Comments on: Easy Proof of the 1-Dimensional Fixed Point Theorem http://www.nivi.com/blog/article/easy-proof my personal blog Sun, 13 Dec 2009 19:02:35 -0800 http://wordpress.org/?v=2.8.6 hourly 1 By: Harmy http://www.nivi.com/blog/article/easy-proof/comment-page-1#comment-58844 Harmy Tue, 17 Oct 2006 21:08:12 +0000 http://www.nivi.com/blog/article/easy-proof-of-the-1-dimensional-fixed-point-theorem#comment-58844 <p>In my high school calculus text, this problem is in the section with the Intermediate Value Thm. Not that I've figured out how to solve it that way, but someone else might figure something out.</p> <p>I do like Chad's proof.</p> In my high school calculus text, this problem is in the section with the Intermediate Value Thm. Not that I’ve figured out how to solve it that way, but someone else might figure something out.

I do like Chad’s proof.

]]> By: Li http://www.nivi.com/blog/article/easy-proof/comment-page-1#comment-3779 Li Wed, 14 Dec 2005 17:43:28 +0000 http://www.nivi.com/blog/article/easy-proof-of-the-1-dimensional-fixed-point-theorem#comment-3779 <p>Chad, that's the first thing came to my mind too.</p> Chad, that’s the first thing came to my mind too.

]]> By: Chad Harrington http://www.nivi.com/blog/article/easy-proof/comment-page-1#comment-3499 Chad Harrington Tue, 06 Dec 2005 18:05:30 +0000 http://www.nivi.com/blog/article/easy-proof-of-the-1-dimensional-fixed-point-theorem#comment-3499 <p>I used a graph: distance on the Y axis, time on the X axis. The monk's trip up starts at (0,0) and ends up at (maxX, maxY). The trip down starts at (0, maxY) and ends at (maxX, 0). No matter how you draw the two lines between those points, they must intersect, which implies being in the same place at the same time.</p> <p>Not as immediately obvious as your solution, but it worked pretty quickly for me.</p> I used a graph: distance on the Y axis, time on the X axis. The monk’s trip up starts at (0,0) and ends up at (maxX, maxY). The trip down starts at (0, maxY) and ends at (maxX, 0). No matter how you draw the two lines between those points, they must intersect, which implies being in the same place at the same time.

Not as immediately obvious as your solution, but it worked pretty quickly for me.

]]> By: Ajay Kapal http://www.nivi.com/blog/article/easy-proof/comment-page-1#comment-3304 Ajay Kapal Fri, 02 Dec 2005 17:04:19 +0000 http://www.nivi.com/blog/article/easy-proof-of-the-1-dimensional-fixed-point-theorem#comment-3304 <p>Cool! I didn't read your solution, but I found the same answer (the intuitive one). I just imagined a map of their travels and superimposed the two paths in my head and imagined that they had to walk past each other at some point.</p> Cool! I didn’t read your solution, but I found the same answer (the intuitive one). I just imagined a map of their travels and superimposed the two paths in my head and imagined that they had to walk past each other at some point.

]]> By: Nivi http://www.nivi.com/blog/article/easy-proof/comment-page-1#comment-3275 Nivi Thu, 01 Dec 2005 09:18:34 +0000 http://www.nivi.com/blog/article/easy-proof-of-the-1-dimensional-fixed-point-theorem#comment-3275 <p>Alison, the monk's velocity is not a constant and is a function of time.</p> <p>Sometimes the monk even stops for a break!</p> <p>So the term v1*tau in your equation should be v1(tau) and that makes your equation unsolvable!</p> <p>That is why the proof I discussed is so cool.</p> Alison, the monk’s velocity is not a constant and is a function of time.

Sometimes the monk even stops for a break!

So the term v1*tau in your equation should be v1(tau) and that makes your equation unsolvable!

That is why the proof I discussed is so cool.

