Xandra, Esther and Jane each had some beads and decided to play a game with their beads. In round 1, Xandra lost ¹₂ of her beads to Esther. In round 2, Esther lost ¹₂ of her total number of beads to Jane. In round 3, Jane lost ¹₂ of her total number of beads to Xandra. In the end, Xandra, Esther and Jane had 114, 93 and 67 beads respectively. How many beads did Esther have at first?

	Xandra	Esther	Jane
Before  1	2 b	139
Change 1	- 1 b	+ 1 b
After 1	1 b	186
Before 2	47	2 u
Change 2		- 1 u	+1
After 2		1 u	134
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Xandra, Esther and Jane in the end	114	93	67

Working backwards.

1 p = 67

At the start of Round 3:
Number of beads that Jane had
= 2 p
= 2 x 67 
= 134

At the start of Round 2:
Number of beads that Xandra had
= 114 - 1 p
= 114 - 67
= 47

At the end of Round 3:
Number of beads that Esther had = 93

1 u = 93

At the end of Round 1:
Number of beads that Esther had
= 2 u
= 2 x 93 
= 186

1 b = 47 

At the end of Round 1:
Number of beads that Xandra had = 47

At the start of Round 1:
Number of beads that Esther had
= 186 - 1 b
= 186 - 47
= 139

Answer(s): 139