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?