.....
...#.       .E  ...  ....
..#..       S.  .#E  ..#.
.#..E           S..  .#.E
S....                S...

Fig 1.         Fig 2.

Consider the layout for a grid with ~R = C = 5~ in figure 1. Tom Cows will reach the ending cell in ~5~ minutes (for example by following the sequence of moves →→→→↑). On the other hand, Monkey Freeman will move as follows:

• Up (the cells above and to the right both have equal Manhattan distances of ~4~ to the ending cell, so he'll choose the former)
• Up (similarly choosing it over moving down)
• Right (similarly choosing it over moving down)
• Up (similarly choosing it over moving left)
• Right (similarly choosing it over moving down)
• etc…

This pattern will repeat until Monkey reaches the grid's top-right cell, at which point he'll move straight downwards to the ending cell. For this particular grid, Monkey will require ~11~ minutes, which is slower than Tom by the minimum required amount of ~2C-4 = 6~ minutes. More generally, this pattern can be extended to any square grid, as in the examples in figure 2.

For any such grid, Tom will need ~C~ minutes while Monkey will need ~3C-4~ minutes, yielding the minimum required difference of ~2C-4~. If we need to fill a non-square grid (such that ~R > C~), then ~R-C~ additional rows of vacant cells may simply be added below this pattern without affecting anything.