How can you impress computer programmers with your programming skills?

Well, if you work as a Front-End Developer then you should improve your designing skills to impress others OR if you work as Back-End Developer then you should be expert in the field of Mathematics.

• Write clean, robust code
• Write code in a way that is self-commenting, and provide the minimal comments where needed for clarity
• Don't over-engineer solutions just to appear clever to other programmers, that will only impress other newbies
• Don't use hacks
• When you must use a hack, provide clear code comments explaining why the hack was needed
• Know all the features of your programming language of choice, the strengths, weaknesses, capabilities, and limitations
• Follow the programming conventions of the group; there is nothing so irritating as having to deal with two or more coding styles or structures within the same application because people could not agree - work with your team or leadership to make improvements to your standards
• Be able to document and diagram programming concepts, explain and communicate clearly
• Develop strong debugging skills and learn how to use your tools expertly
• Work quickly, but carefully

Show them your skills, as simple as that.