Alice and Bob started working part-time parking lot attendants. Bob was managing a car when Alice arrived late on the first day at work.
A: I'm sorry for being late. How many cars have come in?
B: I will make a quiz on it. Why don't you guess the number of cars?
A: You are giving me a problem to solve?
B: Right. Every car has its plate number and the numbers have special relation.
A: So, do you want me to figure out the number of cars from that relation?
B: Yes, conditions are as follows.
Condition 1: Any two cars must share only one digit.
Condition 2: There must be exactly one more car with the equal digit that two cars have.
A: (after thinking) Three cars?
B: Why do you think the answer is three cars?
A: For example, if there are three cars with the plate numbers 1, 12, and 13, any two cars have 1, and there are exactly three cars which has 1 in the plate number.
B: Oh, there is one more condition.
Condition 3: There are more than three cars.
How many cars were in the parking lot when Alice arrived?
Added remark: It's not very hard to construct examples for 7 cars. Could you show the reason why other number cannot be an answer?