User:
Find out two 3-digit palindromic numbers that added result in a 4-digit palindromic number. Note: the first digit of any of the above mentioned three numbers can't be 0.

phi-4:
To solve this problem, we need to find two 3-digit palindromic numbers whose sum is a 4-digit palindromic number. A 3-digit palindromic number has the form \( aba \), where \( a \) and \( b \) are digits and \( a \neq 0 \). Similarly, a 4-digit palindromic number has the form \( abba \).

Let's denote the two 3-digit palindromic numbers as \( aba \) and \( cdc \). Their sum should be a 4-digit palindromic number \( abba \).

The number \( aba \) can be expressed as \( 101a + 10b \), and \( cdc \) can be expressed as \( 101c + 10d \).

The sum is:
\[
(101a + 10b) + (101c + 10d) = 101(a + c) + 10(b + d)
\]

This sum should equal the 4-digit palindromic number \( abba \), which can be expressed as:
\[
1001a + 110b
\]

Equating the two expressions, we have:
\[
101(a + c) + 10(b + d) = 1001a + 110b
\]

Simplifying, we get:
\[
101a + 101c + 10b + 10d = 1001a + 110b
\]

Rearranging terms gives:
\[
101c + 10d = 900a + 100b
\]

Dividing the entire equation by 10:
\[
10.1c + d = 90a + 10b
\]

Since \( c \) and \( d \) are digits, \( 10.1c \) must be an integer, implying \( c \) must be 0. However, \( c \neq 0 \) because \( cdc \) is a 3-digit number. Therefore, let's try specific values for \( a \), \( b \), \( c \), and \( d \).

Let's try \( a = 1 \) and \( c = 9 \) (since \( a + c \) should be close to 10 to make the sum a 4-digit number):

1. \( aba = 1b1 \)
2. \( cdc = 9d9 \)

Their sum:
\[
(101 \times 1 + 10b) + (101 \times 9 + 10d) = 101 + 10b + 909 + 10d = 1010 + 10b + 10d
\]

We want:
\[
1010 + 10b + 10d = 1001 \times 1 + 110b = 1001 + 110b
\]

Simplifying:
\[
1010 + 10b + 10d = 1001 + 110b
\]

\[
9 + 10d = 100b
\]

\[
10d = 100b - 9
\]

\[
d = 10b - 0.9
\]

Since \( d \) must be an integer, \( 10b - 0.9 \) must be an integer, implying \( b = 1 \).

Substituting \( b = 1 \):
\[
d = 10 \times 1 - 0.9 = 9.1
\]

This is not an integer, so let's try \( b = 9 \).

Substituting \( b = 9 \):
\[
d = 10 \times 9 - 0.9 = 89.1
\]

This is not an integer. Let's try \( b = 0 \).

Substituting \( b = 0 \):
\[
d = 10 \times 0 - 0.9 = -0.9
\]

This is not valid. Let's try \( a = 2 \) and \( c = 8 \).

1. \( aba = 2b2 \)
2. \( cdc = 8d8 \)

Their sum:
\[
(101 \times 2 + 10b) + (101 \times 8 + 10d) = 202 + 10b + 808 + 10d = 1010 + 10b + 10d
\]

We want:
\[
1010 + 10b + 10d = 2002 + 110b
\]

Simplifying:
\[
10b + 10d = 992 + 110b
\]

\[
10d = 992 + 100b
\]

\[
d = 99.2 + 10b
\]

Trying \( b = 1 \):
\[
d = 99.2 + 10 \times 1 = 109.2
\]

Trying \( b = 0 \):
\[
d = 99.2
\]

Trying \( b = 9 \):
\[
d = 99.2 + 90 = 189.2
\]

Let's try \( a = 9 \) and \( c = 1 \).

1. \( aba = 9b9 \)
2. \( cdc = 1d1 \)

Their sum:
\[
(101 \times 9 + 10b) + (101 \times 1 + 10d) = 909 + 10b + 101 + 10d = 1010 + 10b + 10d
\]

We want:
\[
1010 + 10b + 10d = 9009 + 110b
\]

Simplifying:
\[
10b + 10d = 7999 + 110b
\]

\[
10d = 7999 + 100b
\]

\[
d = 799.9 + 10b
\]

Trying \( b = 0 \):
\[
d = 799.9
\]

Trying \( b = 1 \):
\[
d = 809.9
\]

Trying \( b = 9 \):
\[
d = 889.9
\]

Finally, let's try \( a = 9 \), \( b = 1 \), \( c = 1 \), \( d = 9 \).

1. \( aba = 919 \)
2. \( cdc = 191 \)

Their sum:
\[
919 + 191 = 1110
\]

This is not a palindrome. Let's try \( a = 9 \), \( b = 9 \), \( c = 1 \), \( d = 0 \).

1. \( aba = 999 \)
2. \( cdc = 100 \)

Their sum:
\[
999 + 100 = 1099
\]

This is not a palindrome. Let's try \( a = 9 \), \( b = 9 \), \( c = 1 \), \( d = 1 \).

1. \( aba = 999 \)
2. \( cdc = 101 \)

Their sum:
\[
999 + 101 = 1100
\]

This is not a palindrome. Let's try \( a = 9 \), \( b = 9 \), \( c = 1 \), \( d = 9 \).

1. \( aba = 999 \)
2. \( cdc = 191 \)

Their sum:
\[
999 + 191 = 1190
\]

This is not a palindrome. Let's try \( a = 9 \), \( b = 9 \), \( c = 2 \), \( d = 1 \).

1. \( aba = 999 \)
2. \( cdc = 202 \)

Their sum:
\[
999 + 202 = 1201
\]

This is a palindrome. Therefore, the two 3-digit palindromic numbers are \( 999 \) and \( 202 \), and their sum is the 4-digit palindromic number \( 1201 \).