Here's the original problem (from Sarah):
The
five FAMILY family members and their five dogs (each family member
owned a dog) were hiking when they encountered a river to cross. They
rented a boat which could hold three living things--people or dogs.
Unfortunately, the dogs were temperamental: Each was comfortable only
with its owner and could not be near another person, not even
momentarily, unless its owner was present. Dogs could be with other
dogs, however. The crossing would have been impossible except that
Lisa's dog had attended a first-rate obedience school and knew how to
operate the boat. No other dogs were that well-educated. How was the
crossing arranged and how many trips did it take?
And here is a solution:
Let
the five people be represented by A, B, C, D, and E. Let the five dogs
be represented by a, b, c, d, and e. [Let Lisa and her dog be A and a,
respectively.]
a, b, and c cross the river. a comes back alone and takes back d. a goes back and says behind while B, C, and D cross the river. Now A, a, E, and e are on the starting shore, and B, b, C, c, D, and d are on the destination shore. D and d return; A and a cross. C and c go back, and C, D, and E cross. Now dogs c, d, and e are on the starting shore, and everyone else is on the destination shore. a goes back and returns with c and d, then goes back and returns with e
a, b, and c cross the river. a comes back alone and takes back d. a goes back and says behind while B, C, and D cross the river. Now A, a, E, and e are on the starting shore, and B, b, C, c, D, and d are on the destination shore. D and d return; A and a cross. C and c go back, and C, D, and E cross. Now dogs c, d, and e are on the starting shore, and everyone else is on the destination shore. a goes back and returns with c and d, then goes back and returns with e
I
modified the problem so we had five knights on a quest with their noble
chargers. The knights and chargers had colored strings tied on their
wrists to identify them: green knight and green charger, red knight and
red charger, etc. The black charger was the one specially trained to be
able to operate the boat. I had the boys physically act out the problem,
no more than three at a time crossing from one side of the river to the
other and following all the rules. I did give occasional hints when
they were stuck. The first group ended up with the solution outlined
above, the second group had a similar solution but with a few
extra steps.
