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Let the weight of the puppy be $a$, the weight of the smaller cat be $b$, and the weight of the larger cat be $c$. We have the equations $$\begin{array}{rl}a+b+c& =24\\ a+c& =2b\\ a+b& =c\end{array}$$ From Equation (2), we have $a=2b-c$. Substituting that into Equation (1) to eliminate $a$, we have $$\begin{array}{r}(2b-c)+b+c=24\Rightarrow b=8\end{array}$$ Substituting $a=2b-c$ into Equation (3) to eliminate $a$, we have $$\begin{array}{rl}(2b-c)+b& =c\Rightarrow 3b=2c\end{array}$$ Since $b=8$, $c=\frac{3}{2}b=12$. Finally, substituting the values of $b$ and $c$ into Equation (1) to solve for $a$, we have $a+8+12=24$, or $a=4$. Thus, the puppy weighs $\boxed{4}$ pounds.