Esther, Shannon and Olivia each had some balls and decided to play a game with their balls. In round 1, Esther lost ¹₂ of her balls to Shannon. In round 2, Shannon lost ¹₂ of her total number of balls to Olivia. In round 3, Olivia lost ¹₂ of her total number of balls to Esther. In the end, Esther, Shannon and Olivia had 135, 80 and 53 balls respectively. How many balls did Shannon have at first?

	Esther	Shannon	Olivia
Before  1	2 b	78
Change 1	- 1 b	+ 1 b
After 1	1 b	160
Before 2	82	2 u
Change 2		- 1 u	+1
After 2		1 u	106
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Esther, Shannon and Olivia in the end	135	80	53

Working backwards.

1 p = 53

At the start of Round 3:
Number of balls that Olivia had
= 2 p
= 2 x 53 
= 106

At the start of Round 2:
Number of balls that Esther had
= 135 - 1 p
= 135 - 53
= 82

At the end of Round 3:
Number of balls that Shannon had = 80

1 u = 80

At the end of Round 1:
Number of balls that Shannon had
= 2 u
= 2 x 80 
= 160

1 b = 82 

At the end of Round 1:
Number of balls that Esther had = 82

At the start of Round 1:
Number of balls that Shannon had
= 160 - 1 b
= 160 - 82
= 78

Answer(s): 78