Mindbender 5: solution

Solution backdated to avoid spoiler to Mindbender 5.

At first this looked like a case for trying an iterative solution, but it turned out for once to be simple arithmetic. In terms of strings, the puzzle is:

ABCDEF x 6 = DEFABC

This problem reduces since you can call the two blocks P and Q, so numerically:

(1000P + Q) x 6 = 1000Q + P

so expanding and collecting:

6000P + 6Q = 1000Q + P
5999P = 994Q
Q = (5999/994)P

... which simplifies on inspection to Q = (857/142)P

This means P has to be a multiple of 142. And further, given the condition that the number must lie between 100,000 and 200,000, 142 is the only possibility within range.

So the number is 142857.

142857 x 6 = 857142 QED.