The Torah Reading Sequence Zvi Har’El Department of Mathematics Technion − Israel Institute of Technology Haifa 32000, Israel E ‐ Mail: rl@math.technion.ac.il 1. Introduction. The Torah, or Pentateuch, which consists of the first five books of the Scriptures, is traditionally divided to 54 portions. It is customary to read these portions in Sabbath mornings during the year, reading the last one in a special feast, Simchat‐Torah (Rejoicing the Law). Few of the Sabbaths fall on a holiday, and there are special readings for them. Thus, there is a gap between the number of Torah portions which should be read, and the number of ordinary Sabbaths in which they may be read. This gap is bridged tradition‐ ally by joining selected pairs of consecutive portions for reading in a single Sabbath. 2. The portions. The following table lists the 54 portions of the Torah, complete with book, chapter and verse indications. ‐2‐ ┌───────────────────────────────────┬───────────────────────────────────────┐ │ Portion Verses │ Portion Verses │ ├───────────────────────────────────┼───────────────────────────────────────┤ │ 1 Bereshit Genesis 1:1‐6:8 │ 28 Metzora 14:1‐15:33 │ │ 2 Noach 6:9‐11:32 │ 29 Acharei‐Mot 16:1‐18:30 │ │ 3 Lech‐Lecha 12:1‐17:27 │ 30 Kedoshim 19:1‐20:27 │ │ 4 Vayera 18:1‐22:24 │ 31 Emor 21:1‐24:23 │ │ 5 Chayei‐Sarah 23:1‐25:18 │ 32 Behar 25:1‐26:2 │ │ 6 Toldot 25:19‐28:9 │ 33 Bechukotai 26:3‐27:34 │ │ 7 Vayetze 28:10‐32:2 │ 34 Bamidbar Numbers 1:1‐4:20 │ │ 8 Vayishlach 32:3‐36:43 │ 35 Naso 4:21‐7:89 │ │ 9 Vayeshev 37:1‐40:23 │ 36 Beha’alotcha 8:1‐12:15 │ │10 Miketz 41:1‐44:17 │ 37 Shelach 13:1‐15:15 │ │11 Vayigash 44:18‐47:27 │ 38 Korach 16:1‐18:32 │ │12 Vayechi 47:28‐50:26 │ 39 Chukat 19:1‐22:1 │ │13 Shemot Exodus 1:1‐6:1 │ 40 Balak 22:2‐25:9 │ │14 Vaera 6:2‐9:35 │ 41 Pinchas 25:10‐30:1 │ │15 Bo 10:1‐13:16 │ 42 Matot 30:2‐32:42 │ │16 Beshalach 13:17‐17:16 │ 43 Mase’ei 33:1‐36:13 │ │17 Yitro 18:1‐20:23 │ 44 Devarim Deutronomy 1:1‐3:22 │ │18 Mishpatim 21:1‐24:18 │ 45 Vaetchanan 3:23‐7:11 │ │19 Teruma 25:1‐27:19 │ 46 Ekev 7:12‐11:25 │ │20 Tetzave 27:20‐30:10 │ 47 Re’e 11:26‐16:17 │ │21 Ki‐Tisa 30:11‐34:35 │ 48 Shoftim 16:18‐21:9 │ │22 Vayakhel 35:1‐38:20 │ 49 Ki‐Tetze 21:10‐25:19 │ │23 Pekudei 38:21‐40:38 │ 50 Ki‐Tavo 26:1‐29:8 │ │24 Vayikra Leviticus 1:1‐5:26 │ 51 Nitzavim 29:9‐30:20 │ │25 Tzav 6:1‐8:36 │ 52 Vayelech 31:1‐31:30 │ │26 Shemini 9:1‐11:47 │ 53 Ha’azinu 32:1‐32:52 │ │27 Tazri’a 12:1‐13:59 │ 54 Vezot‐Habracha 33:1‐34:12 │ └───────────────────────────────────┴───────────────────────────────────────┘ 3. The need for joining. The following table summerizes the combinatorics involved in joining por‐ tions for reading in a single Sabbath. Each line in the table is dedicated to a different year type, determined by the length of the year and the week day of its beginning, 1 Tishri. ‐3‐ ┌───┬─────────────────────────────────┐ │ │ a b c d e f g │ ├───┼─────────────────────────────────┤ │ 1 │ 353 2 5 50 2 54 6 │ │ 2 │ 353 7 3 51 4 52 5 │ │ 3 │ 354 3 7 50 2† 54 6† │ │ 4 │ 354 5 2 51 3† 52 4† │ │ 5 │ 355 2 7 50 2† 54 6† │ │ 6 │ 355 5 3 51 3 52 4 │ │ 7 │ 355 7 5 51 4 53 6 │ ├───┼─────────────────────────────────┤ │ 8 │ 383 2 7 54 2† 54 2† │ │ 9 │ 383 5 3 55 3 52 0 │ │10 │ 383 7 5 55 4 53 2 │ │11 │ 384 3 2 55 2† 53 0† │ │12 │ 385 2 2 55 2† 53 0† │ │13 │ 385 5 5 55 3 53 1 │ │14 │ 385 7 7 55 4† 53 2† │ └───┴─────────────────────────────────┘ In addition to the year type, the columns of the table are as follows: a is the length of the year, in days. Common years have 353, 354 or 355 days, and leap years have 383, 384, or 385 days. b is the Week day of 1 Tishri. Since the Hebrew New Year is excluded from being a Sunday, a Wednesday or a Friday, the possible values are 2, 3, 5 and 7 (representing Sabbath). c is the week day of 1 Tishri of the next Year, a+b(mod7). d is the Total number of Sabbaths in the year: The number of full weeks in a, [a/7], plus one if c