1.	Which of the following polygons can tile the plane?
 I: Regular pentagon
II: Regular hexagon	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

2.	Which of the following polygons can tile the plane?
 I: Regular octagon
II: Non-rectangle parallelogram	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

3.	Which of the following polygons can tile the plane?
 I: Non-square rhombus
II: Non-equilateral isosceles triangle	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

4.	Which of the following polygons can tile the plane?
 I: Scalene triangle
II: Equilateral triangle	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

5.	The exterior angle of a regular hexagon has a measure of	
	A)  18°    B)  30°    C)  45°    D)  60°	

6.	The exterior angle of a regular pentagon has a measure of	
	A)  18°    B)  30°    C)  60°    D)  72°	

7.	The exterior angle of a regular octagon (8-gon) has a measure of	
	A)  18°    B)  30°    C)  45°    D)  60°	

8.	The exterior angle of a regular decagon (10-gon) has a measure of	
	A)  18°    B)  36°    C)  45°    D)  54°	

9.	Choose the correct word.
. . . pentagons can form a tiling of the plane.	
	A)  All . . .    B)  Some . . .    C)  No . . .	

10.	Choose the correct word.
. . . hexagons can form a tiling of the plane.	
	A)  All . . .    B)  Some . . .    C)  No . . .	

11.	Choose the correct word.
. . . non-convex quadrilaterals can form a tiling of the plane.	
	A)  All . . .    B)  Some . . .    C)  No . . .	

12.	Choose the correct word.
. . . convex quadrilaterals can form a tiling of the plane.	
	A)  All . . .    B)  Some . . .    C)  No . . .	

13.	Semiregular tilings can tile with
 I: Three different regular polygons.
II: Three different vertex combinations.	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

14.	Squares and equilateral triangles can form a tiling that has at every vertex	
A)	three squares and two triangles.	C)	one square and four triangles.	
B)	two squares and three triangles.	D)	one square and five triangles.	

15.	Regular hexagons and equilateral triangles can form a tiling of the plane that has at every vertex	
A)	three hexagons and one triangle.	C)	two hexagons and three triangles	
B)	two hexagons and two triangles.	D)	one hexagon and five triangles	

16.	Regular octagons and squares can form a tiling of the plane that has at every vertex	
A)	two octagons and two squares.	C)	one octagon and two squares.	
B)	two octagons and one square.	D)	one octagon and three squares.	

17.	Regular dodecagons (12-gons) and equilateral triangles can form a tiling of the plane that has at every vertex	
A)	two dodecagons and two triangles.	C)	one dodecagon and two triangles.	
B)	two dodecagons and one triangle.	D)	one dodecagon and three triangles.	

18.	A semiregular tiling has one equilateral triangle and two regular p-gons at each vertex.  What is p?	
	A)  8    B)  9    C)  10    D)  12	

19.	A semiregular tiling has one square, one regular hexagon, and one regular p-gon at each vertex.  What is p?	
	A)  8    B)  9    C)  10    D)  12	

20.	A semiregular tiling has two regular octagons and one regular p-gon at each vertex.  What is p?	
	A)  3    B)  4    C)  5    D)  6	

21.	A semiregular tiling has four equilateral triangles and one p-gon at each vertex.  What is p?	
	A)  3    B)  4    C)  5    D)  6	

22.	A semiregular tiling has one square and two regular p-gons at each vertex.  What is p?	
	A)  5    B)  6    C)  8    D)  10	

23.	A scalene triangle ABC tiles the plane.  What is a possible configuration of the angles at the vertices?
 I: Three types of vertices: 6 As, or 6 Bs, or 6 Cs
I: One type of vertex: with 2As, 2 Bs, and 2Cs	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

24.	A non-convex quadrilateral ABCD tiles the plane.  What is a possible configuration of the angles at the vertices?
 I: Two types of vertices: 2 As and 2 Cs; 2 Bs and 2 Ds
I: One type of vertex: with an A, B, C, D	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

25.	Which of the following form the faces of a regular polyhedron?
 I: Eight squares
II: Four equilateral triangles	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

26.	Which of the following form the faces of a regular polyhedron?
 I: Six equilateral triangles
II: Eight equilateral triangles	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

27.	Which of the following form the faces of a regular polyhedron?
 I: Six squares
II: Ten pentagons	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

28.	A regular polyhedron with 12 faces has what regular polygon as faces?	
	A)  Triangle    B)  Square    C)  Pentagon    D)  Hexagon	

29.	A regular polyhedron with 20 faces has what regular polygon as faces?	
	A)  Triangle    B)  Square    C)  Pentagon    D)  Hexagon	

30.	A regular polyhedron with six faces has what regular polygon as faces?	
	A)  Triangle    B)  Square    C)  Pentagon    D)  Hexagon	

