Solution: suppose there are x white balls at the beginning, that is, there are x black balls, and then there are y white balls at the back, so there are 14-y black balls.

( 14-y+x)÷(2x+ 14)= 1/6，

14×6-6y+6x=2x+ 14，

84-6y= 14-4x，

4x = 6y-70；

Obviously, the value of y is from 0 to 14, but because x must be greater than 0, there must be:

6y-70≥0，

y≥ 1 1.7，

That is, the values of y are 12, 13,14;

When y= 12, x=0.5.

When y= 13, x=2.

When y= 14, x=3.5.

Must be a positive integer, so y is 13 and x is 2;

Now there are white balls in the box: 2+ 13= 15 (pieces).

So the answer is: 15.

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