A new series of puzzles! I 💕 it.

And now the solutions.

Læs mere: Puzzle 1, braintwister

Highlight to see solutions.

(1) What is the next number after 95 whose digits add to a multiple of 7?

It can’t be 96-99, because I only add 1-4 to the sum here, and can’t get all the way to a new multiple.

It can’t be 100, the sum is 1, not a multiple of 7.

If we add 6 to 100, we should get there. Answer: 106.

(2) Can you find a pair of consecutive numbers whose digits both add to multiples of 7?

My first guess is x9 and y0, where x+1 = y. Then it could be 69 and 70. The digit sum of 70 is 7, but the digit sum of 69 is wrong. When I go from 69 to 70, I add 1 to the sum and subtract 9, -8 in all. Of course I can’t get from one multiple of 7 to another that way.

How about 699 and 700? When I move from one to the other, I add 1 to the sum and subtract 2*9, -17 in all. 17 isn’t a multiple of 7, this won’t work.

So, when I go from 69…9 to 70…0, I add 1 and subtract n*9 from the digit sum. What I want is: 7 | 1-n*9. A quick check reveals, n = 4 works. 69999 and 70000 is a solution. Answer: Yes.

(I use | as “divides”.)

(3) What about a pair of numbers under 100 whose digits both add to multiples of 7, but that are two apart rather than consecutive?

Inspired by what I already did, 68 and 70 works. So, the answer is yes or 68, 70.