SOLUTION

We will show that every positive integer from 8 onwards can be constructed from a smaller positive integer.

If the number is of the form

5a + 1, construct it by adding 2 to the end of a, to give 10a + 2, then halving;
5a + 3, construct it by adding 2 to the end of 2a + 1, to give 20a +12, then halving twice;
5a + 4, construct it by adding 2 to the end of 4a + 3, to give 40a + 32, then halving three times;
10a + 2, construct it by adding 2 to the end of a.
10a + 7, construct it by adding 7 to the end of a.
10a, construct it by adding 0 to the end of a;
10a + 5, construct it by adding 0 to the end of 2a + 1, to give 20a + 10, then halving.

Then, every integer from 2 to 7 can be constructed as follows: (note that -> stands for an arrow!)

2: 1 -> 12 ->6 ->3 -> 32 -> 16 -> 8 -> 4 -> 2

3: 1 -> 12 -> 6 -> 3

4: 1 -> 12 -> 6 -> 3 -> 32 -> 16 -> 8 -> 4

5: 1 -> 10 -> 5

6: 1 -> 12 -> 6

7: 1 -> 12 -> 6 -> 3 -> 32 -> 16 -> 8 -> 4 -> 2 -> 22 -> 11 -> 112 -> 56 -> 28 -> 14 -> 7

Thus every positive integer can be constructed.

Challenge 234: Building Integers