Cheesemonger's puzzle

Question

^{An entry in Fortnightly Topic Challenge #45: Flags}

The cheesemonger wants to arrange the wheels of cheese on display. The display case has two shelves with space for three wheels of cheese on each. He has six wheels of different kinds of cheese to lay out on the shelves and some very specific ideas about their layout.

1. Cheddar has a 45° wedge cut out of it; Parmesan has exactly half of the wheel left; all other cheeses (Brie, Camembert, Emmental and Swiss) have 135° wedges cut out of them. The cheese left in each wheel forms one continuous piece.
2. All cheeses on the bottom shelf have 7 or more letters in their names.
3. The cheeses on the top shelf are placed in order of increasing name length.
4. The cheeses on the bottom shelf are placed in alphabetical order.
5. The sums of angle sizes of wedges cut out of cheeses in the left and the middle columns are equal.
6. Each cheese in the bottom row has a vertical cut.
7. Both cuts on Swiss cheese point at another cheese; for all other cheeses, exactly one of the cuts points at another cheese.
8. The cuts on each cheese with a double letter in its name point to exactly one other cheese with the same property.
9. If you start at Parmesan and follow the cuts on cheeses as they point to other cheeses, you can visit every cheese once and end at cheddar which you'll visit for the second time.
10. No two cuts in the left column point in the same direction.

Can you figure out how to lay out the cheese and what is today's special in the shop?

Answer 1

The answer is:

Top shelf is Brie, Swiss and Cheddar. Bottom is Camembert, Emmental and Parmesan.

The grid:

For the special offer (Big thanks to Jafe in the comments).

By using the flag semaphore chipher and following the directions of the lines of the Cheese from left to right top to bottom we get SMOKED. So your special offer is SMOKED CHEESE.

Reasoning: let us take a look at what each statement tells us:

1. Cheddar has a 45° wedge cut out of it; Parmesan has exactly half of the wheel left; all other cheeses (Brie, Camembert, Emmental and Swiss) have 135° wedges cut out of them. The cheese left in each wheel forms one continuous piece.

There are 6 cheeses: (Cheddar, Parmesan Brie, Camembert, Emmental and Swiss). Cheddar is 315 degrees, Parmesan is 180 degrees, and the rest are 225 degrees.

2. All cheeses on the bottom shelf have 7 or more letters in their names.

Brie and Swiss are on the top shelf for sure.

3. The cheeses on the top shelf are placed in order of increasing name length.

Brie and Swiss are the first two on the top shelf.

4. The cheeses on the bottom shelf are placed in alphabetical order.

Nothing conclusive yet.

5. The sums of angle sizes of wedges cut out of cheeses in the left and the middle columns are equal.

So this means Camembert and Emmental are left and right following 4 and 5. If Cheddar or Parmesan was either it would lead to two different sums. Also because of 4, Parmesan has to be bottom right leaving Cheddar as the top right.

6. Each cheese in the bottom row has a vertical cut.

We know parmesan is cut like this:

7.Both cuts on Swiss cheese point at another cheese; for all other cheeses, exactly one of the cuts points at another cheese.

Nothing conclusive yet.

8. The cuts on each cheese with a double letter in its name point to exactly one other cheese with the same property.

With 7 you must get:

9. If you start at Parmesan and follow the cuts on cheeses as they point to other cheeses, you can visit every cheese once and end at cheddar which you'll visit for the second time.

Nothing conclusive yet.

10. No two cuts in the left column point in the same direction.

With 9 and the conclusion of 8, here is the journey of the cut's directions