Hunt the Mutant

 is a child's game played in Vault 101. A player is selected as the Mutant and hides while other players hunt for them. When the player is found, the game restarts with another player as the Mutant.

At one of Wally Mack's birthday parties in the 2260's, this game is played in the Atrium. Wally later mentions this game at the Lone Wanderer's tenth birthday party on July 13, 2268. By mentioning this game, Wally is unfavorably comparing his party to the Lone Wanderer's party.

Inconsistency
Due to Vault 101's original purpose in the vault experiment (being closed for 200+ years with no outside contact, that is), it is highly unlikely that any of the residents know about the Super Mutant threat or any mutations commonly known to wastelanders (like Ghouls or Radscorpions). The scouting reports found on the Overseer's computer reveal no hint whatsoever to super mutant activity at all. Given the experiment's purpose and the resulting oppressive nature of how Vault 101 has been ruled by generations of overseers, it seems highly unlikely that any resident would have been told anything about the outside world.

It also seems highly unlikely that James - knowing about the Super mutant threat due to his connections to the Brotherhood of Steel - has told the inhabitants (especially children) about the outside world since he could not even tell the the truth to his own son/daughter, the Lone Wanderer himself/herself.

However, there is the possibility that Vault Inhabitants predicted, or at least fantacised about, the exposure to radiation causing mutation of humans into something less than human. The existence of the film "Godzilla" is likely to be canon in the Fallout Timeline, as the mention of Ronald Reagan in the game is also canon (Reagan joined the Democrat party in as late as 1962). In Godzilla, we see the story of a simple creature being mutated by radiation into a massive hulking being; not unlike the Super-Mutants. Therefore, if such a concept was held in popular culture, it is not unbelievable that people would have at least speculated about the existence of mutants, especially imaginative children.

Another theory as to how they know about the existence of mutants is via the science lessons with Mr. Brotch, or even maybe through conversation with Jonas or James; the latter being improbable. It is quite easy to consider that Mr. Brotch teaches his students about radiation and its damaging effects in his often unconventional lessons. A further possible premise is via one of the questions in the G.O.A.T that eludes to the fact of the subsistence of mutants. If older brothers, sisters or cousins etc have taken the examination, they may well relate to their younger relations about the issue on the mutated arm that purportedly grows from out of your stomach; it's only common knowledge that those beset with mutations are often dubbed 'mutants'.