Mathematical-logical challenges test your understanding of the underlying principles of causal systems, or to manipulate numbers, quantities, and operations. Typically, these are the challenges that we refer to as "mental" challenges.
A common example of a causal system is a lock (actual or metaphorical) that prevents you from accessing an area of the game and all you have to do is to bring it the appropriate key. Quake uses gold and silver keys, each of which opens respective colored doors within a level. It's a simple cause-and-effect system. However, locks and keys do not need to be so literal.
In King's Quest VI, when you first travel to the Isle of Wonder, you meet five sense gnomes that throw invaders into the sea. You need to trick each of them, one by one, with a flower of stench, a mechanical nightingale, a mint, a rabbit's foot, and invisible ink. In other words, you need to use the appropriate key (i.e. the items) on the appropriate lock (i.e. the gnomes).
More complex forms of these challenges present a set of objects, often related in ways that are not directly obvious, that need to be manipulated to make a key. Keep in mind that causal systems are not limited to locks and keys. Any system that has predictable events based on specific actions is causal in nature.
Other mathematical-logical challenges require mathematical skill, such as those used in blackjack. Think about it: Each time a card is drawn from the deck, the probabilities of drawing other cards increases. If you draw the Queen of Hearts, then that card is guaranteed to not come up again, and the remaining cards have a higher probability of being drawn, since the deck is now smaller.
Various strategies based on these probabilities have been derived in order to maximize the chances of winning and minimize the chances of losing. The same principle is applied when you min/max your character in Dungeons & Dragons or World of Warcraft. You are trying to minimize undesirable or unimportant traits and maximize the desired ones. The classic application of min/maxing is to make a fighter with very high physical traits (i.e. strength constitution, dexterity) at the expense of mental traits (i.e. intelligence, wisdom, charisma). Both strategies are applications of number manipulation.
Similar principles extend to economics, the flow and management of resources. In many games, the challenge is simply to accumulate money. In Monopoly, you buy and sell properties, participate in auctions, and collect rent with this goal in mind. But economies in games are not limited to money. Any resource that can be created, moved, stored, or destroyed either physically or conceptually can be considered.
Andrew Rollings and Ernest Adams said, "Even a first-person shooter has simple economy: Ammunition is obtained by finding it or taking it from dead opponents, and it is consumed by firing your weapon."8 Likewise, there are health, mana, inventory capacity, real estate, and the ability to save. The degree of the challenge is proportional to the supply of the resource against its demand. Regardless of what the resource is, the game challenges you understand its value.
Bodily-kinesthetic challenges test your ability to control of your bodily motions, capacity to handle objects skillfully, and sense of timing and reaction. One way to understand bodily-kinesthetic challenge is to view it as the "challenge of doing."
At the large scale, gross-kinesthetic challenges test your use of large muscle groups to perform tasks like running, jumping, or balancing. For example, charades uses gross-kinesthetic challenges. You use gestures and your acting abilities in order to make your team guess the movie, book, celebrity, or whatever it is that you are thinking of. If you were thinking of "Harry Potter," you could wave an imaginary wand around or maybe gesture a scar on your forehead. You could play with your hair and then mime forming clay (i.e. being a potter). Regardless of what you gesture, the idea is for you to use your body to communicate instead of verbal or written language.
A similar type of challenge is experienced in Dance Dance Revolution (DDR). During play, arrows scroll upward and pass over a set of stationary arrows. When the scrolling arrows overlap the stationary ones, you tap colored arrows with your feet on the dance platform. The game moves you around in such a way to approximate dancing, demanding gross physical control.
Looking at the smaller scale, fine-kinesthetic challenges test your ability to use smaller muscle groups to perform tasks that are precise in nature. For example, Jenga is a game of manual dexterity that involves removing blocks from a tower and moving those blocks to the top of the tower. Over the course of several small moments, you are required to analyze and adjust the amount of force each individual finger is using. In the same way, when playing Halo, you have to apply the ideal amount of pressure on the analog sticks with your thumbs to move through the space, move your thumb off of the analog stick to press other buttons, and perform numerous minor adjustments.
Bodily-kinesthetic challenges further include the concepts of timing and reaction. Timing is about acting exactly at the right moment, coordinating with something else happening in the game. Running and jumping across a chasm by pressing the jump button at the last second in Prince of Persia is an example of timing. On the other hand, reaction is about acting as fast as possible. Quickly moving your hands before they are slapped in a game of Hot Hands or instantly responding to a prompt from a Quick Time Event are examples of reaction.