Min, Julie and Ivory each had some marbles and decided to play a game with their marbles. In round 1, Min lost ¹₂ of her marbles to Julie. In round 2, Julie lost ¹₂ of her total number of marbles to Ivory. In round 3, Ivory lost ¹₂ of her total number of marbles to Min. In the end, Min, Julie and Ivory had 123, 81 and 52 marbles respectively. How many marbles did Julie have at first?

	Min	Julie	Ivory
Before  1	2 b	91
Change 1	- 1 b	+ 1 b
After 1	1 b	162
Before 2	71	2 u
Change 2		- 1 u	+1
After 2		1 u	104
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Min, Julie and Ivory in the end	123	81	52

Working backwards.

1 p = 52

At the start of Round 3:
Number of marbles that Ivory had
= 2 p
= 2 x 52 
= 104

At the start of Round 2:
Number of marbles that Min had
= 123 - 1 p
= 123 - 52
= 71

At the end of Round 3:
Number of marbles that Julie had = 81

1 u = 81

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

1 b = 71 

At the end of Round 1:
Number of marbles that Min had = 71

At the start of Round 1:
Number of marbles that Julie had
= 162 - 1 b
= 162 - 71
= 91

Answer(s): 91