Problem
Counting squares in a figure
A 5-by-3 grid has hearts in two of its cells. A square's sides must lie on the grid lines. It must hold exactly one heart — not none, not both. Count the squares that qualify.
Reasoning · Grade 3-1 Counting Large and Small Shapes
Your answer
Watch out Holding a heart is not enough — squares with both are also out, and the big ones tend to catch both.
How to solve
Strategy Make a Systematic List — Since a square is named by its size and its top-left corner, I go size by size (1x1, then 2x2, then 3x3) and slide each square across every position it can sit, checking how many hearts fall inside. I keep only the positions that hold exactly one heart. Splitting by size keeps the search complete and tidy.
1STEP 1
Set up the grid and mark the hearts
The two hearts sit one cell apart diagonally.
Naming each cell by its column and row makes it easy to test which squares cover which heart.
2.G.A.2Identify Subproblems2STEP 2
Count 1x1 squares with exactly one heart
At 1×1 only the heart cells qualify: 2.
2 squares
A single cell can hold at most one heart, so the only 1x1 squares that count are the two heart cells themselves.
3.G.A.1Make A Systematic List3STEP 3
Count 2x2 squares with exactly one heart
Of eight 2×2 spots, 4 hold exactly one heart.
8 - 1 - 3 = 4 squares
Only the 2x2 spot that straddles both heart cells swallows both hearts; among the rest, four cover exactly one heart and the others cover none.
3.G.A.1Make A Systematic List4STEP 4
Count 3x3 squares with exactly one heart
Of three 3×3 spots, only 1 holds one heart.
1 square
A 3x3 square sitting at the far left reaches Heart B but not Heart A, while the ones shifted right scoop up both hearts.
3.G.A.1Look For A Pattern5STEP 5
Add up all sizes
Altogether 2 + 4 + 1 = 7.
2 + 4 + 1 = 7
Squares of different sizes are all different squares, so just add the counts from each size.
2.OA.B.2Make A Systematic ListSquares of different sizes can never be the same square, so the counts from each size are simply added.
Why?
Every square in the picture has one definite side length, so it falls into exactly one size group.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The size groups cover every square and never overlap, so the group counts can be added without loss or repetition.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Answer
7 squares
2 + 4 + 1 = 7
Seven is reasonable for a 5x3 grid. There are 15 + 8 + 3 = 26 squares in all; only those covering exactly one heart qualify, and 7 of them do, while squares far from both hearts or large enough to hold both are correctly excluded. Each kept square was checked to contain exactly one heart.
Takeaway
Go size by size and slide each square around - keep only the ones with exactly one heart inside. You'll find seven!
- Set up the grid and mark the hearts
- Count 1x1 squares with exactly one heart
- Count 2x2 squares with exactly one heart
- Count 3x3 squares with exactly one heart
- Add up all sizes