KA-0138Growth patterns and figurate numbers
5 points, difficulty 3 of 3Level 3–4about 90sSquares of dots grow: 1 dot, then 4, then 9, then 16. How many dots are in the seventh square?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
What shape are these numbers?
2The strategy
Look at the counts rather than the pictures. Each one is a number multiplied by itself, and the position tells you which number to use.
3The full solution
1, 4, 9, 16 are 1 x 1, 2 x 2, 3 x 3, 4 x 4. So the seventh is 7 x 7 = 49.
Solution— no shortcut on this one
Write the counts as products: 1 = 1 x 1, 4 = 2 x 2, 9 = 3 x 3, 16 = 4 x 4. The nth square holds n x n dots, so the seventh holds 7 x 7 = 49. Check against a term you can see: the fourth should be 4 x 4 = 16, and it is. The transferable idea: the differences here grow (3, 5, 7, ...), so continuing by a constant step fails - recognise the shape of the numbers instead.
Why this is on the test: It is the first place a constant-difference habit breaks down, and choice C is exactly the answer that habit produces.