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Mathmagic!

Consider this little number trick involving some mental prowess (or hidden calculator) that you can use to astound students and adults of all ages.

Ask each student (or person) to write down three secret digits, each digit being different and non-zero. Now, they are to form a column sum involving each of the six possible permutations of the three digits. This sum is then added to any one of the six permutations (i.e. seven 3-digit numbers produce the sum, with one of the 3-digit numbers being used twice). Finally, each student is to announce their sum and you (using your great mathematical magic) "instantly" tell them not only what their three secret digits were, but also which permutation was used twice!

For example, suppose Stu Dent selected 1, 2, 7 as his three secret digits. Then the magical sum of the permutations is:

127+172+217+271+712+721+127 = 2347
where the permutation 127 was used twice.

When Stu Dent announces the sum of 2347, you pause to swallow some mental dust, then reply that Stu must have selected the digits 1, 2, 7 and that the number 127 was used twice in forming the sum. A-M-A-Z-I-N-G!

Now, you ask, how is this great trick done, given that you do not have the free weekend needed to memorize all possible sums for any group of 3-digit numbers. (Note: How many possible sums are there, being sure to include the possibility of a number being repeated? And, are these sums all unique?)

In case you are getting impatient, the actual method for accomplishing this trick will be revealed on the Resource page.