A hard question on counting shapes for P4–P5, with a step-by-step solution, diagrams and the answer. Every year of the Hua Xia Cup, P1 to P6, has shape-counting questions. The hard part is counting each shape once and missing none.
The question
A sheet of squared paper has 4 rows of 4 equal squares.
The square in row 1 from the top, 4 from the left, has been cut away (the blank square in the picture).
Ming wants to cut a whole square along the grid lines from what is left, so it can't include the missing square. How many different choices does he have? (A different size or position counts as different.)
Show the hints (3)
💡Hint 1: First pretend nothing was cut away and count every square; then take away the squares that cover the gap.
💡Hint 2: Count the squares covering the gap size by size: for one size, how many positions across cover it? How many down?
💡Hint 3: For example, squares 2 wide cover the gap in 1 position across and 1 down: 1 × 1 = 1.
Worked solution
Have a go first, then read the solution step by step.
1.First pretend nothing is cut away: side 1: 4 × 4 = 16; side 2: 3 × 3 = 9; side 3: 2 × 2 = 4; side 4: 1 × 1 = 1; 30 in all.
2.Now the squares covering the gap. Take side 2: its top-left corner must be on an orange dot to cover the gap (the orange frame is one of them), 1 across and 1 down: 1 × 1 = 1.
3.Every size the same way: side 1: 1 × 1 = 1; side 2: 1 × 1 = 1; side 3: 1 × 1 = 1; side 4: 1 × 1 = 1; 4 in all.
4.Squares he can cut: 30 − 4 = 26.
⚠️Common mistake: taking away only the 1 missing square, forgetting that bigger squares cover the gap too; or thinking a gap at the edge changes nothing.
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