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I am trying to create a scoring scale for a game. I want the scale to be non-linear to reflect the realness of the game. The score is determined by time, the less time you take, the greater score you get.

  • If you take more than $5$ minutes to complete the challenge, your score is $0.$

  • If you take $0$ seconds to complete the challenge, your score is $40.$

My current equation is linear: $y = {\rm MAX\ SCORE} - \frac{{\rm MAX\ SCORE}}{{\rm TIME\ TO\ MIN\ SCORE}}\times {\rm time}$

However, I want the score to accelerate more quickly towards $40$ the closer you get to 0. What equation can I use to do this?

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    I made a$n$ edit to e$n$hance readability. Please make sure that I did not change the semantics of your question.2012-03-31

2 Answers 2

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You can solve this:$f(x) = {a \over {1+x}} + b$ $s.t: f(0) = \text{MAX_SCORE and } f(\text{TIME_TO_MIN_SCORE}) = 0$

which results to (for your case with times in minutes you can also solve it for seconds, ...):

$f(x) = {{48 \over (x+1)} -8}$

Update: $f(0) = 40, f(5) = 0 \Rightarrow$ $a+b=40, a + 6\cdot b = 0 \Rightarrow$ $a=48, b = -8$

I assumed your time is distributed in minutes, you can have your other assumptions, e.g for seconds it will be like:$f(0)=40, f(300) = 0, ...$

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    It get's truncated to two decimal places. The actual use case is a game were you serve as a waiter(ess) to a table. The faster you complete your tasks, the greater tip (score) you get.2012-03-31
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An easy approach is to do $score =maxscore (\frac {maxtime-t}{max time})^n$ where you choose $n$ as high as you like to make the score fall quickly at the start. Alternately, $score=0.1maxscore(\frac {maxtime}{0.1maxtime+t})$, again adusting the 0.1 to your liking.