Define the flipflop function, applied to a positive integer, as
the result of having the 10^2i and 10^(2i+1) digits switch places.
Moreover, if the integer has an odd number of digits, append
a leading zero to the left side of the number so that
it can flipflop with the first nonzero digit.
For example, flipflop(9876) = 8967 and flipflop(1234567) is 10325476.
warm-up:
What is the smallest positive integer such that flipflop(m) = m*4?
octuple the number:
What is the smallest positive integer such that flipflop(n) = n*8?
In the case of multiplication by 8, I have found only one solution: are there any others?
(In reply to
re(2): computer solution by Charlie)
Looking over the solution sets, it seems that each of 1782 and 02519748 is a generator for a family of solutions for their respective problems.
A fairly simple observation is if number n is a solution then appending 00 to n is a trivially obtained larger solution.
A more significant observation is if numbers m and n are both solutions AND calculating m+n has no carries then m+n is also a solution.
1782 is a primitive solution for the first problem. Then the first observation 178200 is a solution and by the second observation 1782+178200 = 179982 is also a solution.
We can repeat this to generate 17998200 and 17999982 as further solutions.
Alternatively appending four 0s to 1782 gives solution 17820000, then 1782+17820000 = 17821782 is another solution.
So in this problem an extended family can be formed by appending building blocks of 17[99]82[00], where 17 and 82 are mandatory and [99] and [00] are optional or can occur multiple times. So something like 17820017999999820000179982 is also a solution.
A similar process can be applied to 02519748 for the second problem. 025197480000 is a solution and so is 02519748+025197480000 = 025199999748, etc.
0251999748 seems to be a second primitive but looks like is should be derived from 02519748. Anyways, taking both as primitive solutions does make creating an extended family solution easier.
So in this problem the building blocks are generated from 0251[99]9748[00] which makes something like 025199974800000002519999974800 a solution.
I suppose we still have an open question of whether or not there are any more primitive solutions.
re(3): computer solution
