In last month's issue of Starlink Extra, we included a link to the FIDO puzzle online. We asked anyone who could unravel the puzzle of how it worked to send us an email - and one of our Starlink readers did exactly that. :)
Julia Roberts of Cadro Pty. Ltd. sent us the answer. Here is her explanation:
When you swap 2 digits in a number, you change its value by a multiple of 9. For example when you change 37 into 73 you add 40 (30 into 70) and subtract 4 (change 7 to 4). Or if they are separated by a digit the difference is a multiple of 99 and so on.
Shuffling numbers can be represented as a sequence of such swaps so the resulting difference will be a multiple of 9.
When you add up the digits of a multiple of 9 you get a multiple of 9. E.g. 8672 - 2876 = 5796 whose digits add up to 27.
Therefore if you add the digits you were left with and subtract the total from the next higher multiple of 9, there is the answer. For example if you chose 5 above, the remaining digits add up to 22 which is 5 less than 27.
Thank you Julia! Mystery solved. :)