A three-digit number reads the same forwards and backwards. Its three digits add up to 12, and its middle digit is twice its first digit. Which number is it?

Write the number as a $b$ a. Reading the same both ways is already built in, the digits add to $2a+b=12$, and $b=2a$. Putting the second into the first gives $2a+2a=12$, so $a=3$ and $b=6$, and the number is 363 — check: 3 + 6 + 3 = 12 and 6 is twice 3. The traps: 242 and 484 both have a middle digit that is twice the first, so they pass one test, but their digits add to 8 and to 16, not 12; 636 has the two conditions the wrong way round, with the first digit twice the middle one, and its digits add to 15.