At Least 97 Scientists can be Saved
Assumptions - Additional Rules:
- scientists can only guess integers between 1 and 101
- if a scientist says a number that has already been said, all scientists are killed
First Scientist
Can see $99$ other numbers, so can deduce his two possible numbers $x$ and $y$.
Announces a value $q=r+1$ where $r$ is the remainder of $x+y$ when divided by $101$.
Hence $1 \le q \le 101$.
Any Middle Scientist
Assuming all previous middle scientists have guessed correctly, she has knowledge of 98 correct numbers (the first scientist is unlikely to be correct).
So she has to choose between three numbers: $a,b,c$. She must be able to find a pair, say $a,b$, such that
$$ a+b+1 \equiv q \quad (\textrm{mod} \quad 101)$$
since two of her possible numbers must be the same as the first scientist's.
Assume there is another such pair, say $a,c$. Then
$$ a+b+1 \equiv a+c+1 \quad (\textrm{mod} \quad 101) \quad \textrm{so} \quad b \equiv c \quad (\textrm{mod} \quad 101)$$
leading to $b=c$. So the assumption was false and there cannot be another distinct pair.
Since her number cannot be the same as either of the first scientist's possibilities (which are $a,b$), she is able to deduce her number ($c$).
What then if that number was already said?
Then the number must have been the one announced by the first scientist.
This unlucky middle scientist says the number of the last scientist (which cannot have been announced yet). Since all subsequent middle scientists are able to see the last scientist, each knows when this case has occurred and hence the actual number of this middle scientist. So then all other middle scientists can now assume that all prior middle guesses were correct, so are able to deduce their numbers. None can be blocked from saying their numbers.
Last Scientist
Can deduce his number by the same reasoning as for the middle scientists. If it has already been announced, he announces either of the two numbers that nobody has yet announced.
Possible Outcomes
First Scientist: survives only if the aliens removed the number $100$.
Middle Scientists: all survive if the aliens removed $100$, if the first scientist has $100$ (so says the number removed), or if he says the number of the last scientist. Otherwise, exactly one dies.
Last Scientist: survives only if the first scientist has $100$ (needs all middle scientists to survive) or if aliens removed $100$.