Difficulties

One of the most popular feature requests on Integrity was assigning difficulty levels to questions.

After maintaining for a long while that this would do more harm than good (since it is so impossible to accurately ‘objectively’ assess the difficulty’of a maths problem) …I finally decided to introduce them.

Why?

There are 2 primary reasons for introducing the difficulty levels:

1. This is helpful for customising what kinds of questions you encounter in the Speedrun environment, if you’re looking to practise questions of a particular difficulty level.¹
2. It can help indicate to less experienced maths fans those questions that are much more challenging than they look, so that they don’t end up pouring excessive time into problems way above their ability level.

The latter should pretty quickly become irrelevant, as once you have sufficient confidence and fluency in a topic you‘ll have no qualms about tackling nasty problems!

These difficulty levels are very weakly correlated with the skill and experience required for a question, and even more weakly correlated with how difficult a solution might be to spot and/or execute. They give little indication as to how long a problem might take – because this will vary astoundingly from person to person.²

You’ll also notice they’ve been given somewhat cryptic names, instead of some sort of numerical scale. There’s also only 4 of them, so each one can cover a fairly wide difficulty range. It’s intentionally fuzzy. This significantly simplifies the process of picking a suitable difficulty while giving me a lot of breathing room.³ Also, you’ll build up your own intuition for what each difficulty level feels like as you do questions, and I think that’s part of the fun!

At the end of the day, remember it’s me who’s assigning these difficulties, and while I strive to be as objective and fair as possible, I’m only a purple portal. A lot of this is just based off vibes. You’ll almost certainly find some ratings that you disagree with; if so, I’m sorry.⁶

[^sorry[: It honestly doesn’t even matter, ngl.

The Difficulty Tiers

Based

Baseline. These are entry-level questions, probably suitable for an intermediates. Integrity is not intended for absolute beginners (you’d be much better off finding practice questions elsewhere!) – the questions I write assume basic proficiency in the topic.

Incline

Standard. The vast majority of questions tend to fall into this category, pretty naturally. Naturally, they’ll still probably require a little thought, but probably won’t require that much effort.

Manifold

Rare. You’ll feel pretty satisfied after managing to solve one of these.

Chaos

Exceedingly rare. Reserved for the most ridiculous of questions. The scariest kind are Chaos questions which have literally nothing going on ;)

Further Explanations

Here’s another good way to think about it, in relation to the electromagnetic spectrum:

• Incline is visible light. There’s incredibly richness to be had here, and it’s the most natural level you’d go for.
• Based covers all wavelengths longer than visible light, from infrared to radio waves, all the way up to a theoretical wavelength of $\infin$, or equivalently a frequency of $0$. These questions are easier, and they can get really, really easy.
• Manifold takes you beyond the visible range, through to ultraviolet and X-rays. It’s exciting, exotic stuff.
• Chaos is gamma. The frequency is high, but it can go even higher, all the way to $\infin$. Unlike the other rays which are constrained to a narrow energy band, gamma rays can vary wildly.

You can see it’s the 2 difficulties at either extreme, Based and Chaos, that have the greatest theoretical range – because there’s no lower bound to easiness, nor any upper bound to difficulty. Incline is familiar while Manifold is more alien.

Funny how so many games seem to have a 4-tier difficulty/rarity system.

• EZ / HD / IN / AT charts in Phigros
• N / R / SR / UR craft points in Master Duel^↗
• Common / Uncommon / Rare / Epic blueprints in Asphalt 9: Legends^↗

And hilariously, all of these have the exact same vibe as Based / Incline / Manifold / Chaos on Integrity. Perhaps they’re all just homeomorphic to Easy / Medium / Hard / Extreme, the archetypal difficulty scale.⁴

Finally, it’s worth mentioning that difficulties may change over time. It’s unlikely to happen much or at large scale,⁵ but particularly those questions on the boundary between tiers are more likely to undergo reassignment. The main reason for this is powercreep – as more and more difficult questions are added, and/or the average difficulty of questions of a topic goes up, some questions will inevitably have to be lowered a tier to maintain the distinction between difficulties. After all, the Manifold difficulty wouldn’t really mean that much if 70% of all questions are Manifold.