Expanding Pattern
Problem
The diagram shows the first three patterns in a sequence in which each pattern has a square hole in the middle.
How many small shaded squares are needed to build the tenth pattern in the sequence?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 92 squares
Building the tenth pattern in rectangles
Total 23$\times$4 = 92 squares
Building the tenth pattern from a larger square
Each large square has the centre and corners removed
14$^2-$ 10$^2-$ 4 = 196 $-$ 100 $-$ 4 = 92
Using smaller rectangles and algebra
Break the pattern down into four rectangles, each two squares wide, and four L-shaped corner pieces, of three squares each.
Pattern 1 = $4 \times 2 \times 1 + 4 \times 3 = 20$
Pattern 2 = $4 \times 2 \times 2 + 4 \times 3 = 28$
Pattern 3 = $4 \times 2 \times 3 + 4 \times 3 = 36$
Pattern n = $4 \times 2 \times n + 4 \times 3 = 8n +12$
Pattern 10 = $4 \times 2 \times 10 + 4 \times 3 = 92$ squares
Using larger squares and algebra
Imagine a large square with corners removed, and a square removed from the middle:
Pattern 1 = $5^2 - 4 - 1^2 = 20$
Pattern 2 = $6^2 - 4 - 2^2 = 28$
Pattern 3 = $7^2 - 4 - 3^2 = 36$
Pattern n = $(n+4)^2 - 4 - n^2 = 8n +12$
Pattern 10 = $14^2 - 4 - 10^2 = 92$ squares