The difference between two four digit numbers that have the same digits results in a number that is a multiple of 9 every time. Since he tells you not to pick 0, that leaves only 9 digits to choose from, and only one will fit that solution.

the subtracted value is always a multiple of 9. 614 as a base number will return 27 (9*3),153 (9*17), 198 (9*22), 450 (9*50), 468 (9*52). And a basic property of mulitples nine is that all of their digits summed are a product of nine. The part I couldn't figure is that they were always able to get a subtracted number that is a multiple of nine. However a brilliant coworker figured it out.

Any number can be represented algebraically as a variable and the place, such as 1000a+100b+10c+d. Rearanging them would give you 1000c+100d+10a+b. Subtracting the two gives you 990a+99b-990c-99d. As you can see, this always gives you a resulting number with a multiple of nine 9*(110a+11b-110c-11d), which sets up the second part of the trick which plays on the property of nine.

OK, here is what I got. I can't explain it but it works. Follow the same steps as above. When you get to the step where a number has been circled; add up the remaining numbers. Then go to the first multiple of nine that is greater than that number and subtract. Voila you have your answer.

For example, if the remaining digits are 4-6-0 then adding them together gives you 4+6+0=10. The first multiple of nine that is greater than 10 is 18. Thus 18-10=8. That's your answer 8! Works every time. Try it, enjoy it, be the hit at all the parties!!