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Balanced?

At this time of the year I usually want to post a review of the German Puzzle Championship. There is a secondary issue I would like to discuss, though, so I will keep the first part brief. Yes, this year’s championship did take place (with the puzzles from 2020, which could not be used at the time due to the Covid pandemic). My results were acceptable, I guess – I ended up in a solid fifth place, but the distance to the top was once again enormous.

A few notes regarding the puzzles (primarily for those who have either seen them already or who have access to them): I had some reservations because, as it has been the case so often, the instructions for various innovative puzzle styles were overwhelming and appeared overloaded at times. But it turned out that many of my concerns were unfounded. I have often criticized puzzles, so it seems only fair that I also bring out the good things. The puzzles in Round 2 (“Die wilden Zwanziger”) were surprisingly accessible, and so were the Masyu variants and most of the puzzles in round 7 (Number systems) as well – at least those which I attempted to solve.

The Assorted Round covered much more stuff than one could handle in the given time, which is as it should be. I broke Round 6 (“Bewegung bitte”) entirely, which was of course my own fault (and which cost me pretty much all chances to reach a spot in the A-team). The last of the preliminary rounds, “Fishing for Complements” was hard to manage and overdid things a little in my opinion, but not by much, and in any case it is nice to have a manipulative round – of sorts – in the event.

All in all, it was a tough Championship (and it was a little depressing to see that, after missing the playoff spots, I managed to solve the playoff puzzles faster than three of the four finalists). But then, most of the puzzles were very nice, and there were no major disruptive incidents, which is the most important thing. Congratulations to Philipp and the runner-ups, and many thanks to all the organizers and helpers.

The secondary topic I would like to talk about is balance – not in terms of puzzle categories, which is something I have focused on frequently in the past, but regarding the scoring. There were several rounds which contained easy and hard puzzles, but no intermediate ones. As I see it, this is in general an undesirable round design.

For instance, one of the rounds contained two hard puzzles (70 and 75 points, respectively) and three easy ones with point values between 10 and 25. This gap in difficulty will usually produce a gap in the scoring table. Unless there is reason to believe that is impossible to finish with more than half the points anyway, the strong performers will likely start with the hard puzzles, because each of them is worth more than the entire rest. On the other hand, inexperienced participants may not attempt to solve them at all, since it costs too much time and risks ending up with no points at all.

There were two more rounds with similar traits; each time I started with the expensive puzzles, which was a successful strategy (at least in my case). I do not have the scoring tables available at this point, but my guess is that the results for these rounds broke down into two or three groups each, rather than a continuous spectrum of scores, according to the number of hard puzzles each participant solved.

The round design I prefer is based on a large number of easy puzzles and a gentle slope of increasing difficulty ranges with a decreasing number of puzzles for each range. Imagine, for simplicity: four puzzles worth 10 points, three puzzles worth 20 points, two puzzles worth 30 points and finally one puzzle worth 40 points. (One might call this the “Battleships model”.) I believe such a design has the best chance to produce a continuous scoring spectrum between the minimum of zero and the maximum of 200 points.

Obviously, the numbers must be adapted depending on the available puzzles and the length of the round. Still, I think this is better than having equal numbers of puzzles for each difficulty level or even a gap in the middle.

In fairness to the authors, this is quite an ambitious target, and in practice there are several aspects which should be taken into account. First of all, you have to live with what you have. If there are not enough puzzles available to achieve a perfectly balanced round, the priority should still lie on accuracy instead of balance. (Some special rounds may even require that the difficulty level increases in large steps only.) It would be questionable to distort the point values too much just to obtain the slope I mentioned earlier.

Next, it is not clear if the continuity in the scoring spectrum is really so important. After all, there may be a truly significant disparity in the overall performance between different groups of participants. If memory serves, the difference between my own score and the cut for the playoffs was about 200 points; it would have been a lot more frustrating if the fifth place had been separated from the playoff spots only by a much small margin.

Finally, it is worth noting that even a considerable number of (scoring-wise) unbalanced rounds in an event can be compensated for by one or two large balanced rounds. The lack of intermediate puzzles in some parts would still be there, but the gaps in the overall standing would disappear.

Let me therefore conclude by saying that the unbalanced rounds are not such a big deal for me. In a contest such as a national Puzzle Championship which is comprised of many individual rounds, the lack of balance in a few of them does not play a major role; it is something to keep in mind primarily if one wants to design a single-round event.

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