Concerning the phrase, "three days and three nights" of Matthew 12:40.
The NICNT-Matthew Commentary has this to say (p.491):
. . . The different phrasing of the three-day period compared with the "third day" of Matt. 16:21; 17:23; 20:19; 27:64 and the "after three days" of Matt. 27:63 is due to the LXX wording, but in Semitic inclusive time-reckoning these do not denote different time periods as a pedantic Western reading would suggest. 12
Footnote 12 says:
The same phrase, "three days and three nights," occurs in 1 Sam 30:12 to denote a period which began (literally) "today three days," the day before yesterday (v. 13). Similarly in Esther a period described as "for three days, night and day" (4:16) is concluded "on the third day" (5:1). It is worth noting that the partially Pharisaic delegation which requests the guard at the tomb, and which may reasonably be assumed to be recalling this, the only public pronouncement by Jesus about his resurrection, nevertheless uses the terms meta treis hemeras [literally "after three days" - Bobcat] and he trite hemera [literally "the third day" - Bobcat] to specify the period Jesus had spoken of (Matt 27:63-64). Underlying this flexible usage is the Jewish tendency to speak of a period of twenty-four hours as a day and a night, so that Jesus time in the tomb can be said to embrace (parts of) three "day-nights."
As R. T. France points out, the argument here about time is really a difference of culture: Between a much more relaxed view of time phraseology in the ANE and a modern Western world filled with wrist-watches, or as France puts it, "a pedantic Western reading."