baker sent off his boy with a message to the butcher in the next village, and at the same time the butcher sent his boy to the baker. One ran faster than the other, and they were seen to pass at a spot 720 yards from the baker’s shop. Each stopped ten minutes at his destination and then started on the return journey, when it was found that they passed each other at a spot 400 yards from the butcher’s. How far apart are the two tradesmen’s shops? Of course each boy went at a uniform pace throughout.