Solutions for Act IV Scene 4 - Connect Four

The overarching theme here seems to be the number 4, and with this in mind, looking carefully at the grid reveals we've been presented with a series of examples of the Four Fours game. Going either vertically down a column or horizontally along a row, the mathematical operators distinctly segregate sets of four 4's (they Connect Fours). For example, the top of the first column reads 4^4-4!+sqrt(4), implying the number 234 if we follow the usual order of operations. All the Four Fours are highlighted below:

The fact the 4's in the grid are greyed out implies we might want to write over them, but it's not exactly clear what should be written on them yet. For instance, we might try writing "234" or "0234" over the top of the first column, but after working out few more Four Fours solutions, it turns out these numbers can't possibly be written in without clashing digits.

The key step here is to look back to the theme and realise that a more suitable system would be to enter the derived numbers in base 4. Then the number 234 would become 3222, which seems a more apt number to enter into the top of the first column, and furthermore corroborates with the horizonal numbers that intersect it. Our assumptions thus confirmed, we can fill out the whole grid in this way:

(Note this can also be used to confirm what the operations and their ordering is. In particular, the dot before a 4 represented .4, or 4/10, while the dot-plus-bar represented .4 recurring, or 4/9).

We now have a grid that contains only the numbers 0 through 3, and has lightly coloured edges with heavily marked corners. All signs point towards this being a Slitherlink puzzle, one of the better known Japanese box/Nikoli puzzles. It's clear this Slitherlink is slightly unconventional in that it can't possibly contain only one closed loop, but if we approach it assuming that it contains a set of distinct, disjoint closed loops, the grid is uniquely solvable:

Finally we want to extract the answer. The loops seem to spell out the message LEGEND OF ZELDA LEVEL, but what level can it be referring to? Of course, we want to use the puzzle theme one more time, and find the name of The Legend of Zelda's fourth level.

It turns out this level is named THE SNAKE, which is an especially fitting answer since snakes Slither-, while the Legend of Zelda series stars -Link!

The answer is: thesnake