How to calculate max iterations needed to equally increase row of numbers per some value each iteration? I don't know whether title describes the main idea of my question, so apologize me for it.
I have 6 numbers whose values can vary from 0 to 100, but initial value cannot be more than 35. As example, here is my number list:

20, 31, 15, 7, 18, 29

In one iteration we can distribute some value (5, 7, 10, 15 or so) among these numbers. Let it be 15. And each number must be increased at least once per iteration. So one iteration may look like:

20 + 3 => 23
31 + 2 => 33
15 + 3 => 18
7 + 5 => 12
18 + 1 => 19
29 + 1 => 30

The question is: how to calculate the max amount of iterations for any number row with constant distribution value per iteration? One should know that iterations must stop once one number reaches value of 100.
What math field should I learn to get more info on alike questions?

You start with $a_1, \ldots, a_n$, have to distribute $d\ge n$ in each round and stop at $m>\max a_i$.
Then the maximal number of rounds is bounded by two effects:


*

*The maximal value grows by at least one per round, so it reaches $m$ after at most $m-\max a_i$ rounds.

*The total grows by $d$ each round, so you exceed $n\cdot (m-1)$ after at most $\left\lceil\frac{n(m-1)+1-\sum a_i}{d}\right\rceil$ rounds.


Interestingly, these two conditions are all there is, i.e. the maximal number of rounds is indeed
$$ \min\left\{m-\max a_i,\left\lceil\frac{n(m-1)+1-\sum a_i}{d}\right\rceil\right\}.$$
This can be achieved by (in each round) distributing one point to each term and then repeatedly increase the smallest term until all has been distributed.
The field of math this belongs two would be combinatorics.
