What's MySQL's string compare rule?

Question

Why does SELECT * FROM users WHERE password='a'='mm'; actually return rows?

And why doesn't SELECT * FROM users WHERE password='1'='1'; work?

What's the exact equal operator compare rule used?

UPDATE(INFORMATION OF TABLES AND ROWS ETC):

mysql> show tables;
+------------------+
| Tables_in_shengy |
+------------------+
| users            |
+------------------+
1 row in set (0.00 sec)

mysql> describe users;
+----------+--------------+------+-----+---------+-------+
| Field    | Type         | Null | Key | Default | Extra |
+----------+--------------+------+-----+---------+-------+
| username | varchar(255) | YES  |     | NULL    |       |
| password | varchar(255) | YES  |     | NULL    |       |
+----------+--------------+------+-----+---------+-------+
2 rows in set (0.01 sec)

mysql> select * from users;
+----------+----------+
| username | password |
+----------+----------+
| victim   | 123456   |
+----------+----------+
1 row in set (0.00 sec)

Answer 1

MySQL converts compares to 0 and 1.

So password='a' evaluates to 0 (unless password actually is 'a').

Also MySQL tries to make comparisons between strings and numbers by converting the string to a number from its beginning.

So 0abc would be converted to 0. abc isn't a number at all and is also 0.

That means

password='a'='mm'

results in

0='mm'

and since mm does not have any leading digits, you get

0=0

which is true.