This is a huge question, and I think people can help each other here so I'm going to start with some initial strategy:
I have set up this board:
Which allows for this move:
For a score of 2372 (Edit: confirmed by OP)
My solution was already accepted, but (I assumed) far from optimal. I finally made myself a prime checking algorithm and checked the 32 numbers that allow maximum scoring for the main 'word'.
Edit: Score increased to 3532
Can we remove the magic?
This is the path I took to make a magic square into the grid:
First, let's take the magic square given:
We need to make it to:
So with that,
That helps out a ton, as the rest is pretty trivial:
Well, say goodbye to the magic in the 3 by 3 magic square, and say hello to a normal 3 by 3 square!
Assuming that Bo and Jo cannot set up secret stashes from each other (as otherwise the problem is fairly trivial), but assuming that they can set up a shared stash, one strategy they could use is to:
With the "Use as many and any black or white pieces (both kings included)" restriction, the theoretical maximum becomes
Obviously, there can't be any legal position with that many white pieces, so we throw legality out of the window and start from here:
And as long as we are careful to eat the bottom two pawn rows in good order, we are good to go ...
Variation 1. position must be legal:
As there are at most 15 capturable white pieces this is a hard upper bound on the chain length.
This bound can
Variation 2. position need not be legal:
I can do
Let's start by removing the lower bound - say you can go into stamina-debt as far as you need to, and you keep playing until you escape. Then:
Okay, but what does this say about the actual problem?
In other words:
Given an unbounded number of trials, we could simply test one egg over and over and over again until it breaks, and find our floor.
"But that's too slow", you say, and you're right - so instead we
The only information we need to solidify our method is...
The best strategy is to
This will take
steps, which is optimal: it breaks the least possible number of eggs, and if you were to use some other strategy to choose the floors,
so the expected number of steps won't be any less for that other strategy, either.
This happens because "infinity is really, really big", or more specifically:
And so is, ...
I really hope I'm understanding the rules correctly...
Here's a battle plan, using the original capture-on-adjacent rule (with the stipulation that you don't get extra movement from such captures) and starting on a Monday:
For the harder version where captures have to be exact,
As a supplementary answer I implemented a breadth-first search with duplicate suppression (so e.g. if a state is reachable by swapping A with B and then C with D, it only keeps one of the A<>B C<>D or C<>D A<>B paths), scored by the "total moves left" which monotonically decreases with each swap.
Assuming no errors this ...
If the devil always chases straight after me, following my movements tropistically, then
But if he's smarter than that,
But I must be missing something, because variations in my speed wouldn't seem to matter to this solution, as long as I can move at least as fast as him.
Bo and Jo must:
Bo's travel time is:
Jo's travel time is:
Bars 1 and 2 are the easiest to solve:
From here things get interesting, but essentially:
Which means we must calculate:
Bars 3 and 4:
Summary of Train Station Arrivals
Missed the 5 minutes early being important.
T. Linnell has it done properly.
Bigger question, how they get the bars on the train quickly without someone pushing the other out at the last second?