WHY THE FIDO PUZZLE WORKS

The key lies in a characteristic of the number 9.  If you transpose the digits in a number and subtract the smaller number from the larger, the difference is always evenly divisible by 9.

Example 1
4321 rearranged to 1423
4321 minus 1423 = 2898
I choose to circle the 9, leaving the numbers 2, 8 & 8 to be typed in.
The only number that may be added to the sequence 2+8+8 and be divisible evenly by 9 is 9 (the digit I circled).  [2+8+8+9=27]

Example 2
6734 rearranged to 7346
7346 minus 6734 = 612
I choose to circle the 6, leaving the numbers 1 & 2 to be typed in.
The only number that may be added to the sequence 1+2 and be divisible evenly by 9 is 6 (the digit I circled).  [1+2+6=9]

Example 3
578 rearranged to 857
857 minus 578 = 279
I choose to circle the 2, leaving the numbers 7 & 9 to be typed in.
The only number that may be added to the sequence 7+9 and be divisible evenly by 9 is 2 (the digit I circled).  [7+9+2=18]

The designer has programmed the puzzle to do the math so your computer has no problem supplying the 'missing' number.