Tokens in boxes problem Tokens numbered $1,2,3...$ are placed in turn in a number of boxes. A token cannot be placed in a box if it is the sum of two other tokens already placed inside that box. How far can you reach for a given number of boxes?
For two boxes it's quite straightforward to see that it's impossible to place more than $8$ tokens (box $\#1$ contains $1,2,4,8$ and box $\#2$ contains $3,5,6,7).$ Using a computer to try many solutions [1], I was able to show that $23$ is the maximum for $3$ boxes. After trying more than 10 billion solutions, I have found a solution going up to $58$ for four boxes, but I am unable to show that it's the optimal solution.
Is there a more clever way to find the answer?
[1] https://gist.github.com/joelthelion/89e1a98c73ea6784bcbd4d7450b0bd5e

These numbers are tabulated at https://oeis.org/A072842 – but not very far. All that's given there is 2, 8, 23, 66, 196, and that last number is a little shaky. Several references are given for further reading. Some bounds are given for large numbers of boxes, but the upper and lower bounds are very far apart. 
So, if you find a good way to go beyond 5 boxes, you're onto a winner. 
