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Ah sorry I misunderstood! The concept of lives actually arose from a lot of playtesting and valuable feedback during the beta. Many players were feeling that th

by ramly 3y ago

Ah sorry I misunderstood! The concept of lives actually arose from a lot of playtesting and valuable feedback during the beta. Many players were feeling that there is no real 'cost' to brute-forcing the exploration of every possible path on the board. Granted, the time complexity of doing so is really high. But from an emotional standpoint, beta testers seemed to appreciate more often than not the adrenaline rush that comes with a low number of remaining lives and the renewed focus that comes with it. One playtester mentioned that they only felt they got to the 'flow' state and finally resolved an exceptionally hard level when they were calculating every move on their last life.

BoiledCabbage 3y ago
Ah I see, thanks for the explanation!
- ramly 3y ago
  Sure thing. Keep in mind by the way that it's actually already prohibitively difficult for a player to truly brute-force their way around the board, especially in larger boards. Below is a rough intuition-based proof of why that is. If you think about the chess board in abstracted terms, you can actually map an echo chess puzzle to a graph theory problem. First, you disregard any empty squares, and you consider each piece-square as a node, with its neighboring nodes being any node that can be reached from this node using any number of >0 valid moves without reaching any interim node on the way. Using that process, you can now convert an entire echo chess board into a graph where each chess piece is a single node. Now you can ask the question, "what is the objective of solving an echo chess puzzle in terms of this graph?" It turns out that since the goal is to clear all pieces of the board, and in order to clear pieces we have to reach their corresponding square, then in graph terms our goal is to visit every node. However we can only visit every node once at most (because once a piece is captured, it is taken off the board). In other words, an echo chess puzzle is the same as "visiting every node on the graph exactly once". That's the HPP which is NP-complete, and if you're looking for the shortest HPP, it's even NP-hard. TL;DR: brute-forcing as a player is not the optimal strategy.

by ramly 3y ago

BoiledCabbage 3y ago

Ah I see, thanks for the explanation!

ramly 3y ago
Sure thing. Keep in mind by the way that it's actually already prohibitively difficult for a player to truly brute-force their way around the board, especially in larger boards. Below is a rough intuition-based proof of why that is. If you think about the chess board in abstracted terms, you can actually map an echo chess puzzle to a graph theory problem. First, you disregard any empty squares, and you consider each piece-square as a node, with its neighboring nodes being any node that can be reached from this node using any number of >0 valid moves without reaching any interim node on the way. Using that process, you can now convert an entire echo chess board into a graph where each chess piece is a single node. Now you can ask the question, "what is the objective of solving an echo chess puzzle in terms of this graph?" It turns out that since the goal is to clear all pieces of the board, and in order to clear pieces we have to reach their corresponding square, then in graph terms our goal is to visit every node. However we can only visit every node once at most (because once a piece is captured, it is taken off the board). In other words, an echo chess puzzle is the same as "visiting every node on the graph exactly once". That's the HPP which is NP-complete, and if you're looking for the shortest HPP, it's even NP-hard. TL;DR: brute-forcing as a player is not the optimal strategy.