Question

A farmer went to a market and purchased a wolf, a goat, and a cabbage. On his way home, the farmer came to the bank of a river and rented a boat. But crossing the river by boat, the farmer could carry only himself and a single one of his purchases: the wolf, the goat, or the cabbage. If left unattended together, the wolf would eat the goat, or the goat would eat the cabbage. The farmer's challenge was to carry himself and his purchases to the far bank of the river, leaving each purchase intact.

1. Using visual representation where each state represents position of objects either to the left or right of the river, the total number of possible states for this problem is:
(a) 7
(b) 16
(c) 15
(d) 8
(e) None of the above

Answer 1

The total number of possible states for this problem, taking into account the constraints given, is 8. This is because we need to exclude the scenarios where the wolf and goat or the goat and cabbage are left together without the farmer present.

Step-by-step explanation:

The problem you have described is a classic puzzle often referred to as the wolf, goat, and cabbage problem. To determine the total number of possible states for this problem, we need to consider the different combinations of positions for the wolf (W), the goat (G), the cabbage (C), and the farmer (F). Since each one can be either on the left bank or the right bank of the river, there are 2 options for each, resulting in 2^4 = 16 different combinations. However, this assumes that all combinations are valid, but we must exclude the cases where the wolf and goat are together without the farmer, or the goat and the cabbage are together without the farmer, which reduces the valid states. After evaluating which combinations are permissible, we find that the correct number is 8 valid states.

The scenarios where the farmer is not present and the wolf and goat are together or the goat and cabbage are together must be excluded from the possible states due to the constraints of the problem. Therefore, the correct answer to the total number of possible states for this problem is (d) 8.