I just heard a delightful claim from a work colleague; so far I have no idea how to check or prove it. The "additive persistence" of a number is how many times you need to apply the operation "n -> sum of the digits when you write n in base 10" before you get to a single digit. For example 199 has additive persistence 3, since 199 -> 1+9+9 = 19 -> 1+0 = 10 -> 1+0 = 1. Sometimes you can get to a single digit more quickly if you're allowed to only insert + signs between *some* of the digits in the base-10 writing, rather than all of them. For example, if you're allowed to choose 199 -> 1+99 = 100, then you can get to a single digit in only 2 steps instead of 3. Here's the remarkable claim: By appropriate choice of where to insert + signs, you can always reach a single digit in at most 4 steps! It is extremely not obvious to me that any finite number of steps suffices, much less that it's 4. Right now I believe it only because the person I heard the problem from has a good record of relaying puzzles with correct solutions :-) --Michael -- Forewarned is worth an octopus in the bush.