# Math Riddles #10 - Car Meter Riddle

Today my car meter reads as 72927 kms. I notes that this is a palindrome. How many minimum kms I need to travel so my car meter find another palindrome?

For every choice of the three first digits of a 5-digit number, there is exactly one way to complete it so it becomes a palindrome.

The next palindrome is the one that starts with 730, namely 73037, which will be reached in $73037-72927=110$ km.

The number is 72927 and you want to get to the next palindrome forward. To find the next palindrome forward you want to separate your number into a left and right side so that the 2 sides have an equal number of digits like so, 72 9 27. It's fine if you have a single digit left in the middle like we do here. Next, if there is a middle number, just add +1 to it (If there was not a middle number, you'd add the +1 to the left side), and in this case we turn the 9 to a 10 so the 9 becomes a 0 and the 1 gets added to the left side because it needs to be carried over (72+1) 0 27. Now just adjust the right side appropriately to make it a palindrome, 73 0 37. The next palindrome forward is going to be at 73037.

After that, to find the difference, you subtract the smaller number from the larger number and get your difference, 73037 - 72927 = 110. You will need to travel another 110 kms to get to the next palindrome.

If you were in a magic time travelling car, that could travel backwards in time & distance, the next closest palindrome would be 72827 and would only be 100 kms away.

You can find the next palindrome of any palindrome in this way. for example,

Palindrome: 891737737198891737737198

Half it: 891737737198 891737737198

+1 left side & adjust right side: 891737737199 991737737198

Next Palindrome: 891737737199991737737198

The answer is 110. since the next palindrome will not come in any twenties we will try thirties. so the next obvious answer is 73037. so now we are asked to find the distance that had to be travelled to get the next palindrome. so the initial thing must be subtracted from the final thing basically. so subtract 73037-72927 which gives 110 which ultimately is the answer.