I have been playing Puzzle Storm on Lichess lately. Essentially it involves solving chess puzzles within a given time limit. Additionally, the player gets a time bonus for a certain number of moves. Chess puzzles are essentially problems where the player needs to find the best move (or sequence of moves) in a given position. I am trying to work out the time period within which I need to make one correct move so that I can go on playing the game infinitely.
Now the exact rules are as follows :
The player gets 3 minutes on the clock. For every wrong move the player loses 10 seconds. For every 5 correct moves, a player gains a 3 second bonus. For every 12 correct movies, the player gains a 5 second bonus. For every 20 correct moves, the player gains 7 seconds. For every 30 correct moves, the player gains 10 seconds. After 30 correct moves, the player gains 10 seconds for every subsequent 10 correct moves.
For simplicity I am assuming the player only makes correct moves. I want to figure out what is the longest time period per move in order to play this game infinitely/ never run out of time.
My attempt at a solution :
I tried to solve a simpler version of the same problem first. I assumed that the player has 3 minutes at the start and gains 10 seconds for every 10 correct moves. I am not too sure if a linear in-equation would be used to model this problem or a linear equation.
If I let $x$ be the time period to make one correct move, then $1/x$ would be the number of correct moves per unit time. Now, if $y$ is the total number of correct moves (which seems paradoxical since I am looking for a way to get infinite moves!) then $x*y$ is total time taken. Now since I get 10 bonus seconds after every 10 correct moves, $10*x$ should be the time taken to get one 10 second bonus. So if I can always have enough time leftover to get a 10 second bonus then I should be able to play without ever running out of time. Hence I think this should work... $$ xy - 180 \ge 10x $$ Plot for this in-equation
I am not sure if this is correct, hence I have not moved on to the real case with the variable bonus times. Although in the event my approach is correct, I am a little unsure about how to create expressions for the bonus times (if adding the initial time bonuses directly to -180 works, then the only change would be to make the equation $xy - 180 + 3 + 5 + 12 + 7 + 10 \ge 10x $)
