Math Word Problem?

Question

Ten people are fishing from a canoe. The seats in the canoes are just wide enough for one person to sit on, and the center seat is empty. The five people in the front want to change seats and fish from the back, and five people in the back want to change seats and fish from the front. ONly one person may move at a time. A person changing seats may move to the next empty seat, or step over one other person to reach an empty seat. What is the minimum number of moves needed to exchange the five people in the front with the five in the back?

Answer 1

35 moves

Here's a site that explains the problem:

http://www.cut-the-knot.org/SimpleGames/FrogsAndToads.shtml

From the site:

"Each of the N frogs must move M + 1 positions to the right. Each of the M toads must move N + 1 positions to the left. The total number of positions to move over is thus given by N(M + 1) + M(N + 1). However, on a jump, a piece moves over 2 positions. How many jumps are there?

It's easier to think of jumps as orientation swaps. Who ever does the move, a pair of neighboring pieces frog-toad converts to the pair toad-frog. To reach the goal, each of the toads must swap orientation with each of the frogs. Therefore, there is the total of MN swaps, or jumps. Therefore, the number of moves needed to solve the problem is

N(M + 1) + M(N + 1) - MN = MN + M + N"

In this case, you have 5 people on each side, thus m=5, n=5, and mn+m+n=35

Answer 2

15 times

Answer 3

11 moves
you take the first person and move them to the centre seat and then move all the other people to each other's seats with the person in the centre occupying the last empty seat on the other side of the boat.