Non-zero positive integers, not necessarily distinct, are written on the squares of an $8$ × $8$ chessboard. Unorthodox number theory problem.

Here is the full problem : Non-zero positive integers, not necessarily distinct, are written on the squares of an $8$ × $8$ chessboard (one number per square). At the beginning, five grasshoppers are on five differentsquares and hide the numbers. Gabriel calculates the sum of all the visible numbers and he obtains $100$. Simultaneously, each grasshopper jumps onto an adjacent square (it crosses a side shared by two squares). Gabriel calculates the sum of all the visible numbers and he gets $1000$. And so on, until he can no longer obtain a sum ten times greater than the previous one (when Gabriel calculates a sum, two grasshoppers are never on the same square). The total of the sixty-four numbers written on the chessboard is divisible by $35$ and it is the largest possible. What is this total?

This was the only problem on a math competition that I couldn't figure out and I still don't know how to solve it (I don't even know if I'm supposed to use number theory or combinatorics...) . The detailed solution isn't available on their website, the only thing that is said is that the answer is $11110785$. If anyone could help me figure out how to solve it, it would be appreciated!

Solution 1:

Hint: each time the grasshoppers jump a new large value must be exposed so the sum increases so much. There were only five squares hidden at the start, so there are at most five steps upward. You can think of the grasshoppers all being next to each other in a row and jumping one square in the same direction. They can turn the corner if they are going to leave the board. So each time one new square is revealed and one deleted. Now use the factor $35$ to refine the problem.