Description
[EMERGENCY TRANSMISSION - GALACTIC COMMAND]
Subject: Mission Omega-7 - Navigation through the K-94 Tesseract Anomaly.
Background:
The Omega-7 probe has entered a non-Euclidean hyper-dimensional region known as a Tesseract Grid. The probe’s onboard computer is failing, and you must provide a sequence of movements to reach the "Stability Core" before the probe's energy reserves deplete or it is crushed by a Quantum Anomaly.
1.The 4D Toroidal Spacetime:
The grid is a 4-dimensional hyper-cube of size 
. Coordinates are given as 
where 
. The space is toroidal in all four dimensions: moving from 
in any direction with a step of 
results in 0.
. Coordinates are given as where . The space is toroidal in all four dimensions: moving from in any direction with a step of results in 0.2.Goal Decoding (The Eigen-Lock):
The coordinates of the Stability Core 
are hidden within a 4 x 4 Quantum Matrix M. To decode 
:
are hidden within a 4 x 4 Quantum Matrix M. To decode :- Find the four eigenvalues of
. - Take the absolute value of each eigenvalue.
- Sort these values in ascending order:
- Apply a permutation vector
( where P contains indices 1 to 4) such that 
- The final coordinates are
3.Dynamic Quantum Anomalies (4D Cellular Automata):
The grid is infested with anomalies. Let 
be a logical 4D array where 
means an anomaly exists. The field evolves every second 
based on the number of neighbors K(where each cell has 
- 1 = 80 neighbors):
be a logical 4D array where means an anomaly exists. The field evolves every second based on the number of neighbors K(where each cell has - 1 = 80 neighbors):- Birth: A cell becomes active if it was dead and has exactly B neighbors.
- Survival: An active cell remains active if it has between
and 
neighbors. - Energy Flux: Each active neighbor of a cellB at time t increases the "Action Cost" of moving into that cell by 0.1
4.Space Warping ( The Wormhole Effect ):
There are two special coordinates, 
and 
. If the probe enters at time t, it is instantly transported to at time 
(and vice versa) with a fixed cost of 0.5, regardless of local anomalies.
and . If the probe enters at time t, it is instantly transported to at time (and vice versa) with a fixed cost of 0.5, regardless of local anomalies.5.Energy and Action Constraints:
- Max Energy: The probe starts with
. - Consumption: Each move (including staying still) consumes 1.0 unit of energy. If
, the mission fails. - Action Cost: The total "Action" to be minimized is the sum of costs:
- Movement: In one second, the probe can move to any of its 80 neighbors or stay in place. However, it cannot enter a cell that is currently "Active"
at time t.
Goal: Find the minimum total Action to reach 
steps and before energy runs out.
steps and before energy runs out.Solution Stats
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Testcases have been updated
A bizarrely complex and poorly explained problem with a trivial set of 5 static test cases, every one of which was simple enough that apart from decoding the goal, I could find and score the least cost path in my head, including the 4-D toroidal space. I don't have great visualization in 4-D, but I can track a vector of 4 integers and a subset of their adjacent coordinates. IIRC, dimension 3 never played a critical role.
More details about the testing:
* 4D Toroidal Spacetime: In every test case, there is a minimal path which does not "wrap" in any dimension. Annoyingly for Matlab, stated coordinates start at zero. I have asked The Math Works for an "array-friendly" version of mod() which never returns a zero or negative value (e.g., modf(0,N) returns N), but so far, no joy. (This is not just for Cody; in my real world problems, I have written several variants of this function to make my higher-level code more clear). I'm pretty sure I wrote something similar in FORTRAN 40 years ago as well. Clear code is maintainable code.
* Goal decoding: This one is just silly, and untested as the permutation matrix is uniformly 1:4.
* Dynamic anomalies: Only used in one test case, and then only for the first second because it's empty afterwards.
* Space Warping: This one's actually used twice! but its action cost is unclear; does it replace the non-warp action cost, or add to it? From the test cases, I see the answer is replacement.
* Energy constraint: never used. Energy used is never more than 5, and energy available is never less than 50.
* Action Cost: nothing new, but poorly explained, even more poorly tested.
* Movement: "Active" cells cut off some optimal paths, but never ALL of the optimal paths. As such, unused.
* T<=100 steps: clearly unused. Every problem is solved by T==5 seconds.
Ignoring the problem and focusing on the test cases, I have developed a solution of size 21, consisting of 240 characters, only a few characters more than the size of the leading solution, which I suspect took some shortcuts.