I'm sure this has been asked before somewhere, though I do not know the name of the problem/algorithm, so here goes.
We are asked to test mailboxes with firecrackers, and find the resistance of a new type of mailbox. We need to find how many firecrackers it can take at once. We are to test with up to $k$ firecrackers, and have $i$ mailboxes for our tests. What is the minimal number of firecrackers you need, to run a test such that you for sure know how many the mailbox can resist? Once a mailbox has blown up, it can't be reused.
I formulate this as $T(i,k) = x$. If $i$ is one, this is just the sum of the integers, since we can't afford to guess, and blow it up accidentally.
Example; $T(2,10) = 28$. The optimal place to start is with $7$ firecrackers in the box. If it blows up, we only have one mailbox left, and you have to test from 1 up to 6, hence a sum of 28 firecrackers in total, in the worst case. Looking closely, this is just
$$T(2,10) = 7 + T(1,6) = 7 + \frac{6(6+1)}{2} = 28$$
So we expect some recursion here. If it didn't blow up, you can reuse it and try with 9, and then in the worst case 10, thus 26 in total, clearly less. So, to be completely sure we find the actual resistance, we need $x=28$.
Q: How can we in general calculate $T(i,k) = x$? E.g. $T(6,67) = $?
Edit - SOLUTION: Number of tests with $i$ boxes in interval $[k,l]$ (including ends) is
$$T(i,k,l)=\min_n\left(n+\max\left(T(i-1,k,n-1),T(i,n,l)\right)\right)\; $$
Base cases for $i = 1$ is computed as above. If $k+1=l$ we just get $2k+1$. Memorizing the values of $T$ helps the computations go faster. We simply iterate over all possible $n$'s. More base cases are possible, once you start thinking about it, but not necessary.
EDIT: Found a link to the problem, and it has a name too https://www.ida.liu.se/projects/progcontest/progsm/2002/prognm2002-hints.pdf.