A bookkeeper sets up two number-columns: DEBITS (expenditures) and CREDITS (what expenditures "bought"). The TOTALS of these two columns should AGREE.

They may disagree for several reasons. But one of the most common is the entry (usually in CREDIT) of THE PERMUTATION of DIGITS, rather than their intended order. Say, the item is "$267.89", but the incorrect value "$627.85" was entered, and now "messes up" the CREDIT TOTAL.

The DECIMAL PATTERN IMPOSED ON NUMBERS results in THE TRICK OF CASTING OUT NINES, which is equivalent to ANY PERMUTATION OF A "DECIMAL" DIFFERS FROM ITS ORIGINAL BY A MULTIPLE OF NINE.

Take 254. 2 + 5 + 4 = 11, and 1 + 1 = 2. Clearly, all the other permutations -- 245, 425, 452, 542, 524 -- will yield the same sums. Note what this does to a "total": 254 + 384 = 638. Now permute 254 to 245 and add it in: 245 + 384 = 629. Not the same total. But look at the difference between these two totals: 638 - 629 = 9 -- a MULTIPLE OF NINE. Take another permutation: 254 to 524; then total: 524 + 384 = 908, and 908 - 629 = 279 -- a multiple of 9, since 2 + 7 + 9 = 18, and 1 + 8 = 9. You will find this for all of the other permutations of 254. Similarly, with any number written in DECIMAL NUMERATION.

So what? The bookkeeper learned this is school. If the two columns disagree, before looking for an error, she/he tests the type of blunder by taking the difference between the two columns. If their DIFFERENCE IS A MULTIPLE OF NINE THEN MAYBE(!!!) THE BLUNDER WAS DUE TO PERMUTATION OF DIGITS, and this can be looked for. Thus, $627.85 - $267.85 = $36.00, clearly a MULTIPLE OF 9.

However, IF THE DIFFERENCE IS NOT A MULTIPLE OF NINE, THEN THE BLUNDER CANNOT BE DUE TO PERMUTATION AND ANOTHER TYPE OF BLUNDER MUST BE LOOKED FOR.

This is typical of problem-solving in general. THE NEGATION IS CERTAIN, but the ASSERTION IS A SOME-TIME THING.

When I worked in an insurance brokerage, I had to teach their bookkeepers this. Here's an example. Take that number above, $267.89; write it as 26789, and add its digits: 2 + 6 + 7 + 8 + 5 = 28, 2 + 8 = 10, 1 + 0 = 1. (Or, casting out nines, 2 6 7 8 5 → 2 6 7 8 5 → 6 8 5 → 6 + 8 + 5 = 1 9 → 1.) The DIGITAL ROOT OF THIS NUMBER is 1 (yields 1 REMAINDER WHEN DIVIDED BY 9). Now, consider another number with the same DIGITAL ROOT, say, $100.00. Now, $267.85 - $100.00 = $167.85, and 1 + 6 + 7 + 8 + 5 = 27 → 2 + 7 = 9, a MULTIPLE OF 9!

What does this mean? PERMUTING CONSERVES THE DIGITS ROOT OF A NUMBER, SINCE ADDITION IS COMMUTATIVE AND THE SUM REMAINS THE SAME. THE DIFFERENCE BETWEEN ANY PERMUTATIONS IS A MULTIPLE OF 9. AND THE DIFFERENCE BETWEEN ANY TWO NUMBERS WITH THE SAME DIGITAL ROOT IS A MULTIPLE OF 9.

By showing this, I convinced bookkeepers that the "POSITIVE" CHECK MIGHT TELL THEM SOMETHING, whereas the 'NEGATIVE" ALWAYS RULED OUT ANY PERMUTATION.