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For part (b), we can observe that the total number of tiles is a square number.

Total number of tiles = (Figure No + 2)^2

144 = (Figure No + 2)^2

Square root both sides.

12 = Figure No + 2

Figure No. = 10 (b)

For part (c), we can just find the total number of tiles minus the white tiles.

15 + 2 = 17

17 × 17 = 289 (total)

White tiles: 15 × 4 = 60

289 – 60 = 229 (c)

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(a) By inspection, for pattern n

total square = ( n+ 2)²

Number of white tiles = 4n

Number of shaded tiles = 4 + n²

Pattern              Number of  white tiles                Number of shaded tiles

1                                  4                                                                5

2                                  8                                                               8

3                                 12                                                              13

4                                 16                                                              20

5                                 20                                                              29

6                                 24                                                              40

 

(b) ( n+ 2)²= 144

n+2  = 12

n = 10

(c) For Pattern 15,

number of shaded tiles = 4 + 15² = 229

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