1. The head of the fish weighs as much as the tail and half of the body, the body weighs as much as the head and the tail together. The tail weighs 1 kg. How much does the fish weigh?
Answer: 8 kg.

Solution 1. The body weighs as much as the head and tail, i.e., two tails and half the body. This means that half the body weighs as much as two tails, i.e., the body weighs 4 kg. Then the head weighs $1+2=3$ kg, and the whole fish weighs $4+3+1=8$ kg.

Solution 2. Let $\Gamma, T, X$ be the weight of the head, body, and tail, respectively. Then, according to the condition, $\Gamma=T / 2+X, T=\Gamma+X$. From this, $\Gamma=(\Gamma+X) / 2+X$, i.e., $\Gamma=3 X$. Therefore, the fish weighs $\Gamma+T+X=3 X+(3 X+X)+X=8 X=8$ kg.