Dale Newfield wrote:
Student picks a number, X.
The master picks a rule that they think will take the student X turns
to figure out.
The Student gets points for guessing in less than or equal to X turns,
and is given one anyway if the master made the rule too hard:
(1 point if not solved at X, 2 if solved at X, 3 at X-1, etc.)
The Master gets two points if it's solved in exactly X turns.
On turn X, as the Student, I will purposely not guess the rule
correctly, because if I guess it correctly, I gain no points over the
Master, while if I fail to guess correctly, the game ends and I gain a
point. You could just give the Master a single point instead of two
points if it's solved in exactly X turns. But then neither player cares
what happens on turn X, because the Student gains a single point either
way. Since that's the best result the Master can hope for, the best
strategy is just to purposely come up with a rule that's too hard. You
could tweak the scoring further to insure that the Student always gains
two or more points over the Master for guessing the rule correctly, and
gains exactly one if the rule is too hard, but then once again the best
strategy for the Master is to purposely come up with a rule that's too
hard. I'm not seeing any obvious way around all of this.
I'm intrigued by Guy's idea of keeping X hidden from the Student. Here's
my variation on that theme. Interestingly, this suggestion actually
works for any number of players, and provides a kind of cap to keep
games from going on too long.
The Master secretly comes up with a rule, and secretly comes up with a
target range of rounds during which he or she hopes a Student will guess
the rule. (A round means a full round of play, with each Student taking
If a Student correctly guesses the rule in fewer rounds than the
beginning of the Master's target range, the game ends and that Student
gets a point per round played.
If a Student guesses the rule correctly on a round within the target
range, the game ends and the Master gains a number of points equal to
the start of the target range minus the size of the target range.
If no Student has guessed correctly after the last round of the target
range, the Master announces this fact, and no one gets any points for
this game. (However, the Students may continue playing for fun.)
So if, as the Master, I choose the range 10-15, the size of my range is
6. I will win exactly 4 points (10 minus 6) if any Student guesses
correctly on a round within my range, while Students can win between 1-9
points by guessing correctly before that. If I choose the range 20-20, I
will gain 19 points if any Student guesses correctly on exactly round
20, and Students can win between 1-19 points by correctly guessing
I can increase my chances of winning by increasing the size of my range,
but that reduces the number of points I might win. Notice that there's
no point in making the size of my range equal to or greater than the
bottom end of my range. Choosing the range 10-30 is silly (and should
probably just be illegal), because I will then win negative points if
any Student guesses correctly within my range.
As a Student, my potential payoff for a correct guess becomes greater
and greater as the game progresses, but of course, I'll wonder whether
or not we've moved within the Master's range. If I become convinced that
we have, I'll just stop making good guesses (and should probably just
explicitly announce this fact and start passing my turns). This will
keep the Master from suddenly becoming "helpful" in too obvious a way
(or simply blurting out the answer!) when the game enters the target
range. However, I would consider it perfectly legal and within the
spirit of this game for the Master to start building more helpful
counter-examples once the game moves into the target range. It's up to
the Students to think about this and decide whether they want to risk
One potential way to salvage Mondo for the single-Student game would be
to institute what I've called "Risky Mondo". You start the game with no
stones, and you need to spend stones to guess. You win a stone by
calling Mondo and answering correctly, but if you answer incorrectly,
you lose all your stones. This wouldn't change the single-Student game a
lot, but it would have some subtle effects, and perhaps it's more
pleasing than just getting an infinite number of stones. A slightly more
complex version of Risky Mondo can also be used in multi-Student games.
I have no idea if this would be fun. I'd like to try it sometime.