Sheriff

So, what follows could be a spoiler for those of you who are also going it alone

I figure I'm a "pretty good" Sudoku-er - I focus on only the hardest problems in whatever book I've grabbed. But sometimes they stump me. I recently had a conceptual breakthrough - still time before the spoiler...

This so-called "breakthrough" of mine is based on the idea that for each puzzle there is only one solution.

Given that "na" means "not applicable" and "(x)" means one of two possible locations,

Let's say you have this rough situation in two of the nine boxes (in the same row or column):

na na na

na na na

(91) 7 (91) <--- previously calculated that only 9 and 1 could go in these two locations

-----------

na na na

na na na

(91) na (91) <--- this is not a valid "possibilities" notation, because it would allow two solutions.

It turns out that this strategy applies a lot when I'm stuck. Is this a no-brainer that I was just slow to get to?

Spot false dilemmas now, ask me how!

(If you're not on the edge, you're taking up too much room.)

You can't have 1's and 9's like this, because then you could switch the 1's and the 9's and there would be two solutions (unless some of the 1's and 9's are pre-printed).

Yes, that's a strategy that I use, rarely. But also yes, it wasn't one of the first strategies I discovered. When I mentioned it to my father he acknowledged that it was a strategy that he used, too.

However I did once find a correct solution to a published Sudoku which wasn't the same as the correct solution in the back of the book...

Sheriff

You have a sphere, and you drill a straight, circular, hole through the centre. (Along a diameter of the sphere.) The length of this hole is 10 centimetres. What is the volume of the remaining material?

Sheriff

*The mind is a strange and wonderful thing. I'm not sure that it will ever be able to figure itself out, everything else, maybe. From the atom to the universe, everything, except itself.*

Stephan: no, the diameter of the sphere is also one of the unknowns. And sorry for the unclarity, the hole does go all the way through the sphere. (So the diameter of the sphere is at least 10 centimetres but can be greater.)

But every year there was one or two students who would say "Look, there's a unique answer to this, or you wouldn't have asked it. So it must be independent of the diameter of the hole. So let's set the diameter of the hole to be zero, then what we have is a sphere whose diameter is 10 centimetres and therefore the answer is just given by the formula for the volume of a sphere."

And it turns out, if you grind your way through the calculation correctly, that the answer is indeed independent of the diameter of the hole.

This always made for an interesting classroom discussion when the homework was graded and returned. Was that a legitimate assumption or not? (But we always gave those smart-asses full credit for showing good mathematical instincts.)

How can it be independent from the size of the drill? Like you said, for diameter zero, the solution equals the volume of the entire sphere. If we use a wider drill (possibly one that is so large that we grind away the entire sphere!) won't the solution be different from the volume of the original sphere? Doesn't that imply that the outcome *does* depend on the diameter of the hole (as one would intuitively think)?

Maybe I'm completely misunderstanding the question.

*The mind is a strange and wonderful thing. I'm not sure that it will ever be able to figure itself out, everything else, maybe. From the atom to the universe, everything, except itself.*

Sheriff

So, if I'm understanding this correctly, you could start with a sphere of diameter 100 cm. In order to have a hole 10 cm high, the bit would have to be almost 100 cm. itself, so the resulting shape would be this thin donut-y thing (10 cm high)... is that the gist?

Spot false dilemmas now, ask me how!

(If you're not on the edge, you're taking up too much room.)

Stephan van Hulst wrote: If we use a wider drill (possibly one that is so large that we grind away the entire sphere!) won't the solution be different from the volume of the original sphere?

If you use a wider drill, then you need a bigger sphere so that the length of the hole is still 10 centimetres. (Remember, that's the only length you know.) Like Bert said, if the sphere is really big then you need to drill a really wide hole so that it's length is only 10 centimetres. Imagine drilling such a hole through the earth, for example -- you'd end up with a ring 12,800 kilometres around which was 10 centimetres high and thinner than a sheet of paper.

*The mind is a strange and wonderful thing. I'm not sure that it will ever be able to figure itself out, everything else, maybe. From the atom to the universe, everything, except itself.*

I suppose the question is whether you want to encourage the "trick" or not. If not, it would be easy enough to rephrase it "find the volume

*and*prove that it is independent of the width of the drill bit" if you wanted to discourage it.

Or even just the "prove" bit, leaving room for some intuitive shortcuts. Just been googling, and found a neat proof of it without going to the trouble of calculating the volume.

Stephan van Hulst wrote:Oh geez... So the premise is "Drill a hole wide enough so that you're left with a ring that's 10cm high". I call shenanigans on the phrasing :P

Eh, I think that part was clear enough. But I don't think the phrasing here actually justifies the assumption that there's a single numeric answer. The answer could have been a formula based on the radius of the sphere, for example. And I think students need to be prepared to consider outside-the-box answers to questions, including "there are infinitely many solutions" and "there is no solution" for a given problem. Training them to assume there is one unique solution isn't good practice, because the real world isn't always like that. Still, I would be inclined to give full credit to students taking the shortcut, because it's good to consider such alternate approaches.

Mike Simmons wrote:But I don't think the phrasing here actually justifies the assumption that there's a single numeric answer. The answer could have been a formula based on the radius of the sphere, for example. And I think students need to be prepared to consider outside-the-box answers to questions, including "there are infinitely many solutions" and "there is no solution" for a given problem. Training them to assume there is one unique solution isn't good practice, because the real world isn't always like that. Still, I would be inclined to give full credit to students taking the shortcut, because it's good to consider such alternate approaches.

Yes, that was the gist of the classroom discussion which would take place. Bear in mind that first-year calculus problem sets basically consist entirely of calculations which result in a single correct number as the answer, so the assumption that there was a single correct number as the answer isn't exactly out-of-the-box. However identifying that as an assumption is in itself a problem-solving methodology; most students just take it for granted and are unable to articulate it as such.

Sheriff

Still, it feels like this is going to be an important tool in my Sudoku toolkit.

Spot false dilemmas now, ask me how!

(If you're not on the edge, you're taking up too much room.)

One of the sudoko books I have listed this as a strategy. Making it a good strategy and not a spoiler .

[OCA 8 book] [OCP 8 book] [Practice tests book] [Blog] [JavaRanch FAQ] [How To Ask Questions] [Book Promos]

Other Certs: SCEA Part 1, Part 2 & 3, Core Spring 3, TOGAF part 1 and part 2