Ten people sit at a round table. The sum of Rupees 10 is to be distributed among them so that each person receives the average of what each of his two neighbours receives.
In how many different ways can this be achieved ?
[Given: 1 Rupee = 100 Paise]
(In reply to
Puzzle Answer by K Sengupta)
Let us denote the respective amounts received by the 10 individuals in Rupee(s) as: P,Q,R,S,T,U,V, X, and Y.
Without any loss of generality, we can assume that P>R
Then, Q= ( P+R)/2, so that: P>Q>R
Again, R is the average of Q and S. Therefore, we must have P>Q>R>S.
Continuing in this fashion we will obtain:
X>Y>P, so that: P>P
This is a contradiction.
Similarly, assuming at the outset that P<R we would similarly obtain P<P by way of reduction ad absurdum.
Therefore, we must have P=Q=R=S=T=U=V=W=X=Y
Consequently, each of the 10 individuals must receive 10/10 = 1 Rupee.
Edited on August 17, 2022, 10:40 pm
Explanation to Puzzle Answer
