A group of adults and children go on a camping trip On the f

A group of adults and children go on a camping trip. On the first day, they come to a river too deep to wade across. Fortunately, the campers find a small canoe, which can carry either one adult or up to two children at a time. That is, the canoe cannot hold one adult and one child at the same time. The campers have only this one canoe to transport the entire group across the river and there is no way to transport the canoe across the river without a passenger.

Problem 1: Regardless of the number of adults and kids on the trip, do you expect the fewest number of one-way trips necessary to get them all across the river to be even or odd? Explain. (Note that sending the canoe across the river and then back counts as 2 one-way trips.)

Problem 2:

a) Suppose one adult and two kids are on the trip. What is the fewest number of one-way trips it will take to get them across the river?

b) What if there are two adults and two kids?

c) What if there are three adults and two kids?

d) If there are an unknown number of adults and two kids on the trip, can you come up with a formula for the fewest number of one-way trips it will take them to get across the river?

e) Suppose it takes 53 trips for a group of campers to cross the river. If there were two kids in the group and they crossed using the fewest possible number of one-way trips, how many adults were in the group?

Solution

Problem 1:

The number of trips will always be even

Reason: Irrelevance of number of person, the canoe will be going to take the people across the river and then come back

Number of trips = 2 * (one side trips)

Hence the number is of the form 2n, hence it will be always even

Problem 2:

a) One adult will take one trip and then 2 children will take 1 trip

Hence the fewest number of one way trips will be equal to 2

b) Two adults will take 2 trips and two kids will take 1 trip

Hence the fewest number of 1 side trips will be equal to 3

c) Three adults will take 3 trips and two kids will take 1 trip

Hence the fewest number of 1 side trips will be equal to 4

d) Let the unknown number of adults be X

then the fewest one side trips will be X+1 ( X trips for adults, 1 trip per adult and 2 children per trip)

Fewest one side trips = X + 1

e) Correct answer will be 52

d)