Question

ABCD is a square. How many squares have two or more vertices in the set {A, B, C, D}?

Answer 1

To determine how many squares can be formed with two or more vertices from the set \{A, B, C, D\}, we start by analyzing squares with varying numbers of vertices selected from this set.

1. **Counting Squares with 2 Vertices:**
Selecting 2 vertices from the square will not form a square, as two points only define a line segment, not a square. Thus, there are no squares with exactly 2 vertices.

2. **Counting Squares with 3 Vertices:**
Similar to the situation with 2 vertices, 3 points selected from a square cannot form a square, as three points define a triangle or a corner of a square, but not a complete square. Thus, there are no squares with exactly 3 vertices.

3. **Counting Squares with 4 Vertices:**
The only square that can be formed using all 4 vertices A, B, C, and D is the square ABCD itself. Therefore, we have:

\[
\text{Number of squares with 4 vertices} = 1
\]

In conclusion, the possible squares that can be formed with two or more vertices from the set \{A, B, C, D\} are as follows:

- 2 vertices: 0 squares
- 3 vertices: 0 squares
- 4 vertices: 1 square (ABCD)

Adding these together, the total number of unique squares that can be formed with two or more vertices in the set is:

\[
\text{Total} = 0 + 0 + 1 = 1
\]

Thus, the final answer is:

\[
\boxed{1}
\]