How many squares are there altogether?
A hard-level "Count" problem.Do not count in the order you spot things — count in the order they are determined. A square is fixed by where its top-left corner sits, so count the possible corners.
GivenA 4 by 4 grid of squares
Hints
- Counting whatever catches the eye always misses some. A little square, a bigger square and the whole outer square all count. So sort them by size first.
- Start with the 1×1 squares: 4 across and 4 down, so 4 × 4 = 16.
- Now the 2×2 squares. The trick is to count where the top-left corner can sit — the red dots. That is 3 across by 3 down, so 9.
- Keep going. The room to place them shrinks by one each time: 3×3 gives 4, 4×4 gives 1.
- Add them all: 16 + 9 + 4 + 1 = 30. Notice the pattern — it is 1×1 ＋ 2×2 ＋ 3×3 ＋ 4×4, the sum of the square numbers up to 4.
Answer 30
More "Count" problems
- How many turns does the small circle make?
- How many cubes have 2 painted faces?
- Into how many pieces do 4 lines cut the paper?
- In how many ways can the three chosen rods form a triangle?
- How many triangles are there in the figure?
- How many squares does the diagonal pass through?