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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link="#0563C1" vlink="#954F72"><div class=WordSection1><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>A question I posed a long time ago, but I have new insights.<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Imagine a cube a cm on a side, tossed like a gaming die. By symmetry, any face has the same probability of coming up (or down): one in six. Likewise, a tetrahedron could be tossed and each face would have a one in four chance of landing face down. The same symmetry argument could be made for any of the five platonic solids.<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Cool but what if we wanted to create an unsymmetrical solid, such as a square based pyramid one cm on a side for the square, and we wanted to make it such that the probability of landing face down on any of the five surfaces was the same. How tall do we need to make it?<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><img width=341 height=245 style='width:3.552in;height:2.552in' id="Picture_x0020_5" src="cid:image001.jpg@01D287DA.74734F60"><o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>I get a CG height of h/4.<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal><img width=460 height=247 style='width:4.7916in;height:2.5729in' id="Picture_x0020_6" src="cid:image002.jpg@01D287DA.74734F60"><o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>So if we want to make a fair pyramid shaped five sided die, what is h?<o:p></o:p></p><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>spike<o:p></o:p></p></div></body></html>