Introduction:
To prove that the maximum value of a third-order determinant whose elements are 1 or -1 is 4, we need to consider all possible combinations of these elements and find the determinant with the maximum value.

Step 1: Determinant Calculation:
Let's start by considering a general third-order determinant with elements a, b, c, d, e, f, g, h, and i. The determinant can be written as follows:

| a b c |
| d e f |
| g h i |

The determinant can be calculated using the formula:

Det = a(ei - fh) - b(di - fg) + c(dh - eg)

Step 2: Considering the Possible Combinations:
Since we are given that the elements can only be 1 or -1, let's consider all possible combinations of these elements for the determinant.

Case 1: All Elements are 1:
If all the elements of the determinant are 1, the determinant becomes:

| 1 1 1 |
| 1 1 1 |
| 1 1 1 |

Using the determinant formula, we have:

Det = 1(1*1 - 1*1) - 1(1*1 - 1*1) + 1(1*1 - 1*1)
= 1(0) - 1(0) + 1(0)
= 0

Case 2: All Elements are -1:
If all the elements of the determinant are -1, the determinant becomes:

| -1 -1 -1 |
| -1 -1 -1 |
| -1 -1 -1 |

Using the determinant formula, we have:

Det = -1((-1)*(-1) - (-1)*(-1)) - (-1)*((-1)*(-1) - (-1)*(-1)) + (-1)*((-1)*(-1) - (-1)*(-1))
= -1(0) - (-1)(0) + (-1)(0)
= 0

Step 3: Determinant with Maximum Value:
From the above calculations, we can see that both cases resulted in a determinant value of 0. Therefore, we need to consider a combination that results in a non-zero value.

Case 3: Combination for Maximum Value:
To find the combination that results in the maximum determinant value, we can choose two elements as 1 and one element as -1. Let's consider the following combination:

| -1 -1 1 |
| 1 1 -1 |
| 1 -1 1 |

Using the determinant formula, we have:

Det = -1(1*1 - (-1)*(-1)) - (-1)(1*(-1) - (-1)*1) + 1(1*(-1) - 1*1)
= -1(0) - (-1)(0) + 1(-2)
= 2

Therefore, the maximum value of a third-order determinant whose elements are 1 or -1 is 2.

Conclusion:
We have shown