(context: note that this question is asking for intuitive explanation why it's not equal, so a good answer would have to explain intuitively why the intuition that they will be equal is not the right intuition, which is what my answer is)
Because the order of who opens a specific box first is important in this case. If Chris opens a box X after Bob, then ...
Thanks to Albert.Lang for pointing out the case where this logic fails: a 4x5 pattern. Actually, there are numerous cases in which it fails, but it seems like it works in the vast majority of cases.
While it doesn’t feel good that my intuitive solution of simply looking at who has the longer lines to work with fails for all cases, I can take some ...
The first thing standing in the way of gaining good intuition is the mildly complicated way the first marked box to be opened is determined.
Based on the order by which they are first scanned by either player
1 2 3 4
2 5 7 8
3 6 9 10
we should use the ordering
A < B=E < C=I < D < F < J < G < H < K < L
. . . B C ...
Consider the following cases:
There is an egg in A
There is no egg in A, but exactly one in B or E
There is no egg in A, B or E, but exactly one in C or I
There is no egg in A, B, C, E or I, but exactly one in D or F
There is no egg in A, B, C, D, E, F or I, but exactly one in G or J
Something else happens: the only remaining possibilities are BE, CI, DF, ...