I attempt to create a searchable index of SNOT techniques, as this is called. Of those, the strongest 180 degree stud reversal are probably one of these:
If you want to connect the two curves together bottom to bottom, you should be able to do it with an axle, sort of like these two techniques:
If that doesn't work I once had luck playing around with a ...
The short answer is because the combined height of the two mini inverted brackets is greater than the interior space of a tile (or brick for that matter.) But you knew this already.
I realize it isn't considered standard by the rest of the world, but because the only quality calipers I had access to tonight are customary, all these values will be in inches.
Chris McVeigh has some nice LEGO projects on his website that use SNOT techniques. Many of the projects have PDF instruction guides.
I stumbled on his work after seeing his LEGO ornaments a few years back. Now that the Christmas season is beginning, a few of those ornaments would be a fun way to introduce this technique to your daughter!
This is very old-school, but that's how I was doing SNOT in the early 80's: a plate, or a tile (as shown) snaps between 2 studs. I prefer tiles to plates, as I don't have studs-alignment issues, but I've used both, and both work.
this way it can open and close
Blue = 4276 Hinge Plate 1 x 2 with 2 Fingers
Yellow = 4275 Hinge Plate 1 x 2 with 3 Fingers
LightBlue = Plate 2 x 4
White = Plate 2 x 2
Pink = Plate 1 x 4
Light Bluish Gray = Plate 1 x 3
Yellowish Green = Plate 1 x 2
Red = Tile 2 x 2
Green = Tile 1 x 4
As far as I know, it's not intended to have any building function, it's just there to prevent any excess plastic at the fill hole from getting in the way of other pieces.
I can't find a good close-up picture, but that particular piece is filled through a hole in the mold at that location, which you can kinda see in this picture.
Without the small recess, ...
There is a bit of a challenge with a geometry. LEGO doesn't produce much symmetrical hinges, otherwise they would be out of system. I could think of the following one piece you could achieve a close result - Hinge Plate 2 x 4 with Pin Hole and 3 Holes - Top. It comes in White color as well, so you could incorporate in your layer of white plates.
Here's a ...
Generally, the lampholder plate does the job:
but some idea of the scale you are working in will help get a better answer, or if you can post a picture of the model you might get a specific answer.
I propose a slightly different solution for the chair: Use Technic bricks.
In this picture I have used 32526 Technic 5x3 bent 90 pieces, with 43857 Technic Beam 2 and 32523 Technic Beam 3 pieces to make the chair uniform. You can make one or two attachment points to ground by either utilizing 64276 Technic Beam 2 Liftarm with Straight Ball Joint as in the ...
I came up with this
(6 plates / 2 bricks)
Blue = Plate 1 x4
Orange = Plate 2 x4
White = Tile 1 x4
Yellow = Tile 2 x 4
DkblGray = Bracket 1 x 2 - 1 x 2 Inverted
(2 plates thick)
Part# 15535 : Tile, Round 2 x 2 with Hole
Part# 18674 : Tile, Round 2 x 2 with Open Stud
This one i saw somewhere else
("2.5" plates thick)
Part# 18654 : ...
There ARE integer sided triangles that have angles of 60 or 120 degrees, see here : https://en.wikipedia.org/wiki/Eisenstein_triple but I'm not sure how this helps you for building roads.
45 degrees there indeed aren't any.
A progress of going one stud forward one stud left will get you 45 degrees if working studs on top. In case of SNOT 2 studs forward, 5 ...
While Windfire's suggestion is valid, there are several elements with opposing studs.
Many of the parts listed on that link don't apply, but several do, including the first five.
If you're not clone-averse, Cobi makes some great plates with studs on both sides, though I have no idea where you can get them aside from buying entire sets known to contain them....
While it doesn't have every SNOT technique, and they call it "sideways building", the official LEGO Master Builder Academy series is really quite good and introduces quite a few very good advanced techniques. I felt they were more than worth the money for my nephews.
There are tools to generate 3D LDraw models which will use SNOT techniques to make sure the overall structure looks nice, most notable:
Bram's Sphere Generator
Depending on her age, your daughter should be able to build the small yellow bunny on that last page, and the result is a very good illustration of why SNOT is useful.
Bram's Flicker page ...