Shiyun, Julie and Linda each had some balls and decided to play a game with their balls. In round 1, Shiyun lost ¹₂ of her balls to Julie. In round 2, Julie lost ¹₂ of her total number of balls to Linda. In round 3, Linda lost ¹₂ of her total number of balls to Shiyun. In the end, Shiyun, Julie and Linda had 137, 84 and 72 balls respectively. How many balls did Julie have at first?

	Shiyun	Julie	Linda
Before  1	2 b	103
Change 1	- 1 b	+ 1 b
After 1	1 b	168
Before 2	65	2 u
Change 2		- 1 u	+1
After 2		1 u	144
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Shiyun, Julie and Linda in the end	137	84	72

Working backwards.

1 p = 72

At the start of Round 3:
Number of balls that Linda had
= 2 p
= 2 x 72 
= 144

At the start of Round 2:
Number of balls that Shiyun had
= 137 - 1 p
= 137 - 72
= 65

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

1 u = 84

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

1 b = 65 

At the end of Round 1:
Number of balls that Shiyun had = 65

At the start of Round 1:
Number of balls that Julie had
= 168 - 1 b
= 168 - 65
= 103

Answer(s): 103