I have cobbled together two sources in the title for this post. “An Inconvenient Truth” is a documentary about climate change narrated by former U.S. Senator, Vice-President and Presidential candidate Al Gore.  “A Beautiful Mind” is the title of a biography written by Sylvia Nasar and a movie directed by Ron Howard.  The subject of both book and film is John Nash, a mathematician whose 1950 Ph.D. thesis established the foundations for non-cooperative, two-person game theory. 

Nash’s theorem on finite games is relevant to Take-Away games in general, and to Carousel Ring Games in particular.  Both “Reals” are assumed to know the number of rings and the layout of riders on the Carousel. The presence of “Takers” means that any ring dispenser will eventually be exhausted.  So Carousel Ring Games satisfy Nash’s “finiteness” criterion.

Nash’s inconvenient truth is this:  it can be determined in advance which rider on the Carousel Ring Game gets the Gold Ring when both Real Players play their best strategies. To see how my game is played see post # 7.

It is easy to devise games where two contenders have an equal chance to win:  Coin Flip; “Evens/Odds”; or “Rock, Paper, Scissors”.  There is clearly no strategy for winning at Heads/Tails with a “fair” coin, i.e., when the probability of each outcome is ½.  Neither player is advantaged at Evens/Odds and Rock, Paper, Scissors, unless his/her opponent “telegraphs” moves by following a detectable pattern.  Even a non-uniform probability distribution can be exploited.  For example, computers use letter frequencies and combinations present in natural languages to decode encrypted messages.

The inconvenient truth proven by John Nash has this implication:  determining which of two Real Players gets the Gold Ring is like solving a puzzle.  That process can be satisfying to the puzzle-solver, but playing a game with a predetermined outcome is not much fun. 

Post #1 claimed that games of Tic-Tac-Toe on a 3×3 grid end with no winner when two informed contestants face each other; why, then, should such players bother to fill nine open squares with “naughts” and “crosses”?

I have developed additional Carousel Ring Games to increase the fun component, subject to the constraint set by Nash’s theorem.  It is unlikely that an amusement park will host any of these new games, for they rely heavily on personal interaction.  Incorporating a social dimension into Ring Games establishes a happier balance between education and entertainment in a home setting.