EventHorizon
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Everything posted by EventHorizon
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Yes...I described your allowed movement....and it doesn't involve chess pieces
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Yup....good job.
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That is the right approach....I'll check the numbers from your next post a little later.
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I wasn't asking how...I was asking when it is possible and when it isn't possible. And for a proof.
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Yes, but there is a more simply way of stating it.....you may get it once you figure out the middle ones.
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Maybe also have a textbox that the OP can edit (originally says "Unsolved"). It would be nice to not only know that a problem has been solved.....but have the topic also say who solved it first. Or instead of a text box, just have a checkbox to click on the post with the correct answer...and let the back end get the name of the member who posted the solution. Might get some extra competition going. It could even be tracked....so there could be a leaderboard on the number of solved puzzles
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Suppose you are dropped onto a random square on an N by M chessboard. You can move in the four compass directions, and you cannot return to a square once you have left it. Under what conditions can you visit all squares? When is it not possible to visit all squares? Can you prove your answers? Easily?
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definitely on the right track
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Suppose you have an N by M chessboard with black and white squares. You can choose a rectangle, and all colors within it will change to the other (white to black and vice-versa). 1. What is the minimum number of rectangles you would need to invert to make the whole chessboard a single color? 2. What is the minimum number of rectangles you would need to invert to make the whole chessboard a specified color (white or black)? 3. Under what conditions are the answers to (1) and (2) equal?
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Way off? Or did I get it?
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Yeah, I was a bit rusty on differentiation as well and ended up looking up taking the log too. I think many good puzzles are simply math or logic exercises wrapped in a story. I'm sure if I took the time that I could hide much of the mathiness from the description of it.
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yup, good job. I do want to mention that I got the problem from another website (why the problem is in quotes) and I said I did, but the name of the site was removed... As for the original problem, I thought I would give my proof for it...
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Pythagorean triples are two distinct squares that sum to another square. This problem (I took from another website) is to prove that for any integer n, that you can find n distinct squares whose sum is square. (eg, prove that pythagorean quadruples, quintiples, and so on exist)
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One more follow-up question
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You don't need to come up with a proof to be able to answer the follow-up questions. Just in case proofs scare you.
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Since the last post of this kind was so wildly popular (I was the only one to post....), I thought I would try another. "For any positive integer, k, let Sk = {x1, x2, ... , xn} be the set of real numbers for which x1 + x2 + ... + xn = k and P = x1 x2 ... xn is maximised. For example, when k = 10, the set {2, 3, 5} would give P = 30 and the set {2.2, 2.4, 2.5, 2.9} would give P = 38.25. In fact, S10 = {2.5, 2.5, 2.5, 2.5}, for which P = 39.0625. Prove that P is maximised when all the elements of S are equal in value and rational." Oooh....the British spelling of maximized....this must be a good problem....right? And follow-up questions of my own design....
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"It is well known that there exist pairs of distinct squares that add to make another square. For example, 5^2 + 12^2 = 25 + 144 = 169 = 13^2. But this can be extended to any number of squares. For example, 2^2 + 5^2 + 14^2 = 15^2. Prove that there exists a sum of n distinct squares that is also square."
