Pretty Palindromic Patterns

What is a Palindrome?

A palindromic number reads the same from left to right and right to left. Examples: $66, 848, 575, 12321$.

Reverse-and-Add Method

You can often generate a palindrome from a normal number using this method:

1. Take a number (e.g., $34$).
2. Reverse it ($43$).
3. Add them together: $34 + 43 = 77$. (Palindrome!)

Sometimes it takes multiple steps. Let’s try $69$:

1. $69 + 96 = 165$ (Not a palindrome)
2. $165 + 561 = 726$
3. $726 + 627 = 1353$
4. $1353 + 3531 = 4884$ (Palindrome reached!)

Digit Detectives Puzzle

Problem: I am a 5-digit palindrome. I am odd. My tens (‘t’) digit is double my units (‘u’) digit. My hundreds (‘h’) digit is double my tens digit. Who am I?

Solution:

1. Structure: $u \ t \ h \ t \ u$ (since it is a palindrome).
2. I am odd $\rightarrow u$ must be $1, 3, 5, 7, 9$.
3. $t = 2 \times u$. Since $t$ must be a single digit ($<10$), $u$ must be small.
   • If $u=1$, $t=2$.
   • If $u=3$, $t=6$.
   • If $u=5$, $t=10$ (Too big).
   • So $u$ is $1$ or $3$.
4. $h = 2 \times t$.
   • Case $u=1 \rightarrow t=2 \rightarrow h=4$. Number: 12421.
   • Case $u=3 \rightarrow t=6 \rightarrow h=12$ (Too big).
5. Answer: $12421$.