The diagram shows 3 squares, the length of which are 1 cm, 2 cm and 3 cm respectively. Each square is made up of small squares which has a length of 1 cm and alternate squares are coloured grey and white.

1. How many small grey squares are needed to create a square of side 19 cm?
2. Find the length of the side of a square if 625 small grey and white squares are used to create it.
3. Find the number of small white squares when a square of side 26 cm is created?

(a)
Patterns for the number of grey squares for odd-numbered length
1 cm: (1 x 1 + 1) ÷ 2 = 1
3 cm: (3 x 3 + 1) ÷ 2 = 5
5 cm: (5 x 5 + 1) ÷ 2 = 13

Formula for number of small grey squares for odd-numbered length:
Number of grey squares = (Square length² + 1) ÷ 2 

Number of grey squares to create a square of side 19 cm
= (19 x 19 + 1) ÷ 2 
= (361 + 1) ÷ 2 
= 181

(b)
Patterns for the number of grey and white squares
1 cm: 1² = 1 x 1 = 1
2 cm: 2² = 2 x 2 = 4
3 cm: 3² = 3 x 3 = 9

Formula for total number of grey and white squares:
Total number of grey and white squares = Square length²

Length of the square side when 625 grey and white squares is
= √625
= 25 cm

(c)
Patterns for the number of white squares for even-numbered length:
2 cm: 2 x 2 ÷ 2 = 2
4 cm: 4 x 4 ÷ 2 = 8
6 cm: 6 x 6 ÷ 2 = 18

Formula for number of white squares for even-numbered length:
Number of white squares = Square length² ÷ 2

Number of white squares when a square of side 26 cm is created
= 26² ÷ 2
= 26 x 26 ÷ 2
= 676 ÷ 2
= 338

Answer(s): (a) 181; (b) 25 cm; (c) 338