How Many Fence Poles Does a Farmer Need Around a Square Plot?

Imagine a farmer who wants to protect his square plot with a fence. This plot is quite large, requiring a total of 27 fence poles on each side of the square. Let's delve into the mathematical puzzle to discover how many fence poles the farmer actually needs in total.

The Basic Calculation

At first glance, one might think that the total number of poles needed is 27 * 4, as there are four sides to a square. However, we need to account for the poles at the corners where each side meets.

The Core Question

Description of the Problem

The farmer has a square plot with 27 fence poles on each side. This means that each side of the square has 29 poles in total, accounting for the two corner poles. We need to calculate the total number of poles required to enclose the entire plot, keeping in mind that the corner poles are shared between two sides.

Step-by-Step Solution

Calculate the poles in between corners: For the 25 poles in between the corners, we can calculate by using the formula for one side: 27 - 2 25 poles. Since there are 4 sides, 25 * 4 100 poles. These are the poles only in the middle of the sides. Add the corner poles: Each corner of the square plot has one pole, and there are four corners. So, 4 * 1 4 poles.

Final Calculation

The total number of fence poles needed is the sum of the middle poles and the corner poles: 100 4 104 poles. This is the accurate count of fence poles the farmer needs.

Including Mathematical Variations

Let's consider a more generalized approach. If we use the variable n to represent the number of corners at which a pole is placed (with n being one of 0, 1, 2, 3, or 4), we can write a formula to calculate the number of poles required.

For any value of n:

If n 0, the farmer needs 108 poles. If n 1, the farmer needs 107 poles. If n 2, the farmer needs 106 poles. If n 3, the farmer needs 105 poles. If n 4, the farmer needs 104 poles.

Conclusion

The farmer needs a total of 104 fence poles to enclose his square plot. This solution accounts for the shared corner poles, ensuring an accurate count.

By understanding the mathematical nuances of this puzzle, we can apply this knowledge to real-world scenarios where precision is crucial, such as construction, gardening, and surveillance.