Here's an original puzzle I created. Enjoy!
A teacher had infinitely many students, and he referred to them by the designations $\text{Student } 0$, $\text{Student } 1$, $\text{Student } 2$, and so on. It happened that all of these students had limited eyesight. Specifically, for each $n$, Student $n$ could only see distances of $n$ yards or less. This teacher set up his classroom in a special way so that every student could see the board. He arranged all the desks in a straight line, with Student $0$'s desk immediately against the board (a negligible distance away), Student $1$'s desk $1$ yard away, Student $2$'s desk $2$ yards away, and so on, so that each of them could see the board from where they were sitting.
One day, the students had a pop quiz. They were instructed to turn their desks around so that they were all facing away from the board. (Looking at the board would be cheating, since useful information was written on it.) Right after he handed out the test, the teacher was called out of the room for an emergency, and the students were left unsupervised.
Simultaneously, some (or maybe none) of the students turned back to their original orientations to face the board, in an act of academic dishonesty. Immediately before the teacher reentered the room, all of these cheaters pivoted once more, so that the teacher could not see who had turned.
Suspicious of treachery, the teacher interrogated the students to find out who, if anyone had cheated. They made the following statements:
$\text{Student }0$: "I saw at most $1$ student cheat."
$\text{Student }1$: "I saw exactly $1$ student cheat."
$\text{Student }2$: "I saw at most $2$ students not cheat."
$\text{Student }3$: "I saw exactly $2$ students not cheat."
$\text{Student }4$: "I saw at most $3$ students cheat."
$\text{Student }5$: "I saw exactly $3$ students cheat."
$\text{Student }6$: "I saw at most $4$ students not cheat."
$\text{Student }7$: "I saw exactly $4$ students not cheat."
$\text{Student }8$: "I saw at most $5$ students cheat."
$\text{Student }9$: "I saw exactly $5$ students cheat."
$\text{Student }10$: "I saw at most $6$ students not cheat."
$\text{Student }11$: "I saw exactly $6$ students not cheat."
$\text{Student }12$: "I saw at most $7$ students cheat."
$\text{Student }13$: "I saw exactly $7$ students cheat."
$\text{Student }14$: "I saw at most $8$ students not cheat."
$\text{Student }15$: "I saw exactly $8$ students not cheat."
…
Student $4n$: "I saw at most $2n+1$ students cheat."
Student $4n+1$: "I saw exactly $2n+1$ students cheat."
Student $4n+2$: "I saw at most $2n+2$ students not cheat."
Student $4n+3$: "I saw exactly $2n+2$ students not cheat."
…
The teacher knows that every student who cheated told a lie and that every student who didn't cheat told the truth.
My challenges for you are:
a) Determine whether $\text{Student }20$ cheated.
b) Prove that the situation described above is not contradictory.
c) Bonus challenge: Determine precisely which students are those for whom the teacher can determine whether or not they cheated. (I don't know the answer to this!)
Notes:
- The students all turned simultaneously, so they only saw the actions of the people that they faced during the moment where some students were cheating.
- No student saw themself.
- The students have eyesight limitations. This means that $\text{Student }0$ didn't see any students, and that, for $n >0$, $\text{Student }n$ saw $\text{Student }0, \dots, \text{Student } n-1$ if $\text{Student }n$ cheated and that $\text{Student }n$ saw $\text{Student }n+1, \dots, \text{Student }2n$ if $\text{Student }n$ didn't cheat.