On Fri, Sep 21, 2012 at 3:06 PM, meekerdb <meekerdb@verizon.net> wrote:
On 9/21/2012 9:34 AM, Allan Wechsler wrote:
If you make a longish pentagonal prism, and cap the ends with pentagonal pyramids, the resulting 15-faced die has 0 probability of landing on one of the triangular faces, and 1/5 of landing on each of the rectangular faces. All this is by symmetry, and doesn't depend on the physical model except that we expect the die to topple off an unstable face. This of course is not a solution to the D5 problem Warren proposed.
I wonder if there is a pentahedron that is fair regardless of physical model. My intuition is "no", but I don't see any way to prove that.
I think the answer is yes based on a kind of continuity argument. Consider a three sided prism with equilateral triangular endcaps. If the prism in long compared to the triangular ends, then each rectangular face will come up with probability 1/3. If you make the prism short enough the probability of the rectangular sides will go to zero and the ends, by symmetry, will each be 1/2. So there must be a length at which the rectangular sides each have probability 1/5 and that will imply the ends also have probability 1/5. Note that this kind of reasoning generalizes to higher N.
Sure, but it doesn't seem obvious that the fair height for a die rolled, say, underwater would be the same as for a die rolled in air. --Michael
Brent Meeker
Andy, I haven't frequented casinos. Don't they require you to throw dice from a cup?
On Fri, Sep 21, 2012 at 9:59 AM, Andy Latto<andy.latto@pobox.com> wrote:
On Fri, Sep 21, 2012 at 9:13 AM, Veit Elser<ve10@cornell.edu> wrote:
I bet if one "rolled" a cubic dice more like the standard coin toss one could strongly bias the outcome (against the faces intersected by the rotation axis).
Provided you don't think this bias is almost completely undone by requiring the die to bounce against an irregularly-shaped surface before it lands, there are any number of casinos willing to take your bet. You don't even need a strong bias; a bias of 5% or so on the individual dice should be big enough to get the 2% bias you need on the total to make money at the craps table.
Andy andy.latto@pobox.com
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