10. (16 points) On a certain day in the twentieth century (1900 to 1999), the younger brother said to the older brother: "Brother, look, if you add up the four digits of your birth year, it equals my age." The older brother replied: "Dear brother, you are right! The same applies to me; if you add up the four digits of your birth year, it equals my age. Additionally, if we swap the two digits of our ages, we get each other's age." It is known that the brothers were born in different years. In which year did this conversation take place? $\qquad$
10. (16 points) On a certain day in the twentieth century (1900 to 1999), the younger brother said to the older brother: "Brother, look, if you add up the four digits of your birth year, it equals my age." The older brother replied: "Dear brother, you're right! The same applies to me; if you add up the four digits of your birth year, it equals my age. Additionally, if we swap the two digits of our ages, we get each other's age." It is known that the two brothers were born in different years. In which year did this conversation take place? 

【Solution】Let the older brother be born in the year $19ab$, and the younger brother be born in the year $19cd$. Then, at the time of the conversation, the older brother is $10+c+d$ years old, and the younger brother is $10+a+b$ years old;

The tens digit of the older brother's age = the units digit of the younger brother's age, and the units digit of the older brother's age = the tens digit of the younger brother's age,
(1) When $c+d=10$,

The older brother is 21 years old, and the younger brother is 12 years old, $c+d=11, a+b=2$;
(3) Since $a+b=2$,
The older brother's birth year has 3 possible scenarios: 1911, 1902, 1920,
Since the older brother is 9 years older than the younger brother (21-12=9),
The younger brother's birth year has 3 possible scenarios: 1920, 1911, 1929,
Since $1+9+2+0=12 \neq 21, 1+9+1+1=12 \neq 21, 1+9+2+9=21$,
The younger brother was born in 1929,
Since $1929+12=1941$ (year),
The conversation took place in 1941.
Answer: The conversation took place in 1941.
Therefore, the answer is: 1941.