mathematics – Math formula for: the higher the stress, the higher the difficulty

We can throw your example numbers into a spreadsheet and chart them, say using Google Sheets, and get a look at what the progression looks like:

Difficulty vs Stress curve

We can add a trend line to the graph to try to fit a formula to the data. In Google Sheets, this can be found under:

Chart editor > Customise > Series (Difficulty) > Trend Line

I used these settings, and got a pretty decent fit with a polynomial of degree two (a quadratic):

Trend line settings

We could use exactly the equation that the trend line found for us, but it has a dip in the middle which isn’t ideal. So let’s see if we can make our own version using that as inspiration.

Custom formula

Here I tried squaring the stress value, and dividing by 10000 (bringing a maximum of 100 * 100 stress down to an output value of 1). Then I multiplied the result by 49 and added 1, to get an output between 1 and 50.

That gets pretty close to your numbers, but it over-estimates a little in the middle. Let’s try cubing stress instead. Here I’ll normalize stress between 0 and 1 by dividing by 100, then cube this normalized value, multiply by 49 and add 1 as before. And then let’s take the ceiling (next integer higher than the number we get out). That reproduces your example values perfectly.

Better formula

Moreso than any particular formula though, I hope this demonstrates for you a method that you can use to explore and find formulas on your own.

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