Shulchan Aruch Chapter 631: A Succah who's Sun is More than It's Shade and Other Laws of the Sechack (10 Paragraphs)

1.  A Succah who's sun and shade are equal above is invalid since the sun spreads out with distance and below the sun will be greater than the shade.  However, if  the sun and the shade are equal below then it is valid.

2.  If parts of the Succah have the sun greater than the shade and parts of it have the shade greater than the sun; if you combine them and the shade is a little greater than the sun combined then it is valid. [However, one should be strict if it is a large Succah and there is a place 7 by 7 handbreaths of sun greater than shade even if when you combine everything the shade would be greater.]

3.  It is optimal to build the Sechach thin in order than you can see the large stars through it.  If one made it thick like a house roof, even though one can not see the stars it is valid.

4.  Consider where one made the Sechach very thin such that there are empty patches.  As long as there is not an empty space of 3 handbreaths in one spot and as long as when combined the shade is greater than the sun it is valid.

5.  If the Sechach was uneven such that parts were higher than others then it is valid so long as there is not 3 handbreaths of height between the two levels.  If there is a handbreth of space parallel between the uneven Sechach, even if there is more than 3 handbreaths of height between them we view it as the two joining together as one level and it is valid. [He means if there is air of 1 handbreath  between them that is enough in theory to bring down the higher portion then it is valid.  Even if the sun is greater than the shade on that air space it is valid if the shade is greater than the sun between the spaces.]

6.  Suppose one has an extra middle wall of the Succah with branches above it.  If they are valid by having more shade than sun then this portion is a valid Succah even though the extra wall was only built as an accessary to the surrounding Succah (loose translateion).

7.  Ponder a case where one has branches sticking outside of the Sechach over a Succah wall and there is an outside wall opposite this. This is a valid Succah if the Secach is valid.  This is true even though one originally only intended the Sechach to cover the Succah walls which he built 7x7 handbreaths wide.  And is true even though one did not intend the Sechach to cover an outside wall (loose translation).
8.  Invalid Sechach makes a Succah invalid with 4 handbreaths.  In other words, 4 handbreaths of invalid Sechach can not be found on a Succah.  If one separated the bundles of invalid Sechach and placed a little Kosher Sechach between them it is invalid since if one would remove the invalid Sechach and replace it with air, there would not be enough valid Sechach left for the Succah to be Kosher.  However, if one mixed a lot more valid Sechach with invalid Sechach; or one weaved it together in such a way that there is an excess of Kosher Sechach then it is a valid Succah.  This applies so long as there is enough valid Sechach to make the Succah valid if the invalid Sechach were removed.  [This only applies for a large Succah, but for a small succah, a 3x3 area of invalid Sechach makes it invalid.]

9.  A house with a roof made out of planks of wood that are covered with roofing materials [meaning tar and pebbles] and one wishes to make this roof valid for a Succah.  It is good enough if one simply removes all the boards having in mind in order to do so for the Mitzvah of Succah.  It is also good if one removed two neighboring planks of wood and put in their place valid Sechach, then it is all Kosher even if there are planks remaining that are wider than four handbreaths.  Others say that it is invalid if the planks are 4 handbreaths in size or wider.

10.  Ponder the case of a Succah that has no roof.  For example, if the tops of all four walls are attached like a triangle or  if one leaned the walls of a Succah against a wall.  In such cases the Succah is invalid.  However, if the Succah had a roof, even if it was as small as a handbreath it is valid.  This also applies if one leaned the Succah walls against a wall but left a roof of a handbreath on top, then it is valid.  [Some say that the handbreath of Sechach must be directly under the floor of the Succah and not above air outside of the Succah.  They also say that the Sechach which counts as the roof must be 7 by 7 handbreaths wide.  The walls also must be at least 10 handbreaths high.  And if parts of the walls are at an angle and are to be counted as Sechach, they must be made out of something that is Kosher for Secach.  In such a case it is permitted to sleep even under the walls {1}.]
{1} I am not 100% sure of my interpretation of the REMA in this paragraph, but I gave my best shot at it and hope it is correct.

Translated by Dr. Jay Dinovitser
Free for personal use only  -8/09