Starting with the Mishna in Ta'anis 1.4 we learn that there are occasions (lack of rain in this case) when we fast BeHaB, i.e. Monday, Thursday, and Monday.
הִגִּיעַ שִׁבְעָה עָשָׂר בְּמַרְחֶשְׁוָן וְלֹא יָרְדוּ גְשָׁמִים, הִתְחִילוּ הַיְחִידִים מִתְעַנִּין שָׁלשׁ תַּעֲנִיּוֹת. אוֹכְלִין וְשׁוֹתִין מִשֶּׁחֲשֵׁכָה, וּמֻתָּרִין בִּמְלָאכָה וּבִרְחִיצָה וּבְסִיכָה וּבִנְעִילַת הַסַּנְדָּל וּבְתַשְׁמִישׁ הַמִּטָּה:
If the seventeenth of Marheshvan came and no rain fell, individuals begin to fast three fasts. They eat and drink after it gets dark and they are permitted to do work, to bathe, to anoint themselves with oil, to wear shoes, and to have marital relations.
The reason given for Monday & Thursday being auspicious days for prayer is that Moshe went up to Har Sinai (the 3rd time) on a Thursday and came back after 40 days on a Monday.
But why Monday, Thursday, and Monday - why isn't it sufficient for the fasts to simply be Monday and Thursday?
Same question for our post-Yom Tov BeHaB - why do we need to fast twice on a Monday?
Searching for ideas and sources.