Journals similar to The Mathematical Intelligencer?
TL;DR: Go with the Monthly if you are very confident in the quality of your manuscript; go with Math Horizons if you do not mind taking a more informal tone and possible trimming down your exposition somewhat.
I feel compelled to answer this question mostly because of my own experience with the periodicals/magazines/journals in the MAA (which the American Mathematical Monthly is a part of). Hopefully my answer will prove useful to you and others.
You wrote the following in a comment:
Thanks so much. Actually currently I have one article there [Mathematical Gazette], which unfortunately has been processed for almost one year.
I understand this frustration well--I submitted a manuscript to the College Mathematics Journal (another MAA periodical) in November of 2013. It is still being processed. [~ 1.5 years later] What the hell kind of unprofessionalism is that? My article was not even lengthy (9 pages), was perfectly formatted (used their $\rm\LaTeX$ article tample), etc. I finally decided to write the editor. I got this response months later (no joke). Just for some context, the MAA is a very professional publication with the following core periodicals:
- The American Mathematical Monthly
- Mathematics Magazine
- The College Mathematics Journal
- Math Horizons
The above are listed in terms of what I would consider are the prestige and difficulty of getting your manuscript accepted. The targeted audience is different for each one though. It is very difficult to get a paper accepted into the Monthly sometimes simply because of the sheer volume of submissions the MAA gets. At least their issues are current (if you click the link above, you will see their publications are up to date). Both the Mathematics Magazine and College Mathematics Journal seem to have a huge backlog of unprocessed manuscripts. If you click the links for both of those periodicals, you will see that no content for 2015 has even been listed. That's surprisingly unprofessional, in my opinion, for an organization as professional as the MAA.
All hope is not lost though! While Math Horizons is certainly not on the level of the three other periodicals listed, it would still be a very good publication, but you would have to keep your readership in mind. Articles for Math Horizons are typically somewhat informal but very high quality--it's a fairly delicate balance to strike. The great thing is that the editor (the current editor is David Richeson) will get back to you in a reasonably prompt fashion and, assuming your article is accepted, you can expect it to go to print fairly quickly. I have had two articles published in Math Horizons, and I would say my experience overall has been quite positive. However, I would never recommend Mathematics Magazine or The College Mathematics Journal. It's like throwing your manuscript into a black hole, as it seems you have felt with the Mathematical Gazette. If your manuscript is very high quality and you do not want to take an informal tone, then the Monthly may be a good choice, but you cannot bank on your article being accepted. On the other hand, Math Horizons may not carry quite as much prestige, but it would still be a very high quality publication--acceptance is still not guaranteed though, of course, as Math Horizons is a very popular periodical for readers of all math levels, and the volume of submissions is enormous.