Shoftim- Division and Fractional Segments

"When Hashem, your G-d, will cut down the nations whose land Hashem, your G-d, gives you, and you will possess them, and you will settle in their cities and in their houses, you shall separate three cities for yourselves in the midst of your land, which Hashem, your G-d, gives you to take possession of it. Prepare the way for yourself, and divide three times the boundary of your land that Hashem, your G-d, causes you to inherit; and it shall be for any killer to flee there." ~Devarim 19;1-3

Rashi on 19;3:
"And divide three times the boundary of your land"- That there should be from the beginning of the boundary until the first city of refuge like the amount of travel, i.e., like the distance, that there is from it until the second one. And so, too, from the second one to the third one. And, so too, from the third one until the second, i.e., the opposite, the border of the Land of Israel.

The Gemara in Makkot 9b talks about the word ושלשת, which was translated above as "divide three times". The Gemara brings up the idea that this is unclear because it could either be understood as dividing by three or multiplying by three. Rashi explains that it means division, but not just division into three parts. Rather, it means creating three lines of division, which actually creates 4 sections.

Division and the Concept of Physically Dividing:
When we talk about division, we are dealing with separating a unit with a certain measurement or a group with a number of contents into a given number of smaller equal pieces or groups. Some examples:
*A 12 ft. piece of rope divided amongst 3 people. We would calculate 12 ft. + 3 people = 4 ft. per person
*A bag with 30 marbles divided amongst 5 students. We would calculate 30 marbles + 5 students = 6 marbles per student

Now, let's think about if we are asked to cut a loaf of bread into 8 slices. How many cuts would we need to make in order to get 8 slices? The answer is that we would need to make 7 slices. So, when we are asked to cut a certain number of pieces, how can we know how many cuts we will need? 
*If your cuts are not intersecting with each other at all (think of cutting a loaf), then you will always need one cut less than the number of slices that you want:
1 cut = 2 pieces

2 cuts = 3 pieces

3 cuts = 4 pieces

How does this work? When you make your first cut, you are creating 2 pieces out of the 1 original. Once you continue cutting, however, you are only cutting off 1 more smaller section with each cut.

* If your cuts are going to intersect with each other (think of cutting a rectangular cake), then it helps to think of the area as a multiplication grid. If you need 30 pieces, you could make it 6 pieces by 5 pieces- then you would need 5 cuts in one direction and 4 cuts in the other direction. 
To think about grid cuts:

2 cuts = 4 pieces

3 cuts = 6 pieces

4 cuts = 8 pieces or 9 pieces, depending on where you make the cuts

With cutting on a grid, you can adjust your number of pieces depending on how you choose to place your cuts lengthwise and widthwise, as you can see with our example of 4 cuts above. Trying to keep the lengthwise cuts and widthwise cuts as close to the same number as possible will always result in the most number of pieces. Consider that for the loaf cuts, we needed 3 cuts to get 4 pieces, but for the "cake" cut we only needed 2 cuts to get the same 4 pieces.

Investigating with how many pieces you can get from the fewest cuts is an interesting investigation for students to play with. The more intersections of cuts you make, the more pieces you can get from each individual cut. 

Parsha Connection:
In this week's parsha, it seems that we are told to divide the land in three sections to create cities of refuge. However, Rashi and the Gemara explain that it's actually telling us to make 3 divisions (or 3 cuts) in the land, which will actually separate the land into 4 pieces. With the 3 cities of refuge built on each of the 3 division lines, anyone in each of the 4 settled sections of land will have approximately the same distance to travel to get to the nearest city of refuge. In this way, no one living in any of the settled areas will be closer or farther than anyone else from refuge.

The subtlety in the language is so slight- the difference between dividing by 3 (making 3 cities) and making 3 divisions (making 4 settlements, equidistant to the 3 cities).

Everyday Connection:
How often are you cutting something to share with others? Can you maximize your cuts so that you use the fewest number of cuts possible to get the number of pieces that you want?

Further thoughts: How do intersecting cuts differ when you're dealing with cutting circles into wedges? Is it like a loaf? Is it like a cake? Is it completely different?

Re'eh- Overlapping Patterns

"At the end of seven years you shall institute a release. This is the matter of the release: Every creditor shall release his authority over what he has lent his fellow; he shall not press his fellow or his brother, for He has proclaimed a release for Hashem." ~Devarim 15;1-2

Rashi on Devarim 15;1:
Rashi here explains that one might think that each time a person makes a loan, they count seven years on that particular loan. Rather, we learn here that the seven years mentioned are referring to counting of the year according to the shmita cycle, and all loans are released every time we come to a shmita year, regardless of when they were made.

