Barbara, Zara and Lynn each had some beads and decided to play a game with their beads. In round 1, Barbara lost ¹₂ of her beads to Zara. In round 2, Zara lost ¹₂ of her total number of beads to Lynn. In round 3, Lynn lost ¹₂ of her total number of beads to Barbara. In the end, Barbara, Zara and Lynn had 129, 101 and 50 beads respectively. How many beads did Zara have at first?

	Barbara	Zara	Lynn
Before  1	2 b	123
Change 1	- 1 b	+ 1 b
After 1	1 b	202
Before 2	79	2 u
Change 2		- 1 u	+1
After 2		1 u	100
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Barbara, Zara and Lynn in the end	129	101	50

Working backwards.

1 p = 50

At the start of Round 3:
Number of beads that Lynn had
= 2 p
= 2 x 50 
= 100

At the start of Round 2:
Number of beads that Barbara had
= 129 - 1 p
= 129 - 50
= 79

At the end of Round 3:
Number of beads that Zara had = 101

1 u = 101

At the end of Round 1:
Number of beads that Zara had
= 2 u
= 2 x 101 
= 202

1 b = 79 

At the end of Round 1:
Number of beads that Barbara had = 79

At the start of Round 1:
Number of beads that Zara had
= 202 - 1 b
= 202 - 79
= 123

Answer(s): 123