2009 Putnam B2

A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers

and $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers

$c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$?

游戏涉及在实数轴上向右跳跃。如果 $a$ 和 $b$ 是实数

并且$b > a$，从$a$跳转到$b$的成本为$b^3-ab^2$。对于什么实数

$c$ 可以在有限次数的跳跃中从 $0$ 移动到 $1$ 且总成本恰好为 $c$ 吗？