Of the eight statements given by the assistants, four of them are completely useless and can be ignored due to either always being false or always being true.
Statement 2 would be false if and only if the jackpot door's number was a square that had 6 or more distinct divisors. All four squares in the range of 1 to 20 (specifically, the numbers 1, 4, 9, 16) have fewer than 6 distinct divisors, therefore Statement 2 must be true.
Statement 5 is true if and only if there is exactly one false statement among the first four statements. The possible number of false statements, regardless of what the statements are, among the first five statements is therefore either 1, 3, 4, or 5. Because of this, it is impossible to have exactly 2 false statements among the first 5 statements, therefore Statement 6 cannot be true.
Because statement 6 cannot be true, the possible number of false statements among the first 6 statements is 2, 4, 5, or 6. Therefore, it is impossible for there to be exactly 3 false statements among them, and Statement 7 cannot be true.
Because statement 7 cannot be true, the possible number of false statements among the first 7 statements is 3, 5, 6, or 7. Therefore, it is impossible for there to be exactly 4 false statements among them, and Statement 8 cannot be true.
After reducing the number of statements to check the validity of to 4, we must now figure out when a given statement would be false. Statement 5's truthiness has been elaborated above (since it is required in the proof of the truthiness of other statement(s)), and the remaining three statements are of the form "If A, then B".
For any statement of the form "If A, then B" to be false, A must be true and B must be false.
Statement 1 has the condition of "If the door is an even number", so it only applies if the door is an even number (2, 4, 6, 8, 10, 12, 14, 16, 18, 20).
Statement 1 then states that the only "correct" even numbers are squares (4, 16). Note that while 1 and 9 are squares, they aren't even, and don't matter for this statement.
Statement 1 is therefore false on any non-square even number. (These are the numbers 2, 6, 8, 10, 12, 14, 18, and 20)
Statement 3 has the condition of "If the door is a number with less than 6 distinct divisors".
There are three numbers in the range of 1 to 20 that have 6 or more divisors, and therefore don't match the condition: 12 (1, 2, 3, 4, 6, 12), 18 (1, 2, 3, 6, 9, 18), and 20 (1, 2, 4, 5, 10, 20).
all other numbers match this condition.
Statement 3 states that the door must be in the range of 1 to 10 (1, 2, 3, 4, ..., 10)
Therefore, Statement 3 is false on the following numbers: 11, 13, 14, 15, 16, 17, 19
Statement 4 has the condition of the number being in the range of 1 to 10.
Statement 4 states that the door must then be an even number.
Therefore, Statement 4 is false for any single-digit odd number. (1, 3, 5, 7, 9)
There is one final statement that must be considered to determine which door has the prize, and it's given not by an assistant but by the host himself. Specifically, knowledge of exactly how many liars there are reveals the door.
Factoring in statement 2's guaranteed truthfulness AND the false values for statements 3 and 4 having no overlap, the possible number of false statements among the first five statements are 1 and 3. To find the correct door, we must figure out which door has a unique number of false statements among the first five statements.
Given that it is impossible for there to be 0 false statements among the first five (due to statement 5 being false if the first four statements are all true), any doors where exactly one of the first four statements are false cannot be the door, nor can any door where all four of the first four statements are true be the correct door. This eliminates:
1. all odd numbers below 10 (only statement 4 is false),
2. all even numbers up to and including 10 (With the exception of number 4, these make statement 1 the only false statement. In 4's case, the only false statement is statement 5, so it is wrong as well),
3. all odd numbers above 10 (only statement 3 is false)
4. The numbers 12, 18, and 20 (only statement 1 is false), and
5. The number 16 (only statement 3 is false)
Removing those 19 numbers leaves only the number 14. In the number 14's case, both statement 1 is false (it's an even number that's not a square) and statement 3 is false (it's a number greater than 10 with fewer than 6 distinct divisors). Because this is the only number that results in more than 1 false statement among the first five statements (and therefore more than 4 false statements among the 8 assistant statements), the prize must be behind door 14.