What's the best approach for top Puzzle Storm scores?

Puzzle Storm Primer

If you don't already know, Puzzle Storm is a game where you solve puzzles against the clock, receiving a combo bonus for chains of correct moves, and losing 10 seconds for failing a puzzle (as well as losing your combo streak!). Puzzles start off very easy, and get progressively harder. You can see the full rules at this link. You gain +3 seconds for the first 5 correct moves, then +5 seconds for the next 7 moves, +7 seconds for the next 8 moves, and then +10 seconds for every 10 correct moves after that.

What's the question?

If we look at the fractions from the bonus time above: 3/5, 5/7, 7/8, 10/10; we can see that you get the best value in bonus time per correct move after the initial 20 moves - so it's probably good to aim for keeping a streak! Since an incorrect move will reset your combo (putting you at the least efficient bonus zone), and not give you any bonus time for the move, and cause you to lose 10 seconds, accuracy is very important. But speed also matters! We only have 180 seconds to start with so we can't take too long per puzzle. What's the relationship between these factors?

Simulation

To see how variations in speed and accuracy affect the final score, we can simulate attempts of Puzzle Storm. Note, for the rest of this blog I will refer to puzzles and score, but in the simulation they are really individual moves and total correct moves. Estimating and varying puzzle length would be possible but mathematically in the simulation it would be equivalent - a correct move always adds to your combo and can give more time, and an incorrect move always clears your combo and costs you 10 seconds, regardless of which move in the puzzle it is. If you wished to convert from correct moves to puzzle score, puzzles range from an average of 1.5 moves per puzzle at the easiest, to 4 moves at the hardest - a breakdown of moves by puzzle rating is available here. You can also view the full code for simulations and generating the graphs here. I will also refer to accuracy throughout, meaning the target accuracy of the simulation, however it is possible for the final actual accuracy of any simulated run to deviate from the target accuracy.

Analysis

As an initial simulation, let's look at how speed and accuracy affect final score averages, with a range from 50% to 100% accurate, and taking between 0.5 and 9.5 seconds per move. If we look at the top line, even with 100% accuracy you have to solve as fast as 2 seconds per puzzle to complete a run of 137 puzzles (the current upper limit). We can also see that each step up in accuracy has a greater impact on final score than the previous step (most evident at 0.5s per puzzle), due to the importance of preserving the combo bonus. For example, at 0.5s, the gap between scores for 60% and 70% accuracy is around 15, but the gap between 70% and 80% is around 25. For 90% to 100%, we see less of a jump due to the max score being reached.

It's important to note that these are average performances, and we can see in the next chart that there is considerable variation in each run (unless you have 100% accuracy all the time with no errors - which would be nice). Getting a lucky streak can allow you to "over-perform" significantly. This feeds into the overall strategy, if you are willing to make many attempts it can be worth striving for speed over accuracy and hoping that you will get a clean run at a faster-than-comfortable speed. The standard deviations here are of course based on pure (approximated) randomness but hopefully at least represent the potential variation in outcomes.

So which is more important, speed or accuracy? To find the relationship between these factors, we can plot time spent per puzzle against accuracy, with contour lines representing equivalent average final scores. In this plot, the parameters focused on 70% or higher accuracy, and 6.5 seconds per move or faster. A number of interesting features are present in this plot, primarily the wider spread of lines towards the slower and less accurate end of the graph, and the shifting gradient of the contour lines (shallower on the right). What does this tell us? Large variations in speed and accuracy towards the slower and less accurate end of the scale have smaller impacts on final score (at least in absolute terms). This makes sense as improving by 1 second, from 5 seconds per puzzle to 4 seconds per puzzle, is a 20% improvement, but from 2 seconds to 1 second is a 50% improvement!

What about the gradients? The gradient of each line shows (for a given score) the approximate relationship between speed and accuracy. For the line at a score of 20, an change of 10% accuracy is roughly equivalent to roughly 2 seconds per puzzle. For the line at a score of 50, 10% is closer to 1.5 seconds per puzzle.

So where to go from here? For whatever score you currently get in Puzzle Storm, you can find the relevant line, and find the pay-off between speed and accuracy, and decide which you think is more realistic to improve on (as in the examples above). You can also do this by finding the location of your runs on the graph by looking at your accuracy and calculating your average time per move (time / moves) based on the data on your dashboard. From there, look at the distance to the next line above yours. Is it more realistic to make that jump in speed, or jump in accuracy? For example, if you have 80% accuracy, spent 100 seconds, and made 50 moves, you would have a time spent per puzzle of 100/50 so 2, and around the 40 score line. Is it easier to improve by 4% in accuracy or 0.8 seconds per puzzle, to get to the 50 score line?

Of course, both of these factors are crucial to a strong run, as well as a pinch of good fortune, but hopefully this can help you establish which is easier to improve. Happy Storming!