1053. A vendor was selling eggs. To the first customer, she sold half of all the eggs she had and another half-egg, to the second - half of the remainder and another half-egg, to the third - half of the new remainder and another half-egg, to the fourth - half of the last remainder and another half-egg. After this, she had no eggs left. How many eggs did she have initially?
$\triangle$ Let's solve the problem from the beginning.

After the third customer, the seller had one egg left.

After the second $-2 \cdot 1+1=3$. Let's check: she sold half of the previous remainder plus half an egg to the third customer, i.e., $3 / 2+1 / 2=2$, and she was left with $3-2=1$ egg. After the first customer, the seller had $2 \cdot 3+1=7$ eggs left. (Check it!) Finally, at the beginning, she had $2 \cdot 7+1=15$ eggs.

Answer: 15.