Cindy, Natalie and Gwen each had some beads and decided to play a game with their beads. In round 1, Cindy lost ¹₂ of her beads to Natalie. In round 2, Natalie lost ¹₂ of her total number of beads to Gwen. In round 3, Gwen lost ¹₂ of her total number of beads to Cindy. In the end, Cindy, Natalie and Gwen had 116, 85 and 65 beads respectively. How many beads did Natalie have at first?

	Cindy	Natalie	Gwen
Before  1	2 b	119
Change 1	- 1 b	+ 1 b
After 1	1 b	170
Before 2	51	2 u
Change 2		- 1 u	+1
After 2		1 u	130
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Cindy, Natalie and Gwen in the end	116	85	65

Working backwards.

1 p = 65

At the start of Round 3:
Number of beads that Gwen had
= 2 p
= 2 x 65 
= 130

At the start of Round 2:
Number of beads that Cindy had
= 116 - 1 p
= 116 - 65
= 51

At the end of Round 3:
Number of beads that Natalie had = 85

1 u = 85

At the end of Round 1:
Number of beads that Natalie had
= 2 u
= 2 x 85 
= 170

1 b = 51 

At the end of Round 1:
Number of beads that Cindy had = 51

At the start of Round 1:
Number of beads that Natalie had
= 170 - 1 b
= 170 - 51
= 119

Answer(s): 119