Challenge Ready for a Summer Math Challenge?

[StoneTemplePython]

Hey, Mr. StoneTemplePython, thanks a lot for burning up all my spare time! I took a look at the problem and solving it became an obsession.

I didn't look at your solution or anyone else's until after coming up with my own (and I learned that email alerts don't honor the SPOILER codes, so I had to avert my eyes to that portion of your message). Actually, I haven't read any of the other answers except yours.

Yes, I can certainly relate to this. Figuring out how to break the abstraction into something useful can take a little or an awful lot of time. That's kind of the joy, and peril, of these challenges I think.

I also didn't realize that Spoilers aren't hidden in emails... interesting.

I often get caught by misreading the problem or making the wrong assumptions or failing to make the right assumptions.

For example, I assumed all canisters hold the same amount of fuel, since assuming otherwise would definitely put the problem out of my reach...

I came up with something like your solution, but I don't think your solution is complete. Consider the 4-canister case. I place two canisters together and another two together such that the distance between them is greater than two intervals. Your argument fails.

Sorry no. Your assumption isn't justifed. The canisters in general hold a varying amount of fuel. The only constraint is that of course each amount is real non-negative and in aggregate the amount of fuel sums to exactly the amount required to drive around the track. (There is a way to convert from your not justified assumption to the actual problem by changing perspective and just looking at partial sums though... edit: specifically if you consider for any arbitrarily chosen starting position then the ith 'gas station' has $x_i$ of gas and $y_i$ as the gas required to get to the next station -- the value of $z_i:= (x_i - y_i)$ is what is of interest.)

Your 'counterexample' doesn't hold water I'm afraid. If you place two canister touching each other (I guess that's what 'together' means?) then it's just another legal 4 canister configuration. If together means 'exactly the same spot' I'm not really sure that's allowed physically but in the interest of sport I'd point out that if it were allowed you have just reduced it to the 2 canister problem which most people can solve by inspection.

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edit: thread was moved to the right home so suggested place for follow-ups is not needed.
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note: some of your questions and confusions were already asked and answered in that thread.

another note: even if you've been out of the habit of doing math for many decades, in the event you've been doing computer programming in the interim, my suggested solution should feel quite familiar to certain recursive programs.

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All in all I think we agree that this is a nice Basic Challenge

 

[Freixas]

Spoiler: Comments on problem 2

Your 'counterexample' doesn't hold water I'm afraid. If you place two canister touching each other (I guess that's what 'together' means?) then it's just another legal 4 canister configuration. If together means 'exactly the same spot' I'm not really sure that's allowed physically but in the interest of sport I'd point out that if it were allowed you have just reduced it to the 2 canister problem which most people can solve by inspection.

"At the same spot relative to the track" is what I meant. The canisters could be stacked on top of each other, for example.

As for my counter-example, I mixed my problem statements and was still thinking in terms of the one I working on. I interpreted "extra fuel" to mean more than 1/Nth of the total fuel, rather than more than 100% of the fuel needed for the next section. It took me a while to even see this.

Reading your problem description more carefully, I see you asked if there was a viable starting point, not to identify which point that might be. The reduction proof works for identifying that a starting point exists, but not for identifying which it is. Oddly enough, it is not the one with the most fuel or even the most percentage of fuel for its leg. Consider N=4 where all canisters are spaced equidistant. The fuel amount needed to complete the track is 100 and the canister amounts are 26, 51, 12 and 11. Staring at 26 let's the car finish; starting at 51 doesn't. In same cases, of course, there are multiple valid starting points. When reducing, it doesn't matter whether you start with 26 or 51—you still know it can be done, but a little extra record-keeping is needed to determine where to start.

I do think the problem should have explicitly stated that the fuel was not distributed equally among all canisters. If you argue that I should, in my solution, not include restrictions not imposed by the problem statement, then I will simply have the driver walk to the nearest canister, bring it back to the car and repeat as necessary. Problem solved! If you argue that I shouldn't make unreasonable assumptions, then I would respond that reasonable is a subjective term; if something is important to the problem, it should be included. Personally, I don't think it would hurt to include both.

Thanks for taking the time to create this puzzle, though (assuming it's yours).

 

[StoneTemplePython]

Reading your problem description more carefully, I see you asked if there was a viable starting point, not to identify which point that might be. The reduction proof works for identifying that a starting point exists, but not for identifying which it is. Oddly enough, it is not the one with the most fuel or even the most percentage of fuel for its leg.

In mathematics it's called an existence proof. Nothing more and nothing less.

And yes, it is immediate from my suggested inductive proof that such an approach only shows the existence of a viable path -- but you could have to to try all $n$ starting points to find one that works. This is why in our August thread I said mfb's response here (see post 48 and 49 on this thread) is perhaps the most constructive / satisfying -- it gives existence and a thought experiment to identify the starting point.

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I do think the problem should have explicitly stated that the fuel was not distributed equally among all canisters. If you argue that I should, in my solution, not include restrictions not imposed by the problem statement, then I will simply have the driver walk to the nearest canister, bring it back to the car and repeat as necessary. Problem solved! If you argue that I shouldn't make unreasonable assumptions, then I would respond that reasonable is a subjective term; if something is important to the problem, it should be included. Personally, I don't think it would hurt to include both.

Thanks for taking the time to create this puzzle, though (assuming it's yours).

It doesn't not work like that. You should read through these threads and see how people respond in them. As we've said many times in these threads -- you are free to ask questions. You are not free to over-ride the challenges or their intent.

This puzzle actually comes in 2 different problem books I have. It is not an original of mine. As I said in post #49 of this thread -- a slight variant of this has been asked on many occasions as part of mathematics admissions interviews at Cambridge. I think what you've said above... would have been quite poorly received. What you are suggesting here seems to be in between linguistics and personal philosophy, not math.
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Additional points:

Point 1:
two posts up I gave a direct way to convert from your unjustified assumption to the actual problem -- if you view the problem in said light, then as I've said, whether the amounts of fuel in each canister are constant or not is irrelevant. Given that you are arguing about linguistics I suppose you didn't understand this.

Point 2:
If you need something even more direct: you are free to ask questions and even ask about making certain assumptions. If the person who submitted the question says ___ assumption is fine, then it is fine. If you are told it is not fine then it is not fine. That's how the challenges work. Period. This is a math challenge -- I will not "argue" that you shouldn't make ___ assumption. If it is my problem, I will tell you, as I've already done in this thread. Case closed.