MathDB

In the beginning there are

100

integers in a row on the blackboard. Kain and Abel then play the following game: A move consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers

1,2,\dots,100

and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add

1

to each of the chosen numbers or instead subtract

1

from of the numbers. After that the next move begins, and so on.
If there are at least

98

numbers on the blackboard that are divisible by

4

after a move, then Kain has won.
Prove that Kain can force a win in a finite number of moves.

Problem Statement