Zoe, Diana and Penelope each had some balls and decided to play a game with their balls. In round 1, Zoe lost ¹₃ of her balls to Diana. In round 2, Diana lost ¹₃ of her total number of balls to Penelope. In round 3, Penelope lost ¹₃ of her total number of balls to Zoe. In the end, Zoe, Diana and Penelope had 131, 70 and 78 balls respectively. How many balls did Diana have at first?

	Zoe	Diana	Penelope
		59
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	105
Before 2	92	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	117
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Zoe, Diana and Penelope in the end	131	70	78

Working backwards.

2 p = 78

1 p = 78 ÷ 2 = 39

At the start of Round 3:
Number of balls that Penelope had
= 3 p
= 3 x 39
= 117

2 u = 70

1 u = 70 ÷ 2 = 35

At the start of Round 2:
Number of balls that Diana had
= 3 u
= 3 x 35
= 105

At the start of Round 2:
Number of balls that Zoe had
= 131 - 39
= 92

2 b = 92

1 b = 92 ÷ 2 = 46

At the start of Round 1:
Number of balls that Diana had
= 105 - 1 b
= 105 - 46
= 59

Answer(s): 59