Cathy, Olivia and Joelle each had some balls and decided to play a game with their balls. In round 1, Cathy lost ¹₂ of her balls to Olivia. In round 2, Olivia lost ¹₂ of her total number of balls to Joelle. In round 3, Joelle lost ¹₂ of her total number of balls to Cathy. In the end, Cathy, Olivia and Joelle had 125, 81 and 69 balls respectively. How many balls did Olivia have at first?

	Cathy	Olivia	Joelle
Before  1	2 b	106
Change 1	- 1 b	+ 1 b
After 1	1 b	162
Before 2	56	2 u
Change 2		- 1 u	+1
After 2		1 u	138
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Cathy, Olivia and Joelle in the end	125	81	69

Working backwards.

1 p = 69

At the start of Round 3:
Number of balls that Joelle had
= 2 p
= 2 x 69 
= 138

At the start of Round 2:
Number of balls that Cathy had
= 125 - 1 p
= 125 - 69
= 56

At the end of Round 3:
Number of balls that Olivia had = 81

1 u = 81

At the end of Round 1:
Number of balls that Olivia had
= 2 u
= 2 x 81 
= 162

1 b = 56 

At the end of Round 1:
Number of balls that Cathy had = 56

At the start of Round 1:
Number of balls that Olivia had
= 162 - 1 b
= 162 - 56
= 106

Answer(s): 106