Three. (25 points) Person A and Person B went to the same mall to buy a type of small handkerchief, each costing 0.50 yuan. The mall stipulates that a 20% discount is available for purchasing no less than 10 small handkerchiefs. As a result, A spent 4 yuan more than B, and it is known that the amount A spent does not exceed 8 yuan. Under the condition of fully enjoying (if possible) the discount, how many small handkerchiefs did A and B buy respectively?
Three, let A and B buy $x$ and $y$ pieces of small handkerchiefs, respectively.
(1) When $x<10, y<10$, we have
$$
0.5 x - 0.5 y = 4 \text{, }
$$

which simplifies to $x = 8 + y$.
Thus, only $x = 9, y = 1$ satisfies the equation.
(2) When $x \geqslant 10, y < 10$, we have
$$
0.5 \times (1 - 20\%) x - 0.5 y = 4 \text{, }
$$

which simplifies to $4 x - 5 y = 40$.
Solving this indeterminate equation, we get
$$
\left\{\begin{array}{l}x = 10 + 5 t, \\ y = 4 t .\end{array}\right.
$$ (where $t$ is a natural number)
Since A spends no more than 8 yuan,
$$
\therefore x \leqslant 8 \div \{(1 - 20\%) \times 0.5\} = 20 \text{. }
$$
$$
\therefore t \text{ must satisfy } 10 \leqslant 10 + 5 t \leqslant 20,
$$
which means $0 \leqslant t \leqslant 2$.
$$
\therefore t = 0, 1, 2.
$$
$$
\therefore \left\{\begin{array}{l}x = 10, \\ y = 0 ;\end{array}\right.
$$ (discard) $\left\{\begin{array}{l}x = 15, \\ y = 4 ;\end{array}\right.$
(3) When $x \geqslant 10, y \geqslant 10$, we have
$$
0.5 \times (1 - 20\%) x - 0.5 \times (1 - 20\%) y = 4.
$$
Simplifying, we get $x - y = 10$,
which means $x = y + 10 \geqslant 20$.
Since $x \leqslant 20$ (see (2)),
$$
\therefore x = 20, \text{ at this time } y = 10.
$$
In summary, when A buys 9, 15, or 20 pieces of small handkerchiefs, B buys 1, 4, or 10 pieces of small handkerchiefs, respectively.
(Xie Wenxiao, Huanggang High School, Hubei Province, 436100)