Rosh Hashanah begins on these days with the following frequency:
"At the innovation of the rabbis, the mathematical calendar has been arranged to ensure that Yom Kippur does not fall on a Friday or Sunday, and Hoshana Rabbah does not fall on Shabbat. These rules have been instituted because Shabbat restrictions also apply to Yom Kippur, so that if Yom Kippur were to fall on Friday, it would not be possible to make necessary preparations for Shabbat (such as candle lighting). Similarly, if Yom Kippur fell on a Sunday, it would not be possible to make preparations for Yom Kippur because the preceding day is Shabbat. Additionally, the laws of Shabbat override those of Hoshana Rabbah, so that if Hoshana Rabbah were to fall on Shabbat certain rituals that are a part of the Hoshana Rabbah service (such as carrying willows, which is a form of work) could not be performed.
"To prevent Yom Kippur (10 Tishrei) from falling on a Friday or Sunday, Rosh Hashanah (1 Tishrei) cannot be a Wednesday or Friday. Likewise, to prevent Hoshana Rabbah (21 Tishrei) from falling on a Saturday, Rosh Hashanah cannot be a Sunday.
"This leaves only four days on which Rosh Hashanah can fall: Monday, Tuesday, Thursday, and Saturday, which are referred as the "four gates." Each day is associated with a number (its order in the week, starting with Sunday as 1), and these numbers are associated with Hebrew letters. Therefore the keviyah (Hebrew קביעה for "a setting" or "an established thing") uses the letters ה ,ג ,ב and ז (representing 2, 3, 5, and 7, for Monday, Tuesday, Thursday, and Saturday) to denote the starting day of the year." Wikipedia 2012-06-02
"A popular mnemonic is "lo adu rosh" ("Rosh [Hashanah] is not on adu"), where adu has the numerical value 1-4-6 (corresponding to the numbering of days in the Jewish week, in which Saturday night and Sunday daytime make up the first day)." Wikipedia 2012-06-02