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The Gemara in Pesachim 72a says, "One, who reviews a subject 40 times, is guaranteed it will be rooted in his memory as if it were placed in his pocket."

However, the Mishnah Berurah 114:41 says, "Reviewing one’s studies 101 times ensures that they will not be forgotten."

Does reviewing one's studies 40 times or 101 times, insure that it stays in one's memory?

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    Presumably, the point of the gemara in Chagiga is that, even if someone has already reviewed his studies 100 times, he will still benefit substantially from further study.
    – Fred
    May 1, 2015 at 23:12
  • hebrewbooks.org/…
    – Yishai
    May 3, 2015 at 18:21
  • The Mishna B'rura is citing the Chasam Sofer (OC §20) in the context of how many times a person has to say the amida after we stop saying mashiv haru'ach before he can be assumed to be accustomed to the new formulation. The CS understands the gemara in Chagiga as you indicate, and comes up with a novel calculation whereby there are 101 t'filos over the 30 day period following the switch. Thereby, the CS synthesizes his understanding of the gemara in Chagiga with the Y'rushalmi in Ta'anis (2b, "א"ר יוחנן כל ל' יום חזקה מה שהוא למוד הזכיר").
    – Fred
    May 3, 2015 at 22:42
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    Note, though, that although the CS follows this understanding of the gemara in Chagiga ("וכן הוא מפורסם ומקובל שהשונה מאה פעמים ואחד לא במהרה הוא שוכח"), other opinions follow the contrasting view mentioned in my original comment. For example, when R' Chaim Volozhin was 19, he told the GR"A, "I reviewed seder Mo'ed 19 times, and it's still not completely clear to me!" The GR"A replied, "You expect it to be clear after only 19 reviews?" R' Chaim asked, "How many, then? 101?" To which the GR"A answered: "There is no fixed quantity! Rather, one must spend his whole life reviewing."
    – Fred
    May 3, 2015 at 23:02

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This might answer what the Gemara means in Chagigah 9b. The Gemara there says exactly what the Mishnah Berurah you cited says:

one who reviews his studies one hundred times is not comparable to one who reviews his studies one hundred and one times.

The Baal HaTurim explains the significance of the number 101 in his commentary on Shemos 32:34:

הנה מלאכי ילך לפניך מלאכי עולה בגי' מאה ואחד. ילך עולה כמנין הלכה רמז למה שאמרו אינו דומה מי ששונה פרקו ק' פעמים לשונה פרקו ק''א שעי''כ ילך ההלכה לפניו. גם הנה. עולה כמנין הלכה. גם ישקני מנשיקות ''פיהו'' עולה מאה ואחד. הנה מלאכי אותיות מיכאל שהוא מלאכי המיוחד לי:

The Baal HaTurim explains that the word מַלְאָכִ֖י - angel, has the numerical value of 101.

In the sefer Koveitz Chukai Chaim, a anthology of pratical halachic topics by Moreinu Horav Chaim Aharon Bleier shlit'a, the rav of the Cheder Horo'oh in Yerushalayim, the following is added to this topic:

Indeed, Chazal in Maseches Chagigoh indicate that there is something special about the number 101 when they tell us that "A person who reviews what whe learned 100 times does not compare to a person who reviews 101 times". The Baal Haturim, commenting on the posuk in Shemos 32:34 writes that מַלְאָכִ֖י in gematrio equals 101 and ילך equals הלכה, hinting to this Chazal - if one reviews 101 times, the halochoh will go before him, so to speak.

In fact, these two ideas are intertwined. The Chasam Sofer quotes the Rama MiPano, who says that forgetting comes from the samech-mem, which is 100 in gematrio. Reviewing 101 times defeats the samech-mem, enabling a person to remember what he learned.

Source:

קובץ גליונות ליקוטי ופסקי הלכות חוקי חיים - Koveitz Chukai Chaim - see "The Significance of 101 Issues", p. 14

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