Lots of things, especially with patterns. Let's take scoring for example. Set collection is a common mechanic in games, where you try to collect sets of various things, and generally the more you collect of something, the better you score. But when you're designing the game, how much should that increase?

First there's linear, like 1 is worth 1 point, 2 is worth 2 points, 6 is worth 6 points. For that, there's not a whole lot of incentive to encourage people to make sure they get more cards in that set. But one way in which it could work is if they can only score in certain sets, so they prioritize them because that's the only way they can score it.

But if you want to encourage players to go after a set that they already have the most of, you need to have each one worth progressively more, and a really good pattern for that is the Triangular number pattern. If you recall Pascal's Triangle, the triangular numbers are the third row down, and the pattern is basically 1 (+2) = 3, 3 (+3) = 6, 6 (+4) = 10, etc. to give you a pattern of 1,3,6,10,15,21,28,36. So if you only have 2 of a thing it's only worth 3 points, but if you get 6 of a thing it's worth 21 points, significantly more. You'll see this scoring pattern quite often in set collection games. Offhand, I know Ponzi Scheme and St. Petersburg use it, but I've probably seen it in at least 20 games. If you're not a mathematician or you haven't encountered it in a bunch of other games, you might not recognize the pattern or realize that it's a good way to score set collection.

Another way that is used less often, because it's a much steeper increase, is exponential scoring, i.e. if you have only two items in a set, it's worth 2^2 = 4 points, but if you have 8 of them it's worth 8^2 = 64 points. But if you want the person to win to (almost always) be the person who got the biggest single set, and make people really battle it out just to get one more card in their biggest sets, then this is the way to go. I've seen it work really well in The Rose King, which even includes an exponential chart on the back of its rules going all the way up to like 32 squared, because it's possible to get groups that big (although rare). Apparently an old Alex Randolph game called Good Neighbors also used this.

One I've been playing around with lately is multiplicative, i.e. you multiply the numbers of each of your sets together. So if you have 3 in one set, 4 in another set, and 5 in a third set, your score is 3 * 4 * 5 = 60 points. It opens up an interesting dynamic where sometimes it's better to get cards in a new set than to keep going in one set, but only getting one card doesn't do you any good, and sometimes you'll score more points getting cards in a set you have very little of instead of getting more cards in a set you have a lot of.

Then there's an old one Knizia likes to do where it's basically doing a minimum function on your sets, where you score the set you got the lowest number of points in. He uses it for several of his games, including what many consider his masterpiece "Tigris & Euphrates", as well as "Ingenius". This scoring method forces the user to have to try to get a variety of sets and not to ignore any set, because whichever set they don't get as much of will be the one they score.

Another one I haven't seen too much of, but I've designed a game using it and seen at least one other game that uses it (That's Life! by Wolfgang Kramer, uses it for a single small module in one of its expansions) basically uses a Sine Wave for scoring. If you have an odd amount of a set, you score positive, and an even amount you score negative. Back and forth like a sine wave. It has an interesting property in that the more you get of something the better you score, but at the same time the more you try to get something and fail to make sure it stays an odd amount, the worse you score. So it's like walking a tightrope for that set, and the more you get the better it is and the more dangerous it is at the same time!

Another scoring method Knizia used to good effect in his card game High Society, is kind of the same concept as outliers in statistics (this one is a bit of a stretch, admittedly. There might be a better math concept this applies to). With a set of statistics, you often toss out outliers to get a more even distribution. Well in High Society, it's an auction game where you try to score the most points. However, whoever spends the most money to get those points over the course of the game, loses. Tossed out like an outlier. So you want to do what you can to score the most points, but you have to hold back enough so that essentially you're the second place winner. But of course everyone is trying to do the same thing, so it leads to very tight auctions and gambling when it's safe to bid a little extra to make sure you score something at all.

So yeah, those are several different methods of scoring based on different patterns in mathematics. And that's nowhere near comprehensive, I could probably come up with a half dozen more. As a designer you need to choose which scoring method is best for your game (or you can look into mathematics to find other possibilities).

And that's just scoring. There's all sorts of different aspects of games where knowing mathematics and patterns can help you with your game design.