# Power mean as an indicator of a person’s pessimism

the program itself I won’t write too much, everything is quite simple there, it’s moved on the forehead, the code is one-time, it takes 10-15 minutes to write, it counts 15-20 seconds. Found ratethen it turned out that you could just google.

Then I decided to find two starting words, it was already considered longer, I found a couple carse + doilt, after which, on average, only four words remain. And then Ostap suffered.

These words are the best average. And what words will be the best even in the worst case? In order not to pull intrigue – this Serai and couple tired + loans. Even in the worst case, no matter what word is guessed, after tired loans, no more than 16 words are guaranteed to remain. This is the result for the most notorious pessimists who always expect the worst from life.

These programs sort through – for each possible word from the set of guesses (or for each pair), they sort through all possible hidden words and count how many suitable words should remain after the game’s answer. And consider the average or take the maximum. But besides the average realist and the extreme pessimist, there can be all sorts of intermediate options. Is there any way to generalize this calculation?

It turns out you can (spoiler right in the title). There is such a thing called power meanthough i like the name better generalized mean:

What is the beauty of this formula? At k = one it gives the arithmetic mean. At k = 2 get the root mean square, k = -one will be the harmonic mean, k = 0 (more precisely, in the limit, at k 0) is the geometric mean. All four mediums! And plus infinity will get just the maximum of ai. At minus infinity – the minimum, this is in Worlde for those incorrigible optimists who believe that they will guess right on the first try.

So everyone can choose k corresponding to your own personal level of pessimism and choose the words for Wordle that are optimal at this level. But how to find your level of pessimism?

You can do the reverse process. Estimate the average of several numbers by eye. The closer it is to the maximum, the more optimistic the estimate. Well, or to the minimum, semantics is important here. Let the numbers correspond to the amount of money that you are given, the more the better. Then the closer the estimate is to the maximum number, the more optimistic it is. Having picked up the appropriate k you can get a numerical value of the level of optimism. But since the scale from minus to plus infinity, with the neutral point at unity, is somewhat awkward to handle, one can project the number axis onto a semicircle, getting a scale from -1 (pessimist) to +1 (optimist), with the neutral point (realist) at zero.

And, of course, posting this test onlinebecause what else is it good for?

Something even tells me that after passing this test several times, you can train to guess the average much more accurately. And, therefore, to become much more realistic (actually not).