Missed a Palindrome

Probability Level 1

A palindrome is a number (with no leading zeros) that remains the same when written backwards. For example, 303 and 242 are both palindromes, while 128 and 030 are not.

After hearing an interesting fact that the number of 3-digit palindromes is 90, Alice says, "For each of the 3-digit palindromes, we can add a single digit from 1 to 9 to both ends, so the number of 5-digit palindromes is $90 \times 9 = 810.$ "

Then Bob disagrees and says, "That is incorrect, you missed some of the palindromes."

Which of the following can be a palindrome that Bob says Alice missed?

75457 36363 20802 91019

2 solutions

Áron Bán-Szabó
Jul 4, 2017

When we are talking about 3-digit palindromes, then the first digit can't be 0, but when we are talking about 5-digit palindromes, the second digit from the left, which was the first digit of the 3-digit palindrom, can be 0. So the answer is $2\color{#D61F06}0\color{#333333}802$

William Huang
Jul 23, 2017

If you look at the available answers, there are:

- $75457$

- $36363$

- $20802$

- $91019$

Then, if you remove the digit from the start and end of each number, they become:

- $545$

- $636$

- $080$

- $101$

A palindrome cannot start or end with ' $0$ ' but $080$ (20802) does, so we can conclude $\boxed{20802}$ as our final answer.

Missed a Palindrome

2 solutions

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