Jen, Shiyun and Lynn each had some beads and decided to play a game with their beads. In round 1, Jen lost ¹₂ of her beads to Shiyun. In round 2, Shiyun lost ¹₂ of her total number of beads to Lynn. In round 3, Lynn lost ¹₂ of her total number of beads to Jen. In the end, Jen, Shiyun and Lynn had 120, 88 and 55 beads respectively. How many beads did Shiyun have at first?

	Jen	Shiyun	Lynn
Before  1	2 b	111
Change 1	- 1 b	+ 1 b
After 1	1 b	176
Before 2	65	2 u
Change 2		- 1 u	+1
After 2		1 u	110
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Jen, Shiyun and Lynn in the end	120	88	55

Working backwards.

1 p = 55

At the start of Round 3:
Number of beads that Lynn had
= 2 p
= 2 x 55 
= 110

At the start of Round 2:
Number of beads that Jen had
= 120 - 1 p
= 120 - 55
= 65

At the end of Round 3:
Number of beads that Shiyun had = 88

1 u = 88

At the end of Round 1:
Number of beads that Shiyun had
= 2 u
= 2 x 88 
= 176

1 b = 65 

At the end of Round 1:
Number of beads that Jen had = 65

At the start of Round 1:
Number of beads that Shiyun had
= 176 - 1 b
= 176 - 65
= 111

Answer(s): 111