Kathy, Cathy and Gwen each had some balls and decided to play a game with their balls. In round 1, Kathy 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 Kathy. In the end, Kathy, Cathy and Gwen had 115, 100 and 52 balls respectively. How many balls did Cathy have at first?

	Kathy	Cathy	Gwen
Before  1	2 b	137
Change 1	- 1 b	+ 1 b
After 1	1 b	200
Before 2	63	2 u
Change 2		- 1 u	+1
After 2		1 u	104
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Kathy, Cathy and Gwen in the end	115	100	52

Working backwards.

1 p = 52

At the start of Round 3:
Number of balls that Gwen had
= 2 p
= 2 x 52 
= 104

At the start of Round 2:
Number of balls that Kathy had
= 115 - 1 p
= 115 - 52
= 63

At the end of Round 3:
Number of balls that Cathy had = 100

1 u = 100

At the end of Round 1:
Number of balls that Cathy had
= 2 u
= 2 x 100 
= 200

1 b = 63 

At the end of Round 1:
Number of balls that Kathy had = 63

At the start of Round 1:
Number of balls that Cathy had
= 200 - 1 b
= 200 - 63
= 137

Answer(s): 137