Much of the time, when people are playing roleplaying games, they are not interested in testing their intelligence. They might want to immerse in a different character, tell a story, be told a story, immerse in a separate reality, or maybe just spend time with their friends and roll some dice as an excuse.
There are a few exceptions.
Ascertaining how popular different approaches are would require a difficult scientific study.
A certain branch of old school revival play
The following are all features of a certain branch of OSR play (OSR play, as a whole, is as diverse a cultural movement as any other and also has other branches).
- Creative problem solving: the players are supposed to solve challenges by creative use of their environment, abilities of their characters, and items they have. Redirecting a river to flood a dungeon and thereby slay the orcs is completely valid and celebrated way, as is storing green slime in bottles and using them as biological weapons.
- Drawing maps and navigating by them is done by the players. Megadungeons contain mapping challenges - sloping passages, teleport rooms or corridors, etc.
- Dungeons may contain actual puzzles and brain teasers, which are for the players to solve, ignore, or bypass by the aforementioned creative problem solving.
- Players carefully manage their resources - torches, hit points, food and water, hirelings, and, at domain-level play, even armies.
This kind of play is all about being smart, creative, and having good judgement. Intelligence attribute of characters is typically not used to solve the problems mentioned above. Some games do away with it all together and others rename it as lore or education; Into the Odd, for example, only has will as a mental attribute.
Most OSR games are derivatives of old editions of D&D. Intelligence might influence arcane spellcasting or learning spells (or gaining experience as a magic-user) and languages known. Adventures may also give extra information for characters with high intelligence or use abstractions where, for example, operating arcane machinery is easier for characters of high intelligence. Not everything is a puzzle for players, even if they could be.
Playstyle where tactical combats and moving a figure around carefully and using character powers cleverly is central. You usually do not get to roll intelligence to determine where to move your figure and should you use your fireball spell now or wait for a better opportunity.
Intelligence of a character might affect various numerical characteristics, much as it does in several roleplaying games.
Building mechanically effective characters. This is, again, a test of player intelligence and book-learning. Character intelligence, if present, is one number among many to optimize.
This is a roleplaying game of dungeon exploration, which mechanizes the resource management sides of OSR play (as above), creating an entirely different play experience. Fairly difficult and challenging game, and all about players mastering the rules system and succeeding at risk management.
I recall there is no intelligence attribute.
A specific playstyle related to D&D4. There are at least three scenarios, which contain difficult tactical challenges, problem solving by players, and working under a time limit.
Intelligence is used to derive the mechanical characteristics of a character, much like in other roleplaying games, but it is not used to solve in-game puzzles. It might give better scores at knowledge-based skills, which might help in finding more information about the adventure location.