2. Starting with a list of three numbers, the "Make-My-Day" procedure creates a new list by replacing each number by the sum of the other two.
For example, from $\{1,3,8\}$ "Makee-My-Day" gives $\{11,9,4\}$ and a new "Make$M y$-Day" leads to $\{13,15,20\}$.
If we begin with $\{20,1,8\}$, what is the maximum difference between two numbers on the list after 2018 consecutive "Make-My-Day"s?
2 Claim: 19. In fact, the "MakeMyDay" procedure does not change the maximum difference between two numbers on the list. Suppose our list is $\{a, b, c\}$ with $a$ $<b<c$. The maximum difference between the largest and the smallest number is $c-a$. The "MakeMyDay" operation creates $\{b+c, a+c, a+b\}$.
Since $a<b$, we know that $a+c<b+c$. Since $b<c$, we also know that $a+b<a+c$.
Combining these two inequalities, we have $a+b<a+c<b+c$.
The maximum difference between any number is $(b+c)-(a+b)$ or $c-a$.
So the same as the one we started with. For the initial list of $\{20,1,8\}$, the maximum difference will always be 19 .