Problem Analysis

This is a classic logical puzzle where the key is to realize that on the day the caterpillar reaches the top, it does not slide back down at night.

Step 1: Calculate Net Daily Progress

First, let's determine the caterpillar's net progress over a full 24-hour cycle (one day and one night).

• Climbs up during the day: +5 inches
• Slides down during the night: -4 inches
• Net progress per day: 5 - 4 = 1 inch

Step 2: Determine the Position for the Final Climb

The caterpillar reaches the top of the 75-inch pole during its daytime climb of 5 inches. We need to find out how many days it takes for the caterpillar to be in a position where it can reach the top in that one final climb.

The starting height for the final day's climb must be at least:

75 inches (total height) - 5 inches (day's climb) = 70 inches.

Step 3: Calculate the Time to Reach 70 Inches

Since the caterpillar makes a net progress of 1 inch per day, it will take exactly 70 days (and 70 nights) to reach a height of 70 inches.

Step 4: The Final Day

At the end of the 70th night, the caterpillar is at a height of 70 inches. On the morning of the 71st day, it begins to climb.

It will climb the remaining 5 inches and reach the top of the pole during the 71st day.

Conclusion

The caterpillar will reach the top of the pole on the 71st day.