$7 \cdot 101$ On each card, a number is written: 1 or -1. You can point to 3 cards and ask: "What is the product of the numbers on these 3 cards?" (but it does not tell what numbers are written on the cards)
(1) When there are 30 cards in total, what is the minimum number of questions needed to know the product of all the numbers on the cards?
(2) For 31 cards, answer the same question as (1).
(3) For 32 cards, answer the same question as (1).
(4) On a circle, 50 numbers are written: 1 or -1. If asking one question can reveal the product of 3 consecutive numbers, how many questions are needed to know the product of all 50 numbers?
[Solution] (1) Divide 30 numbers into 10 groups, with 3 numbers in each group, and ask about the product of the 3 numbers in each group. Asking fewer than 10 questions is not possible, because each number must be in a group of three.
(2) Multiply $a_{1} a_{2} a_{3}, a_{1} a_{4} a_{5}$, and $a_{1} a_{6} a_{7}$ to get the product of the first 7 numbers. Then, divide the remaining 24 numbers into 8 groups of three, as in (1), to find that asking 11 questions will yield the product of all the numbers. It is easy to see that asking only 10 questions is insufficient.
(3) Multiply $a_{1} a_{2} a_{3}, a_{1} a_{2} a_{4}$, and $a_{1} a_{2} a_{5}$ to get the product of the first 5 numbers. Then, divide the remaining 27 numbers into 9 groups of three, to find that asking 12 questions will yield the required answer.

Since any number must be in a group of three, when 11 questions are asked, exactly one number will be included in two groups of two. Therefore, when the products of the 11 groups of three are multiplied, all other numbers appear once, but this one number appears twice, so the required product cannot be obtained.
(4) Use 50 questions to ask about the products $a_{1} a_{2} a_{3}, a_{2} a_{3} a_{4}, \cdots, a_{50} a_{1} a_{2}$, and then multiply them to get the required product.

Asking fewer than 50 questions is insufficient. In this case, at least one of the 50 products is unknown, let's say $a_{1} a_{2} a_{3}$. At this point, we can take two sets of numbers: in the first set, let $a_{1} = a_{3} = a_{6} = a_{9} = \cdots = a_{48} = 1$, and the rest equal to -1; in the second set, all are 1. Then, in these two sets of numbers, except for $a_{1} a_{2} a_{3}$, the other 49 products are exactly the same. This shows that asking 49 questions is insufficient, i.e., at least 50 questions are needed.