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Does our Jewish calendar contain a Maleh month with its first day starting on Shabbos (contain parts of six calendar weeks), and if yes, how often does it occur?

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    "how often does it occur?" How precise a number do you need? On average a month can start on one of four days and it's a roughly even distribution – Double AA May 28 '17 at 23:39
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Yes it does. Nisan, Av, Tishrei, Marcheshvan, Shevat or Adar I can be that way. (OC 428:2)

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