Hazel, Natalie and Kathy each had some beads and decided to play a game with their beads. In round 1, Hazel lost ¹₂ of her beads to Natalie. In round 2, Natalie lost ¹₂ of her total number of beads to Kathy. In round 3, Kathy lost ¹₂ of her total number of beads to Hazel. In the end, Hazel, Natalie and Kathy had 125, 84 and 60 beads respectively. How many beads did Natalie have at first?

	Hazel	Natalie	Kathy
Before  1	2 b	103
Change 1	- 1 b	+ 1 b
After 1	1 b	168
Before 2	65	2 u
Change 2		- 1 u	+1
After 2		1 u	120
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Hazel, Natalie and Kathy in the end	125	84	60

Working backwards.

1 p = 60

At the start of Round 3:
Number of beads that Kathy had
= 2 p
= 2 x 60 
= 120

At the start of Round 2:
Number of beads that Hazel had
= 125 - 1 p
= 125 - 60
= 65

At the end of Round 3:
Number of beads that Natalie had = 84

1 u = 84

At the end of Round 1:
Number of beads that Natalie had
= 2 u
= 2 x 84 
= 168

1 b = 65 

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

At the start of Round 1:
Number of beads that Natalie had
= 168 - 1 b
= 168 - 65
= 103

Answer(s): 103