The rest of the time, though, it is the softer-faced man - Leonardo DiCaprio rather than Sylvester Stallone - who is likely to get a woman's attention, particularly as a choice for a life partner.
The new research, by Scottish and Japanese scientists, also leads to the controversial idea that - in evolutionary terms - women are naturally programmed to try to betray a long-term partner by having sex with masculine- looking men when their body knows they are most likely to conceive.
However, at all the other stages of their cycle, women preferred men with rounder, softer faces, described as "more feminine" by the researchers. Thus any DiCaprio lookalike may well get the girl; but some of his children might grow up looking more like Stallone than him.
"Having spoken to a few women about these results, I can say that they at least aren't that surprised," said Dr Bernard Tiddeman, a researcher at the psychology department of the University of St Andrews in Fife, where some of the work was carried out. "It's more of a surprise for men. Women know that their interest in men varies during their cycle," he said. The women were usually unaware of how close to their ovulation they were, he noted.
One confounding factor did emerge: women on the Pill showed no variation in choice, apparently because their hormone levels do not vary during the month.
Previous work by the St Andrews team had found that women choosing a long-term partner preferred a "feminine" face - which raised the question of how masculine-looking men, who would presumably have "stronger" genes, have survived. The answer seems to be that they start looking attractive at just the right time each month to pass on those genes.
The new findings, reported in the science journal Nature, could be explained by evolutionary pressures, the team suggested, since it would allow ostensibly "married" women the benefits of polyandry - a male harem. That is, if anyone can find more than one Sylvester Stallone.