Who knows sixty-eight?
Please cite/link your sources, if possible. At some point in the next few days, I will:
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According to Rabbi Shimon (Sanhedrin 17a), the number of zekainim that assumed their posts in the ohel moed. (70 minus Eldad and Meidad who did not feel they deserved the position and were zochu to another nevuah)
There are 68 bones that make up the essential structure of the human body (the parts that are essential for life, apparently): 22 ribs, 18 spinal vertebrae, 9 bones of the skull, 8 cervical vertebrae, 6 in the "key of the heart" (sternum?), and five at the orifices. (Malbim to Song of Songs 3:7; he explains that the "sixty mighty men standing guard around Shlomo's bed" mentioned there are these approximately sixty organs that guard the soul and keep it from leaving the body.)
CE Year of the Churban Bayis Sheini
The number of Yaakov and his family members that were yordim to Mitzrayim at that time according to Rashi's pshat of the Shiv'im -Rashi includes Yocheved ("bein hachomos"), I am including Yehudah (who was sent ahead) and Yaakov as well. The Torah doesn't include wives.
This one was a toughie!
The Mishnah in Succah 19b states that the Sages allow one to construct a succah by leaning s'chach against a wall. The length of this s'chach must be enough to accommodate the minimum measure of a succah, which is 10 tefachim high and 7 tefachim wide. Some Rishonim say that this minimum measure needs to fit under the s'chach. Others, however, say it is sufficient if the s'chach is just 17 tefachim long, so that 10 tefachim will serve as a wall and 7 tefachim as a roof. Hence, if one fabricates a succah in the form of a 4-sided pyramid, the total length of s'chach he will need (as measured from the base of each side to the top) will be 17 x 4, or 68 tefachim.
(The truth is that in this case it might be enough to have 13.5 on each side, so that each two opposing sides will together contribute 7 tefachim to the roof. I'm not sure how that would work.)