The Phrygian Cap

i just want to comment on that multiplication. It's not a consequence of a deep result, it's actually very simple when you do written multiplication. Additionally, you also have :
11*11 = 121
111*111 = 12321
1111*1111 = 1234321

and so on. If we were using a higher base than 10, for exemple hexadecimal, we would have :
1111111111111111*1111111111111111 =
123456789ABCDEFEDCBA987654321

It also equals 2345678987654321 (doing the math in hexadecimal)
and 111111110000000001 (in binary)
and nothing particularly interesting in octal...

In [1]: 111111111 * 111111111
Out[1]: 12345678987654321L

Finally! A reason to use 64bit apart from the ram issue :P

try:

12345679 * 8 = 98765432

cool :)

the hexadecimal is awesome.

and thanks duda nogueira for the reply! :)

One other intersting number is n = 142857. When you multiply it by 2, 3, 4, 5, and 6, you get a permutation of its digits. A whole research on this number has been done by Lewis Carol...

It comes from the fact that this is the periodic decomposition of 1/7, so n*3 = 428571 is the periodic decomposition of 3/7 = 10/7 - 1, n*2 = 285714 is the periodic decomposition of 2/7 = 100/7 - 14, and so on..

The good question is "are there other such numbers" ? That is numbers such that multiplying them by the first integers you get the same digits in a different order ? If so, what is the greatest such multiplier for which it still works (necessarily < 10, since starting from 10 you're ensured to get one more digit)

Note that 142857142857 n = 1..6 is too easy an answer for the preceding riddle.