Penelope, Cathy and Gwen each had some balls and decided to play a game with their balls. In round 1, Penelope lost ¹₂ of her balls to Cathy. In round 2, Cathy lost ¹₂ of her total number of balls to Gwen. In round 3, Gwen lost ¹₂ of her total number of balls to Penelope. In the end, Penelope, Cathy and Gwen had 115, 84 and 71 balls respectively. How many balls did Cathy have at first?

	Penelope	Cathy	Gwen
Before  1	2 b	124
Change 1	- 1 b	+ 1 b
After 1	1 b	168
Before 2	44	2 u
Change 2		- 1 u	+1
After 2		1 u	142
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Penelope, Cathy and Gwen in the end	115	84	71

Working backwards.

1 p = 71

At the start of Round 3:
Number of balls that Gwen had
= 2 p
= 2 x 71 
= 142

At the start of Round 2:
Number of balls that Penelope had
= 115 - 1 p
= 115 - 71
= 44

At the end of Round 3:
Number of balls that Cathy had = 84

1 u = 84

At the end of Round 1:
Number of balls that Cathy had
= 2 u
= 2 x 84 
= 168

1 b = 44 

At the end of Round 1:
Number of balls that Penelope had = 44

At the start of Round 1:
Number of balls that Cathy had
= 168 - 1 b
= 168 - 44
= 124

Answer(s): 124