The ratio of the number of coins in boxes A and B was 17:7. After 108 coins were shifted from box A to box B, this ratio became 37:20. The number of coins that needs to be shifted further from A to B, to make this ratio 1:1, is

Let the initial total number of coins in box A be $$17x$$, since the ratio of the number of coins in boxes A and B is $$17:7$$, the initial total number of coins in box B would be $$7x$$.

After $$108$$ coins are shifted from box A to box B, the number of coins in box A would be $$17x-108$$ and in box B would be $$7x+108$$. We have,

$$\dfrac{17x-108}{7x+108} = \dfrac{37}{20}$$

$$\Rightarrow 340x - 2160 = 259x + 3996$$

$$\Rightarrow 81x = 6156$$  or  $$x= 76$$

Therefore, the total number of coins in the two boxes combined are $$(17+7)*76 = 1824$$, half of which is $$912$$. The number of coins in box B currently is $$7*76+108 = 640$$. The number of coins needed to make the coins equal in both the boxes is, $$912-640 = 272$$.

Thus, $$272$$ coins will be transferred from box A to box B to make the number of coins equal (1:1) in both the boxes.