Kathy, Diana and Tammy each had some balls and decided to play a game with their balls. In round 1, Kathy lost ¹₂ of her balls to Diana. In round 2, Diana lost ¹₂ of her total number of balls to Tammy. In round 3, Tammy lost ¹₂ of her total number of balls to Kathy. In the end, Kathy, Diana and Tammy had 125, 98 and 75 balls respectively. How many balls did Diana have at first?

	Kathy	Diana	Tammy
Before  1	2 b	146
Change 1	- 1 b	+ 1 b
After 1	1 b	196
Before 2	50	2 u
Change 2		- 1 u	+1
After 2		1 u	150
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Kathy, Diana and Tammy in the end	125	98	75

Working backwards.

1 p = 75

At the start of Round 3:
Number of balls that Tammy had
= 2 p
= 2 x 75 
= 150

At the start of Round 2:
Number of balls that Kathy had
= 125 - 1 p
= 125 - 75
= 50

At the end of Round 3:
Number of balls that Diana had = 98

1 u = 98

At the end of Round 1:
Number of balls that Diana had
= 2 u
= 2 x 98 
= 196

1 b = 50 

At the end of Round 1:
Number of balls that Kathy had = 50

At the start of Round 1:
Number of balls that Diana had
= 196 - 1 b
= 196 - 50
= 146

Answer(s): 146