Dr. Reiner Knizia is an extremely prolific and well-regarded German game designer. A signature element of his designs is that they tend to present the game's mathematical underpinnings rather openly, but those underpinnings have a twist that makes the game engaging and fun rather than a dry exercise in arithmetic.
In print for nearly 20 years, High Society (3-5 players, 20-30 minutes, ages 10+) is an auction game that's a good example of his style of game design. Players in High Society take on the role of nouveau-riche twits attempting to out-ostentate each other through extravagant purchases. This is done through a series of auctions for cards representing the trappings of wealth (fine art, a castle, jewels, etc.) that are worth varying numbers of points.
All the players start with the same hand of money cards, denominated so that no two cards in a player's hand have the same value. One at a time, cards worth various amounts of points come up for auction, and players bid to obtain them. There are also some cards that make you lose points where everyone but the person who takes the card pays into the auction.
You make or raise a bid by laying down one or more of your money cards on the table, and here's the first twist — you can't pick up any cards you've already laid down when raising a bid, and there's no making change. This makes the relationship between the face value of your money and its strategic worth less than straightforward. If you don't hold onto some low-value cards, you'll be forced to overbid when raising later on in the game, possibly all the way into bankruptcy.
Speaking of bankruptcy, the second twist is that regardless of score, the player with the least money at the end of the game cannot win. This means that simply outbidding all your opponents for everything is guaranteed to lose you the game – you have to budget your cards, figure out the combinations from your hand that will leave you with the most flexibility later on, and keep track of how much, and what denominations, your opponents are spending.
These two twists take what would be a dull exercise in guessing which cards will randomly come out next and budgeting accordingly, and turn it into a tense, interactive scramble to get just enough stuff to win while spending just enough not to lose, while encouraging your opponents to overspend on the things you don't want.
The game does have a couple of weaknesses that may put some people off. A player who spends all their money has no way to come back, so they have to spend the rest of the game unable to do anything and knowing they're going to lose (although generally a person will only make this mistake once). Also, the completely open scoring can mean that it can be fairly clear who the winner will be before the end, although there's a clever mechanism that randomizes the timing of the game end and ensures not every card will be auctioned that mitigates this somewhat.
Despite these flaws, High Society gets a lot of play around here as a short game that's mentally engaging, yet light and interactive enough to be very social with lots of table talk. I've found that people with a business or accounting background seem to take to it especially well, because it makes them exercise a funhouse-mirror version of their skill at determining value and making budgets.