$start_date->strftime("%m%d%y");
print $start_date;
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strftime does not modify the invocant ($start_date) in any way. Rather it returns a string that is the result of formatting the invocant.

Since you throw away the result of calling ->strftime, and then print $start_date again in the next line, I guess you haven't fully understood this principle.

So the answer is to work with the return value of the ->strftime call, not with the original object.