Dana, Hilda and Mary each had some balls and decided to play a game with their balls. In round 1, Dana lost ¹₂ of her balls to Hilda. In round 2, Hilda lost ¹₂ of her total number of balls to Mary. In round 3, Mary lost ¹₂ of her total number of balls to Dana. In the end, Dana, Hilda and Mary had 122, 86 and 50 balls respectively. How many balls did Hilda have at first?

	Dana	Hilda	Mary
Before  1	2 b	100
Change 1	- 1 b	+ 1 b
After 1	1 b	172
Before 2	72	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 Dana, Hilda and Mary in the end	122	86	50

Working backwards.

1 p = 50

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

At the start of Round 2:
Number of balls that Dana had
= 122 - 1 p
= 122 - 50
= 72

At the end of Round 3:
Number of balls that Hilda had = 86

1 u = 86

At the end of Round 1:
Number of balls that Hilda had
= 2 u
= 2 x 86 
= 172

1 b = 72 

At the end of Round 1:
Number of balls that Dana had = 72

At the start of Round 1:
Number of balls that Hilda had
= 172 - 1 b
= 172 - 72
= 100

Answer(s): 100