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Answer

  Box A Box B Box C
At first 78 24 60
Tfr some from A to B -48 +48 0
  30 72 60
Tfr 1/4 fr B to C 0 -18 +18
  30 54 78
Tfr 24 from C to A +24 0 -24
At end 54 54 54

Working backwards,

At the end, equal number in each box.

162 / 3 = 54

If after transfer out 1/4, the remaining B’s button is 3/4 = 54.

54 /3 x 4 = 72

Before transfer out 1/4, no of button will be 72 

If after transfer some from A to B, B’s button is tripled (72)

72 / 3 = 24

Before transfer from A, B’s button is 24

At first, the number of buttons in Box A, B and C was 78, 24 and 60 respectively. 

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