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Problem 4:
Consider the numbers $1, 2, 3, \cdots 160$. What is the maximum number of numbers you can choose from this list so that no two numbers differ by $4$? Show the logic behind your answer.

Let me, divide the set into 40 equivalent subset such that every subset consists of 4 consecutive numbers.Now initialize from the 1st subset.The subset is {1,2,3,4}.the next one is {5,6,7,8} every number of this subset differs 4 with previous one.Like that every element of a subset must differ with the numbers of previous subset.If we should follow the condition we have to remove 1st,3rd,5th,7th............subsets.Or 2nd,4th,6th.........subsets.If we proceed with the 2nd procedure,we have to remove 20 subsets from list,So the number of existing integers is 80.Is my logic acceptable??

Good solution..