So, I was watching this video by MindYourDecisions about finding the smallest number that doubles when you move the last digit to become the first digit. Actually, I just saw the title of the video, and I found it interesting, so I began to work on a solution. But, in fact, I had misinterpreted the question posed by the video.
I was looking for a number that doubles when you switch the first and last digits of a number. I started off my search by writing equations like:
$$ 2(10a + b) = 10b + a $$
This simplifies to:
$$ 19a = 8b$$
Since $a$ and $b$ should be integers between 0 and 9 inclusive, I figured out that there are no solutions amongst two-digit numbers.
Next, I applied this technique to three and four digit numbers, getting this equation for three digit numbers (using the convention that $a$ is the first digit, $b$ is the second, and so on):
$$ 199a + 10b = 98c $$ And for four-digit numbers, $$ 1999a + 100b + 10c = 998d $$
Next, I made tables varying the value of $a$ and the last digit between 0 and 9, looking for numbers that were close enough so that the smaller terms in the middle could make up the difference.
I couldn't find any solutions; however, I was able to find near misses such as 37, 397, 3997, or generally, a 3 followed by a string of 9s followed by a 7, which all differ by just one.
I am wondering if there actually are any solutions to this problem, and if not, how you would go about proving that.