<s>[INST] The License Plate Problem The 7-digit license plate of a car happens to be a palindrome. Its first digit on the far left is twice the digit next to it. The 3rd digit from the right is odd and 5 more than 2nd digit from the left. The middle digit is the average of the first three digits on the far left. What is this license plate number?
HINT: A palindrome is a number that reads the same forward and backward. For example, 123321 and 6776 are palindromes.
 [/INST] The license plate is of the form $abcdcba$, where each $a,b,c,d$ is a digit.
From the information in the question, we know
$$
a=2b \quad\text{and}\quad c=5+b \quad\text{and}\quad c\ \text{is odd}
\quad\text{and}\quad d=(a+b+c)/3.
$$
Since $c$ is odd and $c=5+b$, we know $b$ is even. Thus $b$ is one of $0,2,4,6,8$. It can't be $6$ or $8$, because then $a$ would then not be a digit. This leaves that $b$ is $0$, $2$, or $4$.
If $b=0$, then $a=0$ and $c=5$ so that $d=(0+0+5)/3$ is not a digit.
If $b=2$, then $a=4$ and $c=7$ so that $d=(4+2+7)/3$ is not a digit.
If $b=4$, then $a=8$ and $c=9$ so that $d=(8+4+9)/3=7$.
Therefore the number is $8497948$.
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