Step by step
The question
Question: The pattern X, O, O, X, O, O, ...
keeps repeating.
What is the 101st letter?
1 Find the block
The list starts again after O.
So the block that repeats is X, O, O.
block = X, O, O
The block is 3 long.
length = 3
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
3 times 33 is 99.
So 33 full blocks take us to the 99th letter.
101 minus 99 is 2 more.
101 ÷ 3 = 3 × 33 = 99, 101 − 99 = 2
3 Count into the block
Careful!
The 33 only counts full blocks.
The 2 left over gives the place.
Careful: not 33: the number of blocks ✗
2 left means place 2 in the block: O.
101st letter = place 2 in the block = O
4 Check
Let's check by counting on.
A block ends at the 99th letter.
Then we count on: X, O.
The 101st is O.
So it is right!
99th = O (end of a block), 100th = X, 101st = O ✓
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.
Answer
101st letter = place 2 in the block = O