Kathy, Shiyun and Julie each had some beads and decided to play a game with their beads. In round 1, Kathy lost ¹₂ of her beads to Shiyun. In round 2, Shiyun lost ¹₂ of her total number of beads to Julie. In round 3, Julie lost ¹₂ of her total number of beads to Kathy. In the end, Kathy, Shiyun and Julie had 119, 98 and 55 beads respectively. How many beads did Shiyun have at first?

	Kathy	Shiyun	Julie
Before  1	2 b	132
Change 1	- 1 b	+ 1 b
After 1	1 b	196
Before 2	64	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 Kathy, Shiyun and Julie in the end	119	98	55

Working backwards.

1 p = 55

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

At the start of Round 2:
Number of beads that Kathy had
= 119 - 1 p
= 119 - 55
= 64

At the end of Round 3:
Number of beads that Shiyun had = 98

1 u = 98

At the end of Round 1:
Number of beads that Shiyun had
= 2 u
= 2 x 98 
= 196

1 b = 64 

At the end of Round 1:
Number of beads that Kathy had = 64

At the start of Round 1:
Number of beads that Shiyun had
= 196 - 1 b
= 196 - 64
= 132

Answer(s): 132