The Gemara in Masechet Shabbat discusses the question of Shabbat observance by a person in a remote area who lost track of the days, and does not know which day is Shabbat. Such a person, the Gemara rules, should count six days beginning with the day when he realizes his confusion, and observe Shabbat on the seventh day.
The Halachic authorities address the issue of whether such a person recites the Shabbat or weekday prayers on this day which he observes as Shabbat. The general consensus is that such a person recites the Shabbat prayers, as on a normal Shabbat, though the Kaf Ha'chayim (by Rabbi Chayim Palachi, Turkey, 1788-1868) writes that he omits the Musaf prayer. The authorities disagree as to whether this person lays Tefillin on the day he observes as Shabbat: the Chafetz Chayim (Rabbi Yisrael Meir Kagan, Lithuania, 1839-1933) rules that he wears Tefillin without reciting a Beracha, whereas the Kaf Ha'chayim held that he does not wear Tefillin at all.
Would this person recite Kiddush and Havdala? A fundamental rule in Halacha establishes that "Safeik Berachot Le'hakel" – one does not recite a Beracha if there is some uncertainty as to whether it is required. Seemingly, then, in this case, where one does not observe Shabbat with certainty, he should not recite the Berachot of Kiddush and Havdala.
In truth, however, the Gemara explicitly requires reciting Kiddush and Havdala in such a case. A person's observance of Shabbat in this situation is purely commemorative: the Rabbis required him to observe a Shabbat every seven days so that he does not forget the concept of Shabbat. Therefore, his observance of Shabbat must include all the basic elements of Shabbat, including Kiddush and Havdala, so that he will remember how to observe Shabbat in the future.
A major controversy exists among the Halachic authorities concerning the observance of Shabbat in the Far East, in countries such as Japan. According to many authorities, Halacha does not recognize the International Date Line as the geographic point that determines the beginning of a new day. If the Halachic "date line" is west of the Far East, then the day generally considered Saturday in those areas is actually Friday, and the following day – which the world considers Sunday – is Shabbat. For this reason, some authorities require observing Shabbat on both Saturday and Sunday in the Far East, in order to satisfy both views.
Similarly, the authorities address the question of whether or not Shabbat ends for a person on a ship when it crosses the date line. Must he continue observing Shabbat, given that Shabbat has yet to end at the point where he began Shabbat, or do we say that since a new day has begun in his current location, Shabbat has ended for this individual?
The Minchat Elazar (work of responsa by Rabbi Chayim Elazar Shapiro of Munkatch, early 20th century) ruled that a person's status with respect to Shabbat is determined by the location where he had begun Shabbat. We find with regard to a number of Shabbat laws that the period of Bein Ha'shemashot (between sundown and nightfall) on Friday evening determines Halachic statuses for the entirety of Shabbat. For example, an object deemed Muktzeh during Bein Ha'shemashot on Friday evening remains Muktzeh throughout Shabbat. Similarly, the presence of an Eruv Techumin during Bein Ha'shemashot determines a person's status with regard to Techum Shabbat (the limits on walking outside one's city on Shabbat) for the remainder of Shabbat. Likewise, the Minchat Elazar claimed, the sanctity of Shabbat is determined during Bein Ha'shemashot, and therefore even if a person crosses the date line on Shabbat, he continues his observance of Shabbat in accordance with the location where he had begun Shabbat.
This ruling applies as well in a situation of a person sailing on a ship to Eretz Yisrael for the purpose of living there, and the ship arrives in Eretz Yisrael on Yom Tov Sheni (the second day of Yom Tov, observed only outside the Land of Israel). Now that he has arrived in Israel, he is no longer bound by the obligation of Yom Tov Sheni. May he therefore discontinue his Yom Tov observance that day, once he has arrived in Israel? According to the Minchat Elazar, he must continue observing Yom Tov as if he were still outside the Land of Israel. Since he had begun observing Yom Tov Sheni, he must complete the observance of the entire day, even after arriving in Eretz Yisrael.
Summary:
1) A person in a remote area who loses track of days counts six days from the day he realizes his confusion and observes Shabbat on the seventh. He recites the Shabbat prayers, with the exception of Musaf, as well as Kiddush and Havdala. According to some authorities, he must nevertheless don Tefillin, though without a Beracha.
2) Different opinions exist as to when one observes Shabbat in the Far East, on Saturday or Sunday. According to some authorities, one must observe both days to satisfy all opinions.
3) If a person sails on a ship that crosses the date line on Shabbat, he must nevertheless continue observing Shabbat as if he were still in the location where Shabbat had begun.
4) If a person sails to Israel to live there, and he arrives in Israel on Yom Tov Sheni, he must complete his observance of Yom Tov Sheni as if he were still outside Eretz Yisrael.
