Step by step
The question
Question: The pattern circle, square, circle, square, ...
keeps repeating.
What is the 75th shape?
1 Find the block
The list starts again after square.
So the block that repeats is circle, square.
block = circle, square
The block is 2 long.
length = 2
2 Divide
The block comes again and again.
So we divide by its length.
What is left over tells us the place in the block.
n ÷ (length of the block): what is left = the place in the block
2 times 37 is 74.
So 37 full blocks take us to the 74th shape.
75 minus 74 is 1 more.
75 ÷ 2 = 2 × 37 = 74, 75 − 74 = 1
3 Count into the block
Careful!
The 37 only counts full blocks.
The 1 left over gives the place.
Careful: not 37: the number of blocks ✗
1 left means place 1 in the block: circle.
75th shape = place 1 in the block = circle
4 Check
Let's check by counting on.
A block ends at the 74th shape.
Then we count on: circle.
The 75th is circle.
So it is right!
74th = square (end of a block), 75th = circle ✓
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.
Answer
75th shape = place 1 in the block = circle