# Problem of the MonthHint for Problem 4: January 2024

1. Any list that compresses $$[1:9]$$ must contain $$1$$. Think about the largest possible number of integers in $$f(A)$$ when $$A$$ is a list of length $$k$$.

2. First, try to find a list that compresses $$[1:63]$$ that is as short as possible. It might help to read about the binary representation of positive integers.

3. Work out a few more examples like the one in (b). It is possible to compress $$[1:n]$$ using a list $$A$$ that consists entirely or almost entirely of powers of $$2$$.

4. For $$k\geq 3$$ and $$m\geq 2$$, if $$A$$ compresses $$[m:m+k-1]$$, then $$A$$ must contain $$m$$ and $$m+1$$.

5. The answer is $$39$$. Do not worry about trying to compress $$[5:k]$$ using as short a list as possible. As well, inductive thinking could be useful here. Suppose you can show that there is some $$k$$ with the property that $$[5:k]$$, $$[5:k+1]$$, $$[5:k+2]$$, $$[5:k+3]$$, and $$[5:k+4]$$ are all compressible. Can you deduce that $$[5:n]$$ is compressible for all $$n\geq k$$?