Some time in high school I was playing with my calculator in math class. I made a discovery: If you multiply the number 123456789 with any number that isn't a multiple of 3, you get the same digits rearranged:

123456789 * 2 = 246913578
* 4 = 493827156
* 5 = 617283945
* 7 = 864197523
* 8 = 987654312

Now, from 10 upwards, a zero joins the crew:

* 10 = 1234567890
* 11 = 1358024679
* 13 = 1604938257
* 14 = 1728395046
* 16 = 1975308624
* 17 = 2098765413

At 19, there is a small inconcistency:

* 19 = 2345678991

And then it continues onward as before:

* 20 = 2469135780
* 22 = 2716049358
* 23 = 2839506147
* 25 = 3086419725
* 26 = 3209876514

At 28 and 29, it's inconcistent again:

* 28 = 3456790092
* 29 = 3580246881

But from 31, we're back on track:

* 31 = 3827160459
* 32 = 3950617248
...etc.

If you multiply it with a multiple of three, you also get some sort of pattern. In every number there is a three-digit repeating sequence, and in the last one, a number sneaks in between the last two, being the last number minus a certain number that grows steadily. (It seems that when this number is a multiple of 3, there are four numbers in a row with that number substracted, instead of three.)

123456789 * 3 = 370370367 (the second-to-last # is last #-1)
* 6 = 740740734 (-1)
* 9 = 1111111101 (-1)
* 12 = 1481481468 (-2)
* 15 = 1851851835 (-2)
* 18 = 2222222202 (-2)
* 21 = 2592592569 (-3)
* 24 = 2962962936 (-3)
* 27 = 3333333303 (-3)
* 30 = 3703703670 (-3)
* 33 = 4074074037 (-4)
* 36 = 4444444404 (-4)
* 39 = 4814814771 (-4)
* 42 = 5185185138 (-5)
* 45 = 5555555505 (-5)
* 48 = 5925925872 (-5)
* 51 = 6296296239 (-6)
* 54 = 6666666606 (-6)
* 57 = 7037036973 (-6)
* 60 = 7407407340 (-6)
* 63 = 7777777707 (-7)
* 66 = 8148148074 (-7)
* 69 = 8518518441 (-7)
* 72 = 8888888808 (-8)
* 75 = 9259259175 (-8)
* 78 = 9629629542 (-8)
* 81 = 9999999909 (-9)

I think this kind of thing is kind of beautiful. There's a system in it, although I'm not able to find out what the system is, or, more important, *why* the system is. I'd like to know, though, so if anyone has an explanation, I'd be happy to learn. :-)

Yesterday, I discovered something else, but related, while sitting on a train playing "Snake" on my Nokia 3210 cell phone on highest speed, so you get 9 points for every food piece thingy the snake eats. I got a high score, 567 points. I wondered how many thingies my snake had eaten, so I divided it by 9 and found it to be 63. Somehow, I wound up discovering that if you juggle the digits around, it'll always be a multiple of 9.

567 / 9 = 63
576 / 9 = 64
657 / 9 = 73
675 / 9 = 74
756 / 9 = 84
765 / 9 = 85

Then I thought that it's just a little bit odd. It's not odd that any of those numbers are multiples of three (since the digit sum (18) also is -- this we learn in early school years), but every one of them are also multiples of 3*3.

Then I thought: What if I add a couple of digits?

The digit sum of 234567 is 27, so this number, too, and all other arrangements of those digits, will be divisable by 3.

And the same probably also goes for 123456789.

But I think every arrangement of these numbers (without having tested all of them), will be a multiple of 9. So did I just find out something we didn't learn in school? That if the digit sum of a number is a multiple of 9, the number will also be?

Trav"likes to juggle numbers but lacks the knowledge to deduce things like this on his own"holt.