Olivia, Dana and Gem each had some beads and decided to play a game with their beads. In round 1, Olivia lost ¹₂ of her beads to Dana. In round 2, Dana lost ¹₂ of her total number of beads to Gem. In round 3, Gem lost ¹₂ of her total number of beads to Olivia. In the end, Olivia, Dana and Gem had 110, 81 and 73 beads respectively. How many beads did Dana have at first?

	Olivia	Dana	Gem
Before  1	2 b	125
Change 1	- 1 b	+ 1 b
After 1	1 b	162
Before 2	37	2 u
Change 2		- 1 u	+1
After 2		1 u	146
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Olivia, Dana and Gem in the end	110	81	73

Working backwards.

1 p = 73

At the start of Round 3:
Number of beads that Gem had
= 2 p
= 2 x 73 
= 146

At the start of Round 2:
Number of beads that Olivia had
= 110 - 1 p
= 110 - 73
= 37

At the end of Round 3:
Number of beads that Dana had = 81

1 u = 81

At the end of Round 1:
Number of beads that Dana had
= 2 u
= 2 x 81 
= 162

1 b = 37 

At the end of Round 1:
Number of beads that Olivia had = 37

At the start of Round 1:
Number of beads that Dana had
= 162 - 1 b
= 162 - 37
= 125

Answer(s): 125