Challenge Level

You can always win, if you make sure you say the numbers 3, 8, 13 and 18.

If your opponent says 19, 20, 21 or 22, then you can say 4, 3, 2 or 1 and you win. And if you say 18, then your opponent has to add 1, 2, 3 or 4, so they have to say 19, 20, 21 or 22.

So 18 is a "key" number - you can always win if you say 18.

For the same reason, 13 is a "key" number: if you say 13, your opponent has to say 14, 15, 16 or 17. So you can choose a number between 1 and 4 which will make 18. Once you have said 18, you can win!

Counting backwards, 8 and 3 are also key numbers.

When the range is 1, 2, 3, 4, they "key" numbers are every 5 numbers, because:

1 + 4 = 2 + 3 = 3 + 2 = 4 + 1 = 5

So whichever number your opponent added to the last "key" number, you can add another number so that you've added on 5 in total - which is good, because the next "key" number is 5 more than the last "key" number.

So if the range 1, 2, 3, 4, 5, then the "key" numbers will be every 6 numbers. And if the range is 1, 2, 3, ..., $a,$ then the "key" numbers will be every ($a$+ 1) numbers.