We think of "game" mostly as a noun (Andrea played a great new _game_ today) or as a verb (Bettina enjoys _gaming_). In both contexts, we think very much of the formal idea of what a game is, or is not -- the kind of Caillois-completeness that James Wallis blogged about last year (see Man, Play and Games for Caillois's full thoughts), or the much-debated "magic circle" of Huizinga (Homo Ludens). Both Huizinga and Caillois think of games as bounded by space and time; activities with a defined, agreed-upon starting and ending point, and play area. Some games push those boundaries, but generally when we agree on what a game is, or is not, we end up with something similar to one of those definitions.
What about game as an adjective (She knew the challenge would be tough, but Carol was game), or adverb ("I'll do it," Daphne said, gamely)? This is no longer strictly a formal, bounded activity; rather, it's an attitude we can take to all of life. A game person is willing to try new things, even if those things are challenging, and even if they involve a significant risk of failure. The connection with our noun and verb use of "game" is clear: games frequently involve novelty, challenge, and risk of failure; without all those things, there'd be little fun to be had from games. Again, as per Caillois and Huizinga, there is an intentional aspect here too; just as players of games must freely choose to play, the game person is one who freely chooses to do new tasks that are difficult, and freely risks failure.
It should be clear already how close that flow state is to the state of someone who is gaming, or, for that matter, to the state of someone who has just gamely thrown herself into a new challenge. A game person has a special kind of confidence: not so much confidence of certain success, but confidence that success is possible and that failure is acceptable. She'll try anything. We admire that quality of gameness, almost universally. In some situations, though, it's especially useful -- perhaps even vital.
Carol Dweck researched the effect of praise on schoolchildren, dividing them into two groups. Both groups were given puzzles to solve, ones that were within their capabilities. One group was praised for their intelligence, afterwards, and the other was praised for their effort. Those praised for their effort ended up exhibiting "game" qualities: when offered a further set of puzzles to do, most chose more difficult ones. Those praised for their intelligence were not so willing to take risks, and predominantly selected puzzles well within their capabilities again. The undermining effect of praise will be familiar territory to those who know the aforementioned Kohn book, but I'm interested here about the way the type of praise given (and Kohn would probably argue that the "good effort" type is more like feedback than praise, anyway -- subtle difference, but crucial) frames the children's attitude towards the puzzles.
A puzzle can be work or play; a task or a challenge. If it is perceived as work, it will be seen as something to get through, to finish off as rapidly as possible, so as to get the extrinsic reward (praise, or pay, or just the end of the task and the ability to get on with something more fun). If it is perceived as play, it is intrinsically rewarding, like any other fun challenge; acquiring new skills, as mentioned in a comment on the previous post here, is an intrinsic reward too. There should be no reason we can't make learning fun, if we just frame the tasks as challenges, and the work as play. That doesn't necessarily mean gamifying everything; in fact, when badly done, gamification may keep the focus on extrinsic rather than intrinsic rewards, just replacing one set of demotivators (grades or marks) for another set (experience points or levels).
This all fits well with the work of James Paul Gee (What Video Games Have to Teach Us About Learning and Literacy, et al), who writes of digital games granting us permission to fail. Failure is accepted and acceptable, in a well-made videogame; the consequences are not severe. It's not so acceptable in most schools, and is more severely punished. We can encourage a game attitude by ensuring that we give permission to fail, because permission to fail means permission to take risks, to try something new; permission to learn, in fact, rather than to only ever accept tasks that are within our capabilities.
Outside of education, what can this mean for us when we design games purely to be fun? Well, challenges are still more fun than tasks. Any time when you, as a designer, feel the need to offer an extrinsic reward for players, because the tasks they do are so boring, it's perhaps time to think about reframing the tasks as challenges -- in other words, make them fun, rather than work.