What happens if I accidentally take the Columbia Science Honors Program test with a number 3 pencil even though they specifically tell you to use a number 2 pencil?
Most likely, you'll be fine - as I wrote in this answer about taking the ACT with a number 3 pencil, the main reason number 2 pencils are the standard is because older machines couldn't read lighter leads like number 3. If your answer sheet was double-sided, then the machine reading your answer sheet must be a new machine, which means that as long as you made dark marks on the test you should be fine.
If you're worried that some of your answers may not have been recorded correctly, you should definitely contact the SHP and ask them what to do.