This is basically a follow-up question to Hiding/Obscuring position information in a board game which technically answered the question, but raised some well deserved critique.
To quickly summarize the question: Consider a board game with a rectangular 10x10 grid, the player has a position $(p_x p_y)$ and a physical token on this grid. Additionally one or more AI opponents roam the board but have no physical token that the player could observe. The question now is if there is a way to tell the player if he is within distance $d$ of any AI without disclosing the actual location of the AI. From a game perspective the only interesting distances are $d=0$ (AI has "found" the player) and an arbitrary, "close" distance of $d<3$ (AI is near the player and he can avoid or engage it).
As this is a board game, no computer is to be involved. All calculations have to be performed by the player (ie have to relatively simple). The (accepted) answer to the previous question involved hashing the players and AIs position, but this would of course disclose the position of the AI to the player.
In the answer I wrote up on the BGG forum, I "pre-scrambled" both the players and the AIs position, obscuring the relationship between $(x,y)$ position and the numbers that get hashed to "lookup" the distance. The critique was that any observant player would quickly find out that the pre-scrambled AI numbers corresponded to locations on the map.
My "solution" to that would be to obscure the relation between input and output further, by having only using the current turn order and player field id as the input, effectively hiding the current AI locations behind the turn number.
Still, I feel like there might be a clever scheme to disclose just the current distance (or "something is close") to the player without disclosing the actual AI position. Having the game playing out in a heavily distorted coordinate system or similar.
I would be interested in what you guys can come up with.