A man selling fruit has only three weights; but with them he can weigh any whole number of kilograms from 1 kg up to 13 kg, inclusive on his balance.
What weights does he have?
To solve this problem, we can use the concept of binary numbers to represent the weights. The man can have three weights that can be assigned values of 1, 2, and 4.
The weight 1 kg can be represented as 2^0 (2 to the power of 0), the weight 2 kg can be represented as 2^1, and the weight 4 kg can be represented as 2^2.
Using these three weights, the man can create any whole number of kilograms from 1 kg up to 13 kg.
To see how this works, let's consider all the possible combinations of weights:
- By using the weight of 1 kg (2^0), he can measure 1 kg.
- By using the weight of 2 kg (2^1), he can measure 2 kg.
- By using the weight of 1 kg (2^0) and 2 kg (2^1) together, he can measure 1 kg + 2 kg = 3 kg.
- By using the weight of 4 kg (2^2), he can measure 4 kg.
- By using the weight of 1 kg (2^0) and 4 kg (2^2) together, he can measure 1 kg + 4 kg = 5 kg.
- By using the weight of 2 kg (2^1) and 4 kg (2^2) together, he can measure 2 kg + 4 kg = 6 kg.
- By using the weight of 1 kg (2^0), 2 kg (2^1), and 4 kg (2^2) together, he can measure 1 kg + 2 kg + 4 kg = 7 kg.
From here, the pattern repeats. By adding the weight of 4 kg (2^2) to any previously mentioned combinations, he can measure the weights from 8 kg to 13 kg:
- By using the weight of 4 kg (2^2), he can measure 8 kg.
- By using the weights of 1 kg (2^0) and 4 kg (2^2) together, he can measure 1 kg + 4 kg = 9 kg.
- By using the weights of 2 kg (2^1) and 4 kg (2^2) together, he can measure 2 kg + 4 kg = 10 kg.
- By using the weights of 1 kg (2^0), 2 kg (2^1), and 4 kg (2^2) together, he can measure 1 kg + 2 kg + 4 kg = 11 kg.
- By using the weights of 1 kg (2^0) and 4 kg (2^2), he can measure 12 kg.
- By using the weights of 2 kg (2^1) and 4 kg (2^2), he can measure 2 kg + 4 kg = 14 kg. However, the problem specifies that he can only weigh whole numbers up to 13 kg, so this combination is not necessary.
Therefore, the man has weights of 1 kg, 2 kg, and 4 kg.