]]> By: Alison Chaiken http://www.nivi.com/blog/article/easy-proof/comment-page-1#comment-3263 Alison Chaiken Wed, 30 Nov 2005 16:12:03 +0000 http://www.nivi.com/blog/article/easy-proof-of-the-1-dimensional-fixed-point-theorem#comment-3263 <p>Let the first day's velocity be v1 and the second's be v2 and let the distance to the top be d. What we're requiring is that there exist an elapsed time tau so that v1<em>tau = d - v2</em>tau. Clearly tau = d/(v1+v2) has solutions for all reasonable values of d, v1 and v2.</p> <p>The lesson is, express problems as equations. The advantage of math is that you can solve problems essentially without thinking. When you're done (usually with far more difficult manipulations than shown here), you can consider whether the answer makes sense, especially in limiting cases. Barbie was wrong: math is your friend!</p> Let the first day’s velocity be v1 and the second’s be v2 and let the distance to the top be d. What we’re requiring is that there exist an elapsed time tau so that v1tau = d – v2tau. Clearly tau = d/(v1+v2) has solutions for all reasonable values of d, v1 and v2.

The lesson is, express problems as equations. The advantage of math is that you can solve problems essentially without thinking. When you’re done (usually with far more difficult manipulations than shown here), you can consider whether the answer makes sense, especially in limiting cases. Barbie was wrong: math is your friend!

]]> By: mikepk http://www.nivi.com/blog/article/easy-proof/comment-page-1#comment-3252 mikepk Tue, 29 Nov 2005 22:07:16 +0000 http://www.nivi.com/blog/article/easy-proof-of-the-1-dimensional-fixed-point-theorem#comment-3252 <p>Hey Nivi, I saw this post, followed it to the puzzle and decided to <a href="http://daredevilplanner.blogspot.com/2005/11/solution-how-many-hands-did-she-shake.html" rel="nofollow">try and solve it</a>. Not as elegant as the solution you posted for the monk problem though.</p> <p>Enjoy reading your posts, especially regarding "web 2.0".</p> Hey Nivi, I saw this post, followed it to the puzzle and decided to try and solve it. Not as elegant as the solution you posted for the monk problem though.

Enjoy reading your posts, especially regarding “web 2.0″.

]]> By: raghav http://www.nivi.com/blog/article/easy-proof/comment-page-1#comment-3251 raghav Tue, 29 Nov 2005 19:07:59 +0000 http://www.nivi.com/blog/article/easy-proof-of-the-1-dimensional-fixed-point-theorem#comment-3251 <p>this is one a puzzle by Martin Gardener</p> this is one a puzzle by Martin Gardener

]]> By: Cyrus http://www.nivi.com/blog/article/easy-proof/comment-page-1#comment-3250 Cyrus Tue, 29 Nov 2005 16:28:15 +0000 http://www.nivi.com/blog/article/easy-proof-of-the-1-dimensional-fixed-point-theorem#comment-3250 <p>I've heard a two dimensional version of this problem: prove that there are always at least two antipodal points on the globe with exactly the same temperature and pressure. I don't have a solution for this - do you know if it's possible to do without invoking crazy topology?</p> I’ve heard a two dimensional version of this problem: prove that there are always at least two antipodal points on the globe with exactly the same temperature and pressure. I don’t have a solution for this – do you know if it’s possible to do without invoking crazy topology?

]]> By: Nivi http://www.nivi.com/blog/article/easy-proof/comment-page-1#comment-3246 Nivi Tue, 29 Nov 2005 08:19:24 +0000 http://www.nivi.com/blog/article/easy-proof-of-the-1-dimensional-fixed-point-theorem#comment-3246 <p>Again, from <a href="http://www.marginalrevolution.com/marginalrevolution/2004/08/kakutani_is_at_.html" rel="nofollow">Marginal Revolution</a>: "Here is an intuitive proof of the monk problem. Imagine that there are two monks, one going down and one going up, each beginning on the same day at sunrise. At some point in the day the hiker's must meet! QED."</p> Again, from Marginal Revolution: “Here is an intuitive proof of the monk problem. Imagine that there are two monks, one going down and one going up, each beginning on the same day at sunrise. At some point in the day the hiker’s must meet! QED.”

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