That’s right, some real numbers can be easily defined while being arbitrarily difficult to calculate the game result with. But there is another reason why we want to tighten the restriction for a submission beyond the standard of being able to “figure out whether it’s the closest easily”.
The point of the game is for people to try to submit 2⁄3 the average guess. In order to calculate 2⁄3 the average guess, you need two operations: addition and division with nonzero divisors. The rational numbers form a dense set (for all a<b there exists c such that a<c<b) that is closed under these two operations. It is the natural playing field for this game.
The real numbers are constructed in a way that is unrelated to the structure of this game. (I think one typically invokes the concept of sets of rational numbers to construct the reals.)
If you want to see why allowing all real numbers adds nothing to the game play, just note that every real number is equal to a terminating decimal plus a real number smaller than Epsilon, where epsilon is made to be much smaller than the difference between any two submitters’ numbers.
This is the only type of scenario when submitting an irrational number could help you:
You are playing against the submissions π, 2π, 7π. If you submit 2π, you tie for the win, while if you submit a rational number very close to 2π then the 2π submitter wins. The scenario can happen with any irrational number, not just π. The irrational numbers just serve to add pointless additional elements to our already well-structured rationals.
That’s right, some real numbers can be easily defined while being arbitrarily difficult to calculate the game result with. But there is another reason why we want to tighten the restriction for a submission beyond the standard of being able to “figure out whether it’s the closest easily”.
The point of the game is for people to try to submit 2⁄3 the average guess. In order to calculate 2⁄3 the average guess, you need two operations: addition and division with nonzero divisors. The rational numbers form a dense set (for all a<b there exists c such that a<c<b) that is closed under these two operations. It is the natural playing field for this game.
The real numbers are constructed in a way that is unrelated to the structure of this game. (I think one typically invokes the concept of sets of rational numbers to construct the reals.)
If you want to see why allowing all real numbers adds nothing to the game play, just note that every real number is equal to a terminating decimal plus a real number smaller than Epsilon, where epsilon is made to be much smaller than the difference between any two submitters’ numbers.
This is the only type of scenario when submitting an irrational number could help you:
You are playing against the submissions π, 2π, 7π. If you submit 2π, you tie for the win, while if you submit a rational number very close to 2π then the 2π submitter wins. The scenario can happen with any irrational number, not just π. The irrational numbers just serve to add pointless additional elements to our already well-structured rationals.
I hope I have clarified my previous comment.