Let's say instead of seeing statues, each prisoner was given a piece of paper that had an integer on it (ranging from -infinity to +infinity) and on your paper was written 12.
Would the problem be unsolvable? If so, what has materially changed?
What has materially changed is that there no longer are any situations where someone knows the answer instantly. The original puzzle only works because everyone has a non-negative integer and their sum is constrained to a finite number of possibilities. So someone who sees a number larger than the second-highest possible total knows that the value of the total is the highest possibility.
The key to understanding the puzzle is that knowing, knowing they know, knowing they know you know, etc. are increasing strengths of knowledge. You can always remove "knows" from the stack, but you can't easily add them (because how do you know they know?). Passing on a given day is the mechanism that introduces knowledge of infinite depth (because everyone knows they are neither freed nor dead and they all know they know they know ...) Because knowing the answer implies that they don't pass, observing a pass eliminates all possibilities where the answer is known (more precisely, it adds one depth of "knows" to the knowledge that that possibility isn't the case.)
Let's say instead of seeing statues, each prisoner was given a piece of paper that had an integer on it (ranging from -infinity to +infinity) and on your paper was written 12.
Would the problem be unsolvable? If so, what has materially changed?