The expression, "three days and three nights," has been a topic of interest for quite some time. It is used several times in the Bible, and, Matthew 12:40 in particular. The issue relates to how people understand this expression in connection with the death and resurrection of Jesus. If Christ died on Friday and rose on Sunday, then, mathematically, the prophecy that Jesus would spend three days and three nights in the grave—just like Jonah in the belly of the fish—is not entirely accurate.
The prophecy's accuracy is a major concern for those who hold that the Bible is without error. If there is even one error in the Scriptures, then certainly, this raises major questions about the Bible's integrity. For some, it would not be enough to hold to the resurrection of Jesus as a historic reality if the very prophecy concerning that resurrection, mentioned by no less that Jesus Christ himself, is suspect. How are to resolve these issues?
The Pope Made it Up
There are a few possible ways that some have sought to work around this issue. One possible solution is to argue that Jesus either did not die on Friday or else he did not rise on Sunday. If that is the case, where did we as Christians get the tradition of celebrating Easter on Sunday and recognizing the crucifixion on Friday; Good Friday, that is?
Some have argued that the traditional view was that the pope (whichever one it was) invented Good Friday as the day of the crucifixion. And then invented Sunday morning as the day of the resurrection. Popes are known for making things up, after all.
Another possible explanation is that when the Bible refers to the "Passover", the "Day of Preparation" before the Sabbath, and the "High Sabbath," this all indicates that the Sabbath could've taken place on any day of the week, or probably Thursday. The day of Preparation would actually be Wednesday.
Therefore, Jesus died on Thursday and rose on Sunday, thus creating an accurate fulfillment of the three days and three nights motif.
A Third More Likely Solution
The issue is that we have a different way of thinking about the passage of time than first-century Jews did. Not only Jews but also their Ancient Near Eastern neighbors. We should not just assume that because we think about something a certain way, that's how it has always been. This is especially true even in terms of the way we classify what a day is and what a night is.
One commentator has described the three days and three nights motif this way:
Jonah’s miraculous escape authenticated his preaching; the resurrection of Jesus will do the same. Three days and three nights was a Jewish idiom for a period covering parts of three 24-hour ‘days-and-nights’ (cf. 1 Sa. 30:12–13; Est. 4:16–5:1).
A.T. Robertson agrees,
“Three days and three nights” may simply mean three days in popular speech. Jesus rose “on the third day” (Matt. 16:21), not “on the fourth day.” It is just a fuller form for “after three days” (Mark 8:31; 10:34).
Another commentator notes it this way:
Three days and three nights is a special phrase used in the ancient world with the meaning ‘long enough to be definitely dead’. It derives originally from the ancient pagan notion that the soul’s trip to the after-world took three days and nights. Jesus’ use of the same phrase for the duration of his death before his resurrection (Mt. 12:40) carries a similar force: it is a way of saying that he would really die, not that he would be literally dead for exactly seventy-two hours 
This is Jesus’ first unambiguous prediction of his death in Matthew, still not as explicit as it will become, and probably understandable only in retrospect. “Three days and three nights” represents a Semitic idiom for any portion of three calendar days. 
As there does seem to be clear evidence that the motif is a Hebrew idiom, then the expression does not require the full passage of all days and night in order for the timeframe to be established, "The Hebrew idiom “three days and three nights” only requires a part of the first and third days." 
In addition to all of this, Ethelbert William Bullinger comments,
Jonah 1:17 (2:1), quoted in Matt. 12:40. The expression, “three days and three nights,” is an idiom which covers any parts of three days and three nights. In 1 Sam. 30:11 (12), it is said that a certain Egyptian had not eaten bread and drunk water for “three days and three nights,” and yet it was only three days since he fell sick (ver. 13), not four days. In Est. 4:16, Esther says she and her maidens will fast “three days and three nights,” and yet it was on “the third day” that Esther went in to [sic] the king; not the fourth day, which it must have been if the expression were literally understood. It may seem absurd to Gentiles and to Westerns to use words in such a manner, but that does not alter the fact.
The most important thing to recognize here is that there are possible explanations for the motif and the way it was fulfilled by Jesus. Whenever there is, what appears to be, a contradiction in Scripture, we should not be too quick to embrace an overly simplistic approach and merely identify this as a contradiction. The issue is not with the text but our understanding of it. We are very far removed from the original writing of the books of the Bible. This should not disparage us in any way, but it should cause us to recognize that we may need a little bit of extra work to do in order to ensure that we properly understand what the Bible is saying.
 Richard T. France, “Matthew,” in New Bible Commentary: 21st Century Edition, ed. D. A. Carson et al., 4th ed. (Leicester, England; Downers Grove, IL: Inter-Varsity Press, 1994), 920.
 A.T. Robertson, Word Pictures in the New Testament (Nashville, TN: Broadman Press, 1933), Mt 12:40.
 Douglas Stuart, “Jonah,” in New Bible Commentary: 21st Century Edition, ed. D. A. Carson et al., 4th ed. (Leicester, England; Downers Grove, IL: Inter-Varsity Press, 1994), 819.
 Craig Blomberg, Matthew, vol. 22, The New American Commentary (Nashville: Broadman & Holman Publishers, 1992), 206–207.
 Mark Water, Hard Questions about the Bible Made Easy, The Made Easy Series (Alresford, Hampshire: John Hunt Publishers Ltd, 2000), 36.
 Ethelbert William Bullinger, Figures of Speech Used in the Bible (London; New York: Eyre & Spottiswoode; E. & J. B. Young & Co., 1898), 845–846.