It is a little hard to understand what you mean by "solve the problem of women's suffrage." Are you asking how they managed to persuade people to support giving the vote to women? If so, two of the most important ways were:
- Picketing and other protests by women. This raised the profile of the issue.
- Emphasizing that women would be likely to vote for progressive candidates. Progressives, for example, argued that middle class women would be more likely to vote for progressive candidates than immigrant men would be. Thus, giving women the vote would benefit progressivism.