Question

I recently interviewed with Microsoft and they asked me the following puzzle, for which I had to write an algorithm and accompanying test cases. I wasn’t able to crack it and it still is a puzzle to me.

Problem Statement :

A champagne pyramid is a pyramid made of champagne glasses , each of equal capacity say , n.
The pyramid begins with one glass at the top level , two glasses at the second level , then three below that and so on up to infinite levels. A level x of the pyramid thus has x no. of champagne glasses.

A steady stream of champagne is poured down from the top level,which trickles down to the lower levels. What is the distribution of champagne in the glasses at a given level i.

The problem is quite abstract and those are all the inputs I was given.

Answer 1

The answer is Normal Distribution I believe.

Have a look at the diagram:

           |1|
           ---
        |2|   |3|
        ---   ---
     |4|   |5|   |6|
     ---   ---   ---
   |7|  |8|   |9|   |10|
   ---  ---   ---   ----

Let’s say you have a flow of X

1 will flow into 2,3 uniformly, thus each gets 1/2X

each will flow uniformly to the glasses below it, so 4 gets 1/4X, 6 gets 1/4X and 5 gets 2*1/4X= 1/2X

At next level – the same principle applies:

7 gets 1/8X
8 gets 1/8X from 4 and 1/4X from 5, totaling 3/8X,
9 gets same as 8 and 10 same as 8.

At infinity – it should converge into normal distribution.

At any finite number i – it should be f(i,n)/ 2^(i-1) where f(i,n) is the nth binomial number for level i polynomial. As @veredmarald indicated in comments, that distribution function is actually Binomial Distribution for p = 1/2, thus giving you flow(i)~Bin(i-1,1/2)