Insert Plus Signs and Add

Question

If you take any integer (in base 10) and insert plus signs, "+", in between its digits (as few or as many as you like), and carry out the indicated sum, you will end up with a smaller number (unless you chose to insert none). If this resulting number is not a single digit, and you again perform on it the above operation, you are now likely to obtain a single digit.

It is known that at most three such operations are necessary to convert any number into a single digit.

What is the smallest number that requires precisely those three steps?

Answer 1

The answer is

289

Reasoning

I went backwards. Assuming the 4th (last) number is 1, 3rd is 10, 2nd is 19.

Then

The three digit numbers like 667, 577, 478, 388, 289 all work. 199 does not because 1+99 = 100 which goes to 1, not 19.

In particular

289 ==> 2+8+9 =19 ==> 10 ==> 1 or
289 ==> 2+89 = 91 ==> 10 ==> 1 or
289 ==> 28+9 = 37 ==> 10 ==> 1

The 2nd number

must be 19 when the digits of the 1st number is added individually.
why? the most it could be otherwise with 3 digits is 27. Between 1 and 27, the only number that adds to a 2 digit number (10) is 19. We know that the minimum is 3 digits or less because 289 works.

And since

289 is the smallest number which adds to 19 AND has all of its iterations work, this must be the minimum number that works.