$2 \cdot 66$ Write a number, +1 or -1, on each card. You can point to 3 cards and ask: What is the product of the numbers on these three cards (but not tell what numbers are written on the cards)? When there are (1) 30 cards; (2) 31 cards; (3) 32 cards, what is the minimum number of questions needed to know the product of all the numbers on the cards? (4) In a circle, 50 numbers: +1 or -1, are written. If asking one question can reveal the product of 3 consecutive numbers. How many questions must be asked at least to know the product of all 50 numbers?
[Solution] (1) Divide 30 numbers into 10 groups of 3 numbers and ask 10 questions to find the product of the 3 numbers in each group, which will allow us to know the product of all the numbers. On the other hand, since each number should be in one of the 3-number groups, at least 10 questions must be asked.
(2) Multiply $a_{1} a_{2} a_{3}$, $a_{1} a_{4} a_{5}$, and $a_{1} a_{6} a_{7}$ to find the product of the first 7 numbers. Then, divide the remaining 24 numbers into 3-number groups as in (1), and ask a total of 11 questions to find the product of all the numbers. Clearly, asking fewer questions in this scenario is not possible.
(3) Multiply $a_{1} a_{2} a_{3}$, $a_{1} a_{2} a_{4}$, and $a_{1} a_{2} a_{5}$ to find the product of the first 5 numbers. Then, divide the remaining numbers into 3-number groups, and ask a total of 12 questions to find the product of all the numbers.

On the other hand, since any number should be in some 3-number group, it is clear that at least 11 questions must be asked. If 11 questions are asked, then there must be a number that is in exactly two 3-number groups (if it is in more than 3 groups, then there exists a number that is not in any 3-number group). The square of this number will be in the product of all 11 3-number groups, so using 11 questions, we cannot determine the product of all the numbers.
(4) Using 50 questions, we can find the product of $a_{1} a_{2} a_{3}, a_{2} a_{3} a_{4}, a_{3} a_{4} a_{5}, \cdots, a_{50} a_{1} a_{2}$. Multiplying them together, we get the cube of the product of the 50 numbers, which is equal to the product of these 50 numbers.

If fewer than 50 questions are asked, suppose we do not know the product of $a_{1} a_{2} a_{3}$. Then there exist two sets of numbers: in one set, take $a_{1}=a_{3}=a_{6}=a_{9}=\cdots=a_{48}=1$, and the rest equal to -1; in the other set, all are 1. In the first set, the product of all three consecutive numbers, except for $a_{1} a_{2} a_{3}$, is 1; in the second set, all products are 1. Thus, the two sets of numbers have the same product for all three consecutive numbers except for $a_{1} a_{2} a_{3}$, but the product of all the numbers is different. This shows that asking one fewer question cannot determine the product of all the numbers, so at least 50 questions must be asked.