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Sum of product and sum



Write a list from 1 to 100.

Pick two numbers at random

Sum the two numbers, find the product of the two numbers

Sum the above two numbers

Erase the two chosen numbers from the list

Add the sum of the two numbers to the list

Repeat until there is only one number.


Repeat the above process multiple times.  Is it the same number? Do the numbers center around a point? Bimodal?  analyze.


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a ~ b = a + b + ab


associativity: (a ~ b) ~ c = a ~ (b ~ c)

  => (a + b + ab) + c + c(a + b + ab) = a + (b + c + bc) + a(b + c + bc)

  => a + b + c + ab + ac + bc + abc = a + b + c + ab + ac + bc + abc


The ~ is also obviously commutative. Thus, our number can be written unambiguously as 1~2~...~99~100.

It is always the same number.

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