Analysis of River Crossing Problem

I need to determine how a farmer and sheep can cross a river using a small boat in the fewest trips.

Understanding the Constraints

• The boat can hold one human and one animal
• Initially, the farmer and sheep are on the same side
• The goal is to get both across with minimum trips
• A "trip" is one crossing of the river (one way)

Solution Path

Let me trace the steps:

1. Initial state: Farmer and sheep on starting side, boat on starting side
2. Trip 1: Farmer and sheep cross together in the boat (both now on far side)
3. Trip 2: Farmer returns alone in the boat (farmer on starting side, sheep on far side)

At this point, the sheep is on the far side, and the farmer has returned to the starting side.

1. Trip 3: Farmer crosses alone (both now on far side)

Verification

• The farmer and sheep are now both on the far side
• We made 3 trips total
• This is minimal because:
  • The sheep cannot operate the boat
  • The farmer must accompany the sheep on the first crossing
  • The farmer must return to get to the far side

Therefore, the minimum number of trips required is 3.