The solution to Sam Loyd's Puzzling Scales puzzle determines that nine marbles are needed to balance a toy top by itself. This is deduced through algebraic substitution: 1. From the first scale, three blocks ($B$) and one top ($T$) balance twelve marbles ($M$): $3B + T = 12M$. 2. From the second scale, one top balances one block and eight marbles: $T = B + 8M \implies B = T - 8M$. 3. Substituting the value of $B$ into the first equation yields $3(T - 8M) + T = 12M$, which simplifies to $3T - 24M + T = 12M$. Combining terms gives $4T = 36M \implies T = 9M$.