Depth · one concept from first principles · step 2 of 4

Count all shapes hidden in a grid

How many squares, large and small, can you find in the figure below?

The figure is made of equal-sized unit squares joined together. The top row has $3$ unit squares, the middle row has $4$ unit squares, and the bottom row has $4$ unit squares, all aligned at the left edge (so the top row is missing its rightmost cell, giving a stepped shape). Count every square that can be traced in the figure — the $1$ -cell squares, the $4$ -cell ( $2\times2$ ) squares, the $9$ -cell ( $3\times3$ ) square, and so on.

GeometryOperations

Your answer

squares

How to solve

Strategy Make a Systematic List — To avoid missing or double-counting, I organize the count by square size (a systematic list): first all 1x1 squares, then 2x2, then 3x3. Splitting by size turns one tricky count into a few easy subproblems, and sketching the grid lets me check which big squares actually fit inside the stepped shape.

1STEP 1

Count the 1x1 squares

Every unit cell is a 1x1 square. The figure is made of exactly 11 unit cells, so there are 11 squares of size 1x1.

3 + 4 + 4 = 11

2STEP 2

Count the 2x2 squares

Rows 2-3 are full so all three 2x2 blocks fit (3 squares); rows 1-2 lose the columns 3-4 block to the missing top-right cell (2 squares).

2 + 3 = 5

3STEP 3

Count the 3x3 squares

Columns 1-3 fill all three rows, so that 3x3 block fits; columns 2-4 would need the missing top-right cell, giving 1 square of size 3x3.

4STEP 4

Add up all sizes

Total squares = (1x1 count) + (2x2 count) + (3x3 count). There is no room for a 4x4 square because the figure is only 3 rows tall.

11 + 5 + 1 = 17

Answer

17 squares

11 + 5 + 1 = 17

There must be more small squares than big ones, and indeed 11 is greater than 5, which is greater than 1 — that fits the picture; the total 17 is larger than the 11 unit cells, as expected once overlapping bigger squares are included. No 4x4 square is possible since the figure is only 3 rows tall, so we have not missed a larger size.

Takeaway

Count squares one size at a time, smallest to biggest, and you will never miss one - that's Grade 3 thinking you already have!

Count the 1x1 squares
Count the 2x2 squares
Count the 3x3 squares
Add up all sizes

Where next?

Level upsuggested

Count segments between points on a line Express counts of dots, lines, faces

Another one like thissuggested

10 practice problems

Same look, different logicsuggested

Find a side from equal perimeters Perimeter plus tiles needed to build it

14 reasoning problems grow out of this type — the bridge toward competition →

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▶ Practice — 10 problems

The same shape shows up later on the AMC, as Casework by Position / Vertex Type, Lattice / Grid Pattern from Small Cases, Overlap / Complement Counting, Small-Domain Constrained Search. Why the shapes repeat →