91Ó°ÊÓ

A man must ferry a wolf, a goat, and a head of cabbage across a river. The available boat, however, can carry only the man and one other thing. The goat cannot be left alone with the cabbage, nor the wolf with the goat. How should the man ferry his three items across the river?

A hare is 150 paces ahead of a hound that is pursuing him. If the hound covers 10 paces each time the hare cover 6 , in how many paces will the hound overtake the hare?

Prove Proposition 34 of the Maasei Hoshev: $$ \begin{aligned} [(1+2+\cdots+n)+(2+3+\cdots+n)+\cdots+n] \\ \quad+[1+(1+2)+\cdots+(1+2+\cdots+(n-1))] \\ =n(1+2+\cdots+n) \end{aligned} $$

Two men have some denarii. The first said to the second, if you will give me one of your denarii, then mine will equal yours. The other responded, and if you will give me one of your denarii, then I will have ten times as many as you. How many does each man have?

Show that under the assumptions of the mean speed theorem, if one divides the time interval into four equal subintervals, the distances covered in each interval will be in the ratio \(1: 3: 5: 7\). Generalize this statement to a division of the time interval into \(n\) equal subintervals and prove your result.

Compare Levi ben Gerson's use of "induction" to that of alKaraji. Should the methods of either be considered "proof by induction"? Discuss.