How are online tactic exercises ratings calculated?

There are several sites that give you a tactics rating changing while you solve or fail the exercises.

How do they calculate the problems rating?

Do they use engines to calculate the difficulty or strong players evaluate the difficulty exercise by exercise?

• Are there specific sites you are interested in? different sites may do it differently. – Michael West Jan 16 '20 at 11:57
• @Michael West I do exercises on lichess, chesstempo and chess.com Lichess may do it using stokfish is my guess, not sure about chess.com as they have a lot of titled staff. Chesstempo may do it also "exercise by exercise"? I was wondering. Your answer says all about how chesstempo does. – user18196 Jan 16 '20 at 12:01
• Lichess uses the same method as described by Michael. – Annatar Jan 16 '20 at 12:43

A typical way to do it is to treat puzzles the same as players and rate them based on whether they "win/lose" and the rating of the "opponent"

ChessTempo has a nice explanation here copied below:

The rating system is inspired by an idea implemented at the Chess Tactics Server. CTS treats both problem solvers and the problems as opponents with their own ratings and adjusts their rating after problem completion (or failure) based on the Glicko ratings system.

For blitz style games, time taken to solve problems is considered and an extra bonus is given for problems solved quickly. The solver gets a larger rating increase for problems solved quicker than one standard deviation from the average time taken for that problem. For problems solved faster than average but less than one standard deviation from the average, the normal Glicko based adjustment is made. If the problem is solved in slower than average time then the Glicko rating adjustment is reduced in proportion to how much slower than average the problem was solved. Very slow solutions may result in losing rating points, however you will never lose more points than you would if you simply got the problem wrong.

When calculating the average time taken on a problem, only correct solutions are included, time taken from incorrect attempts are not used to calculate the average time a problem takes to solve.

It is important to note that after a recent change to time handling in blitz mode, time taken after the first move is punished more severely than time taken before making your first move. The intention of this feature is to discourage 'incremental' guessing to try and solve a problem, rewarding users who sum up the situation before making their move sequence. Users who prefer to use an incremental approach may still do so, however their rating will be lower than users at the same level who think out the solutions first.

Normal Glicko adjustments are made for standard ratings with no penalty for time taken. Problems marked incorrect cause the rating of that problem to increase and the solver's rating to decrease as dictated by normal Glicko rating calculations. Normal Glicko adjustments are also made for blitz problems that are marked wrong (i.e. the time taken in blitz is ignored when making rating adjustments for incorrect problems).

I believe that the tactics rating start on 100n rating. After solving puzzles, the change in rating decreases, so that eventually, you'll reach a rating which reflects your accuracy.