At the edge of a circular lake, there are stones numbered from 1 to 10, in a clockwise direction. Frog starts from stone 1 and jumps only on these 10 stones in a clockwise direction.

a) If Frog jumps 2 stones at a time, that is, from stone 1 to stone 3, from stone 3 to stone 5, and so on, on which stone will Frog be after 100 jumps?

b) If on the first jump, Frog goes to stone 2, on the second jump to stone 4, on the third jump to stone 7, that is, in each jump he jumps one more stone than in the previous jump. On which stone will Frog be after 100 jumps?
Solution

a) After 5 jumps, Frog returns to stone 1 and starts the same sequence. Since 100 is a multiple of 5, on the $100^{\text{th}}$ jump, he goes to stone 1.

b) On the $1^{\text{st}}$ jump, he moves 1 stone; on the $2^{\text{nd}}$, 2 stones; on the $3^{\text{rd}}$, 3 stones, and so on until the last jump when he moves 100 stones. The total number of movements is $\frac{(1+100) \cdot 100}{2}=$ 5,050. Since every 10 movements, he returns to stone 1 and 5,050 is a multiple of 10, after 100 jumps, Frog returns to stone 1.