The Discrete Difficulty of Size

The difficulty of a Skyscrapers isn’t determined by its size. You certainly have a lot more to think about with a 6x6 Skyscraper than a 5x5, but there are 5x5 Skyscrapers that might stump you for days while you can speedrun a 6x6 in sub-180. Just because a puzzle’s larger doesn’t mean it’s more difficult.

What is true is that larger Skyscrapers tend to be more difficult, or perhaps are more likely to be difficult. More importantly, increasing the puzzle size increases the difficulty ceiling exponentially¹ – i.e. the maximum difficulty a puzzle can reach (not that it can be measured). A 6x6 has complexity that a 5x5 just can’t replicate because of all the extra space and possibilities the increased size introduces.

When that complexity is leveraged, then a puzzle can become much more difficult. Far more influential than the size is the particular placement of clues, their interactions with each other, and how intricate the deductions will have to be.

All that being said, there are still very apparent general trends in difficulty across different Skyscrapers sizes. I’ll give my own perceptions here.

4x4

With no disrespect, the vast majority of 4x4 puzzles are near-trivial. This is simply a result of the small puzzle size – there is inherently not enough complexity to unlock the rich and diverse problem-solving that you see in larger Skyscrapers.

However, this by no means makes 4x4 Skyscrapers worthless. Firstly, they’re great for introducing people to Skyscrapers. Compared to a 5x5, a 4x4 Skyscrapers appears much simpler, which really reduces cognitive load and eases beginners into the puzzle. I find people pick up the mechanics of Skyscrapers much quicker when trying a 4x4 (even Open puzzles) than a 5x5. This is especially true for people less accustomed to puzzles (or just games and rules in general) – the last thing you want is too much cognitive load and them just quitting from frustration.

[!Note] This probably sounds a bit like a given, but 5x5 is often viewed as the ‘standard’ size for Skyscrapers, so going any smaller would be absurd.

Secondly, 4x4 Skyscrapers are phenomenal practice for novices. The numbers are small, but the reasoning is the same, and the tight, hyper-focused environment really hones your understanding and familiarity with all the techniques and deductions. An added bonus is the narrower range of possible difficulties means you can easily use 4x4s for speedrun practice.

Depending on how strong your logical reasoning is though, you can pretty quickly move on from 4x4s.

5x5

5x5 Skyscrapers instantly become far more interesting and dynamic. I would say the increase in the range of possible difficulty jumps the most between 4x4 and 5x5.

5x5s take the core techniques in 4x4 and level them up big-time – where you previously had situations with just 2 numbers, now there’ll be 3 or 4 involved. The whole puzzle feels far more ‘open’. The extra space also allows information to feel a lot more ‘sparse’, so for the toughest puzzles marks and extended thinking will have to do a lot of heavy lifting.

6x6

6x6 ups the ante even more, and at this point the grid really starts to feel quite spacious. Deducing marks with only 2–3 possible digits are more uncommon than common now, and conclusions tend to be much more granular and isolated. You might notice you spend a lot more time making marks than actually entering digits before you finally start to make concrete progress.

What I find interesting is that I started solving 6x6s at the same time I started solving 5x5s. Even with my relative lack of experience, it didn’t feel like there was that much of a gap in difficulty between them – the 6x6s for sure always took far longer, but the process felt pretty much the same. It really was just adding another number.

7x7

While 6x6s feel “comfortably challengingly” large, 7x7s (at least for me) seem to be on another level. The distance from one end of the grid to the other reallyyy feels like a trek now.

A part of this may just be psychology – y’know, a result of unfamiliariy with $7$ and a fear of the larger numbers.

But there is a noticeable step up in difficulty because even [Closed] 7x7 Skyscrapers take significant effort to complete, despite the abundance of clues. There are just so many things to keep track of, so many directions and possibilities to consider. Pencil marks become quite a crux as intuitive deductions become much harder.

Currently, I’ve only solved very few open 7x7 puzzles, and they’ve all taken …a rather long time. Lots of staring and pondering involved, far less action than in a 6x6.

8x8

I haven’t tried 8x8s yet. But I will someday.

9x9

Pfft.

10x10

These exist 💀

Closing Thoughts

Although difficulty can’t be quantified numerically, I do wonder what a (discrete) graph of difficulty against size would look like. It would of course inevitably shoot up as the size quickly reaches a point of unmanageability, but the shape up to that point is hard to say, really hard to say. Even just in $4 \leq x \leq 7$ it’s hard to tell if difficulty is plateauing, increasing, or inflecting.

One could argue time is an indicator of difficulty, and I do agree. However, it’s a pretty unreliable and misleading predictor, as some puzzles are ‘grindy’ while others might require spotting a trick. The variability of time spent solving Skyscrapers varies drastically, and is not always correlated with difficulty.