My former colleague Toby Berger (now at U of Va)
has been looking at this problem. Has anyone seen
anything like this before?
%I A134939
%S A134939 0,2,64,1274,21760
%N A134939 Consider a 3-pole Tower of Hanoi configuration which begins with n rings on pole 1. Moves are made at random, where the 1-step transition probabilities out of any state are equal. Let e(n) be the expected number of transitions to reach the state in which which all rings are on pole 3. Sequence gives a(n), the numerator of e(n).
%C A134939 Both allowable transitions out of any of the three special states in which all the rings are on one of the poles have probabilty 1/2, and each of the three allowable transitions out of any of the other 3^n - 3 states have probabilty 1/3.
%C A134939 It appears that the denominator of e(n) for n>=1 is 3^(n-1).
%e A134939 The values of e(0), ..., e(4) are 0, 2, 64/3, 1274/9, 21760/27.
%K A134939 nonn,frac,more,new
%Y A134939 Cf. A134940.
%O A134939 0,2
%A A134939 Toby Berger (tb6n(AT)virginia.edu) Jan 23 2008
Neil
----- Original Message ----
From: David Gale <gale(a)math.berkeley.edu>
To: math-fun <math-fun(a)mailman.xmission.com>
Sent: Thursday, August 9, 2007 3:17:48 AM
Subject: RE: [math-fun] How we "do mathematics"
...
>A (very) old one I like :
>
>Two Death Valley marathonians friends complete their run
>together and seek shadow in their tent nearby the finish
>line.
>
>The first one had prepared a fresh 5-liter tank of water
>in a cooling box and his friend a similar 3-liter one.
>
>They were about to start drinking when a journalist enters
>the tent desperately asking for water.
>
>OK, the water is divided into three equal parts -- and drunk.
>
>The thankful journalist insists to pay his share and leaves
>8 dollars to be split between the two friends.
>
>How?
>
>----
Why is this problem so fascinating? The two friends are paid $8. The guy who contributed 5 l gets $5, and the guy who contributed 3 l gets $3. Or maybe the $8 goes towards paying for their campsite. I must be missing something.
Gene
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