<body><script type="text/javascript"> function setAttributeOnload(object, attribute, val) { if(window.addEventListener) { window.addEventListener('load', function(){ object[attribute] = val; }, false); } else { window.attachEvent('onload', function(){ object[attribute] = val; }); } } </script> <div id="navbar-iframe-container"></div> <script type="text/javascript" src="https://apis.google.com/js/plusone.js"></script> <script type="text/javascript"> gapi.load("gapi.iframes:gapi.iframes.style.bubble", function() { if (gapi.iframes && gapi.iframes.getContext) { gapi.iframes.getContext().openChild({ url: 'https://www.blogger.com/navbar.g?targetBlogID\x3d12969692\x26blogName\x3dLearning+Strategies\x26publishMode\x3dPUBLISH_MODE_BLOGSPOT\x26navbarType\x3dBLUE\x26layoutType\x3dCLASSIC\x26searchRoot\x3dhttp://larrydavidson.blogspot.com/search\x26blogLocale\x3den_US\x26v\x3d2\x26homepageUrl\x3dhttp://larrydavidson.blogspot.com/\x26vt\x3d53093167121198245', where: document.getElementById("navbar-iframe-container"), id: "navbar-iframe" }); } }); </script>

Monday, May 30, 2005

An argument from continuity

Two sophomores approached my colleague Josh with a question: “How can we construct a fair 5-sided die?”

Josh posed a prior question: Is it even possible to construct such a die? He fashioned an interesting argument from continuity: Consider two square pyramids with the same height — one with a tiny base (much smaller than the lateral faces) and one with a very large base (much larger than the lateral faces). Clearly the first has a very small chance of landing on its base and the second has a very large chance of doing so. By gradually moving from the one to the other, there must be an intermediate point that makes the pyramid a fair die.

Josh used this teachable moment to explain the difference between an existence proof and a constructive proof. We now know that a fair 5-sided die is possible, but we have no idea how to construct one!

Labels: ,


ARCHIVES

This page is powered by Blogger. Isn't yours? Made with Macintosh