How Many Squares Do You See Quiz
Questions: 16 · 10 minutes
1. In a complete 4 × 4 grid, how many squares are at least two unit cells wide?
14
13
15
16
2. A complete 2 × 2 grid consists of four equal unit-square cells with every grid line present. Counting squares of every size, how many squares are there?
6
4
5
8
3. A complete rectangular grid has two rows and three columns of unit-square cells. Counting every axis-aligned square, what is the total?
6
7
8
9
4. A complete 5 × 5 grid contains squares ranging from 1 × 1 through 5 × 5. What is the total number of squares?
50
55
60
45
5. Which complete grid contains exactly 14 axis-aligned squares when every available size is counted?
A 2 × 2 grid
A 2 × 3 grid
A 2 × 4 grid
A 3 × 3 grid
6. Counting 1 × 1, 2 × 2, 3 × 3, and 4 × 4 squares, how many squares are in a complete 4 × 4 grid?
29
30
25
32
7. A four-sided figure is tilted like a diamond. Its four sides are equal, and every interior angle is 90°. How should it be classified in a square-counting puzzle?
It is not a square because its sides are not horizontal and vertical.
It counts as a square only when one diagonal is vertical.
It is a rhombus but cannot also be a square.
It is a square because rotation does not change its defining properties.
8. Someone examining a complete 4 × 4 grid counts its 16 unit squares and nine 2 × 2 squares, then stops. How many larger squares did they omit?
4
6
5
9
9. To count squares measuring exactly 3 × 3 cells in a complete 5 × 5 grid, you track possible top-left corners. There are three horizontal and three vertical starting positions. How many placements does that produce?
6
9
8
15
10. How many squares measuring exactly 3 × 3 unit cells can be placed within a complete 5 × 5 grid?
9
8
6
12
11. A complete 3 × 3 grid has three rows and three columns of unit-square cells. How many squares of all sizes does it contain?
13
14
12
15
12. A complete rectangular grid has two rows and four columns of unit-square cells. How many axis-aligned squares of all possible sizes does it contain?
8
9
10
11
13. Which rule gives the total number of squares in a complete n × n grid?
Add the integers from 1 through n
Multiply n by itself
Add the squares of the integers from 1 through n
Multiply n by one less than n
14. Using the complete-grid counting pattern, how many squares of all sizes are in a 6 × 6 grid?
91
86
96
84
15. In a complete 4 × 4 grid, how many squares measure exactly two unit cells on each side?
9
8
12
4
16. A complete 3 × 3 grid contains 14 squares. It is expanded to a complete 4 × 4 grid. How many additional squares does the larger grid contain?
9
14
18
16