How can a man use his intelligence to be sexy?

Any time you (man or woman) use your intelligence toward an action, think about the vector and magnitude of this use. When your intelligence is oriented toward the opening and deepening of a connection between you and others, When your intelligence opens hearts and minds of others, in a way that deepens the way they feel about themselves and their lives – Then your intelligence is perceived as incredibly valuable, and this elevation in value from a sexual standpoint means you appear highly desirable as a potential partner and mate.On the other hand, When your intelligence is oriented toward the elevation of your sense of status and importance above others, When your intelligence swagger in front of others, parading itself "look at me! look at me! you should want me!" –Then your intelligence is perceived as incredibly arrogant, and this separation of what values you may offer from a sexual standpoint means you appear highly cocky as a potential partner and mate.Now, you may still attract those who have been conditioned to tune into the highly cocky.But this attraction lasts only until a bigger cock appears.

Well, if every woman were attracted to the same thing, the population would collapse, since men are all different.  If there were only one correct answer to your question, all the other men would be out of luck. That said, most women want a man who respects them.  So if your intelligence comes across as not respecting their intelligence (or taste, or opinions, or values, etc) your intelligence will not be sexy. If you use your intelligence to choose the kind of woman who honestly appreciates the kind of intelligence (and personality) that you have, and you use your intelligence to focus on her needs, the rest of your intellectual accomplishments will be easy for her to see. When women (or men) are upset with their partner, it is hard for them to see their partner's accomplishments at all, let alone be attracted by them.  If the women you are interested in feel good in your presence, they will have an easier time seeing your intellectual abilities, and enjoying them. IF your intellectual abilities result in concrete accomplishments (of whatever sort the woman you are trying to attract likes/wants/needs) then it will be sexy. BUT if your intellect is all talk, and nothing to show for it, it will never be sexy.

In my experience, when women talk about intelligence being important, it seems to be more about general competence. For example, who wants to date a brilliant chemist who is terrible with his money, has a pessimistic attitude, and in poor health? Instead both men and women want partners that are overall competent and are intellectually compatible in specific areas. What I mean by competence is things like personal financial management, common sense, ability to read body language, logic, rational thinking, manners, have some awareness of current events, having at least a basic understanding of cultures and religions so that you are more capable to respect others and carry conversations with them (some who might be her friends of different faiths). Intellectual compatibility is what is also important to many. This is why you see so many doctors marrying other doctors, attorneys marrying other attorneys, hedge fund managers marrying others in finance. There is just more to talk about and gain from the conversation when both people can relate to a specific topic that requires education and experience. From personal experience I have never heard a woman admire my knowledge about Baroque era classical music, neoclassicism, algorithms, evolutionary morality, Moorish architecture, etc. But I have been complimented for how my intelligence has landed me the job that I want, how my knowledge of cultures helps me relate to many people, how I can plan out trips, how I was able to learn how to do my own home renovation which saved tens of thousands of dollars, how I can hold conversations, etc. So to answer you in context, it is "sexy" as attractive when you show the practical application of your intelligence.

Intelligence is not about showing off. Use your intelligence to figure out what she likes and offer it to her. Use your intelligence to observe her and offer compliments that actually mean something to her. Use it to learn to do the things she wants from you, and to teach her things that she may find fun or useful in life.