In the Monticello game, a player makes a filled triangle - maybe "pyramid" is the best word - of coins and then takes $1$ more coin. The goal is to then rearrange the big pyramid and the single coin into smaller pyramids. The score - which we want to maximize - is the side length of the smallest pyramid.
For example if the starting pyramid's size is $5$ the player starts with 16 coins and the best score is $3$. We can write this in terms of triangular numbers as follows $$T_5+1=16=T_4+T_3\quad\Rightarrow\quad \text{score}=3$$
What is the largest side length of the starting pyramid for which the best possible score is $100$?
$\bullet\textbf{First steps}$
Let's choose a function - $\mu$ - to represent the best score when starting with $n$ coins. So for the previous example we have $$\mu(16)=3$$ It's an immediate consequence of this definition that $$\mu(T_n)=T_n$$ $$\text{i.e.}\quad \mu(1)=1 \quad \mu(3)=3 \quad \mu(6)=6 \quad \mu(10)=10 \quad ...$$ But regarding the original game, we want to know $\mu(T_n+1)$ in general. The values we get for $n=1,2,3,...$ are quite chaotic $$\mu(T_n+1): 1,1,1,1,3,3,2,3,4,7,4,5,7,7,10,7,7,...$$ When graphed, it looks like $\mu$ obeys some strict linear bounds.
Other properties of $\mu$: $$\bullet\quad \mu(a\cdot b) = \text{max}\big[\mu(a), \mu(b)\big]$$ $$\bullet\quad \mu(a+b) = \text{min}\big[\mu(a), \mu(b)\big]$$ One might also be able to establish an upper bound for the last-occurrence of any integer in the image of $\mu$ as I was trying to do here.
Some graphs of $\mu(T_n + 1)$, which I can't put in a comment, are below.
Here is a plot of the first $1000$ values, showing the clear linear tendency you mention:
This is much more obvious when we look at the plot of $\frac{\mu(T_n + 1)}{n}$:
It's unclear whether there are nice lower and upper bounds on the sequence from this picture; probably it is true that $\frac14 n < \mu(T_n + 1) < \frac34n$, but this doesn't seem to get at the general tendency, which is to converge (from both directions) to something like $\frac35 n$.