Nifty Number Nine & Its Multiples Revisited
This trick is a sequel to Math Trick #14! The number 9 & its multiples have the power to swap the digits of 2-digit numbers! All it takes is some addition or subtraction.
If the bigger number is to the left of 9x, then you should subtract; if the smaller number is to the left of 9x, then you should add.
Furthermore, if a number has more than 2 digits, then all you have to do is multiply the multiple of 9 by a power of 10, depending on which pair of digits you want to swap!
354 + 180 = 534
534 - 180 = 354
However, you can only swap digits that are next to each other!
354 + 180 = 534
534 + 9 = 543
543 - 9 = 534
534 - 180 = 354
354 - 9 = 345
345 + 9 = 354
Just like in the prequel of this math trick, you simply have to multiply the absolute value of the difference of the 2 digits you want to swap by 9; afterwards, multiply the multiple of 9 by a power of 10! (No, I'm not using the exclamation point as the factorial function.)
9(|A - B|)10^X = M
A = 1st digit, B = 2nd digit, M = the multiple of 9 that swaps the digits, X = the column number
|3 - 5| = 2
|5 - 3| = 2
(With the bigger number to the left, the absolute value line symbols are unnecessary, but I put them there anyway to make a point.)
9 × 2 = 18
...And there's the multiple of 9 that had to be used to swap the 3 & the 5, except that it had to be multiplied by 10. So:
18 × 10 = 180
If there were more digits before them, then you would have to multiply by a bigger power of 10. For example, if there are 3 digits before them, then X = 3, which will multiply the multiple of 9 by 1,000.
18 × 103 = 18 × 1,000 = 18,000
35,216 + 18,000 = 53,216
53,216 - 18,000 = 35,216
(|5 - 3| = |3 - 5| = 2; 9 × 2 = 18; 18 × 1,000 = 18,000)
I'll show you 3 more examples with other multiples of 9 multipled by other powers of 10:
16,777,216 + 45,000,000 = 61,777,216
61,777,216 - 45,000,000 = 16,777,216
(|1 - 6| = |6 - 1| = 5; 9 × 5 = 45; 45 × 10^6 = 45,000,000)
29,754,833,146 + 63,000,000,000 = 92,754,833,146
92,754,833,146 - 63,000,000,000 = 29,754,833,146
(|2 - 9| = |9 - 2| = 7; 9 × 7 = 63; 63 × 10^9 = 63,000,000,000)
319,456 + 72,000 = 391,456
391,456 - 72,000 = 319,456
(|1 - 9| = |9 - 1| = 8; 9 × 8 = 72; 72 × 10^3 = 72,000)
Finally, I'll show you 1 counterexample! If you try to swap digits that aren't next to each other, something like this will happen:
217 + 54 = 271
271 - 54 = 217
(|1 - 7| = |7 - 1| = 6; 9 × 6 = 54; 54 × 10^0 = 54)
217 + 45 = 262
217 + 450 = 667
(|2 - 7| = |7 - 2| = 5; 9 × 5 = 45; 45 × 10^1 = 450; however, the 2 & the 7 didn't swap!)
So in conclusion, only calculate the absolute value of the difference of digits that are next to each other. You can only move 1 pair at a time thru addition or subtraction. Also, Brain is here to remind you of something...
...So if there are no digits before the pair of digits you want to swap, then X = 0 in the math trick formula & the multiple of 9 gets multiplied by 1. Since 1 is the number of multiplicative identity, it gets simplified to the original formula from the prequel of this math trick! And that is that!
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© Derek Cumberbatch