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Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

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Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

Bull's Eye

Stage: 3 Challenge Level: Challenge Level:1
We have had correct responses to this problem from a number of students: Andrei Lazanu (aged 12) from No. 205 School in Bucharest (Romania), Chua Zhi Yu (aged 13) from River Valley High School in Singapore, Michael Brooker (aged 10) educated at home, Belinda Guo (aged 14) from Riccarton High School in Christchurch (New Zealand), Prateek Mehrotra, Sim Jingwei (aged 12) from Raffles Girls' Primary School in Singapore, and Fiona Watson from Stamford High School. Well done to you all.

Everyone reasoned in a similar way:

Area of circle $=\pi r^2$

Area of the largest circle $= \pi \times 7^2 = 49 \pi cm^2$

Area of the red ring $= \pi \times 4^2 - \pi \times 3 ^ 2 = 7 \pi cm^2$

$\frac{7}{49}= \frac{1}{7}$

The red ring is $\frac{1}{7}$ of the whole circle.

Area of the green ring $= \pi \times 6^2 - \pi \times 5^2 = 11 \pi cm^2$

The green ring is $\frac{11}{49}$ of the whole circle.