I'd like to use this information about a different aspect of shmita years to build on the shmita cycle information that we looked at in Parshat Behar. 

In the shmita/yovel chart in that post, you can see that every 7 years is a shmita year. At the time of the post, I made the chart based on my understanding of the text- with shmita every 7 years, and a final yovel year in the 50th year (after 7 groups of 7 years). With further research, I found a discrepancy in the understanding of how the cycle works. One explanation of the discrepancy can be seen here in an explanation on Parshat Behar.

In essence, the calculation discrepancy is as follows:
*Rabbi Yehuda understood that the yovel cycle was independent from the shmita cycle. In his view, there was a shmita cycle occurring every 7 years and a yovel cycle occurring every 50 years. The separate cycles overlapped in such a way that yovel fell in the 1st year of every 8th new shmita cycle.

*The sages understood that the yovel cycle was a final year within the larger shmita cycle. In their view, yovel was the 50th and final year of the previous shmita cycle, and the next shmita cycle would start the following year at year 1.

Here is a chart that compares the calculation implications in these differing understandings (bear with me- the chart gets lengthy in order to start to see the difference as we get into the second set of cycles):

Regular Counting year	Rabbi Yehuda	Sages
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Year 7	Shmita	Shmita
Year 8
Year 9
Year 10
Year 11
Year 12
Year 13
Year 14	Shmita	Shmita
Year 15
Year 16
Year 17
Year 18
Year 19
Year 20
Year 21	Shmita	Shmita
Year 22
Year 23
Year 24
Year 25
Year 26
Year 27
Year 28	Shmita	Shmita
Year 29
Year 30
Year 31
Year 32
Year 33
Year 34
Year 35	Shmita	Shmita
Year 36
Year 37
Year 38
Year 39
Year 40
Year 41
Year 42	Shmita	Shmita
Year 43
Year 44
Year 45
Year 46
Year 47
Year 48
Year 49	Shmita	Shmita
Year 50	Yovel (yr 1 of new shmita cycle)	Yovel (yr 50 of old cycle)
Year 51		(yr 1 of new shmita/yovel cycle)
Year 52
Year 53
Year 54
Year 55
Year 56	Shmita
Year 57		Shmita
Year 58
Year 59
Year 60
Year 61
Year 62
Year 63	Shmita
Year 64		Shmita
Year 65
Year 66
Year 67
Year 68
Year 69
Year 70	Shmita
Year 71		Shmita

Just looking at the comparison chart, we can see that after the first full shmita/yovel cycle, the Sages' cycle falls behind Rabbi Yehuda's. For the next 7 shmita cycles, Rabbi Yehuda's shmita years fall one year earlier than the Rabbi Yehuda's. If we continue this pattern, after the next 7 shmita cycles, the Sages' shmita years will fall two years earlier. This discrepancy continues over time. Although the Sage's will fall farther and farther behind, every certain number of cycles, they will be in line again. Can you figure out how frequently that alignment will occur? To connect with counting in different bases, discussed in Parshat Behar, can you count the years in base-7 (as I started in the chart there) to help identify the alignment shift?

We can look at how this discrepancy in understanding changes the meaning of loan release that we learn of in our parsha this week. It says clearly that the loan release takes place after 7 years, which Rashi explains is in the shmita year. According to Rabbi Yehuda, the math works, and every 7 years is a shmita year, even when a yovel year follows a shmita year. According to the sages, though, when a yovel year falls out, there are actually 8 years between the shmita year before and the next shmita year. What happens if someone makes a loan during a yovel year? According to my understanding of the sages, a loan made during the yovel year would not be required to be released until the following shmita year, which would actually be 8 years. Following the sages, of the calculation were to keep to 7 years, then in the 2nd yovel cycle loans would actually be released the year before shmita, and in the 3rd yovel cycle loans would be released two years before shmita, and so on.

Although I did try to research this question, I was not able to find an answer. It seems that the Gaonim followed the Sages calculation of the yovel cycle, but at the time of exile from Israel, yovel was discontinued and the 50th yovel year was removed from the cycle. With this new calculation, the counting cycle looks like the column for Rabbi Yehuda's cycle, but with the yovel label taken out, since the new shmita cycle is bumped up a year.