31.	What is a possible way to tile the plane with a right triangle?
 I: Six at every vertex
II: Four or eight at every vertex	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

32.	What is a possible way to tile the plane with a general isosceles triangle?
 I: Six at a vertex
II: Four or eight at a vertex	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

33.	What is a possible way to tile the plane with a parallelogram with 60° and 120° angles?
 I: Four at a vertex
II: Three or six at a vertex	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

34.	What is a possible way to tile the plane with a pentagon with three 90° angles and two equal obtuse angles?
 I: Three at a vertex
II: Three or four at a vertex	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

35.	What is a possible way to tile the plane with a pentagon with two 90° angles and three equal obtuse angles?
 I: Three at a vertex
II: Three or four at a vertex	
	A)  I only    B)  II only    C)  Both I and II    D)  Neither I nor II	

36.	Can the tile below be used to tile the plane?

A)	No.	
B)	Yes, with translations only.	
C)	Yes, with translations and half-turns only.	
D)	Yes, but reflections must be included.	

37.	Can the tile below be used to tile the plane?

A)	No.	
B)	Yes, with translations only.	
C)	Yes, with translations and half-turns only.	
D)	Yes, but reflections must be included.	

38.	Can the tile below be used to tile the plane?

A)	No.	
B)	Yes, with translations only.	
C)	Yes, with translations and half-turns only.	
D)	Yes, but reflections must be included.	

39.	Can the tile below be used to tile the plane?

A)	No.	
B)	Yes, with translations only.	
C)	Yes, with translations and half-turns only.	
D)	Yes, but reflections must be included.	

40.	Can the tile below be used to tile the plane?

A)	No.	
B)	Yes, with translations only.	
C)	Yes, with translations and half-turns only.	
D)	Yes, but reflections must be included.	

41.	Penrose Tilings are non-periodic.	
	A)  True    B)  False	

42.	Portions of Penrose Tilings can have rotational symmetry.	
	A)  True    B)  False	

43.	Most Penrose Tilings require at least 10 differently shaped tiles.	
	A)  True    B)  False	

44.	Any finite portion of a Penrose Tiling can be found in another Penrose Tiling.	
	A)  True    B)  False	

45.	Escher No. 128 [Bird], Figure 20.10 in the textbook, tiles by	
A)	translations only.	C)	translations and reflections only.	
B)	translations and half-turns only.	D)	translations, half-turns, and reflections.	

46.	Escher No. 67 [Horseman], Figure 20.11a in the textbook, tiles by	
A)	translations only.	C)	translations and reflections only.	
B)	translations and half-turns only.	D)	translations, half-turns, and reflections.	

47.	Escher No. 6 [Camel], Figure 20.15 in the textbook, tiles by	
A)	translations only.	C)	translations and reflections only.	
B)	translations and half-turns only.	D)	translations, half-turns, and reflections.	

48.	Escher No. 88 [Sea Horse], Figure 20.16 in the textbook, tiles by	
A)	translations only.	C)	translations and reflections only.	
B)	translations and half-turns only.	D)	translations, half-turns, and reflections.	

49.	If you attempt to tile a non-Euclidean surface with eight triangles meeting at each vertex,	
A)	you will find your efforts to be impossible.	
B)	you will be tiling a surface with elliptic geometry.	
C)	you will be tiling a surface with hyperbolic geometry.	

50.	If you attempt to tile a non-Euclidean surface with five squares meeting at each vertex,	
A)	you will find your efforts to be impossible.	
B)	you will be tiling a surface with elliptic geometry.	
C)	you will be tiling a surface with hyperbolic geometry.	

51.	The interior angle of a regular hexagon has a measure of	
	A)  60°    B)  108°    C)  120°    D)  135°	

52.	The interior angle of a regular pentagon has a measure of	
	A)  72°    B)  108°    C)  120°    D)  144°	

53.	The interior angle of a regular octagon has a measure of	
	A)  45°    B)  60°    C)  120°    D)  135°	

54.	What is a possible way to tile the plane with a hexagon that has four angles measuring 140° and two angles measuring 80°?	
A)	Three at a vertex only	
B)	Four at a vertex only	
C)	Either three or four at a vertex	
D)	This shape will not tile the plane by itself.	

55.	Which of the following is a regular tiling of the plane?	
A)		C)		
B)		D)		

56.	Which of the following is a semiregular tiling of the plane?	
A)		
B)		
C)		
D)		

57.	At the vertex of a tiling of the plane there are two regular pentagons and one regular p-gon at each vertex. What is p?	
	A)  8    B)  9    C)  10    D)  12	

58.	A semiregular tiling has two squares and three regular p-gons at each vertex. What is p?	
	A)  3    B)  4    C)  5    D)  6	

59.	A semiregular tiling has an equilateral triangle, a regular hexagon and two regular p-gons at each vertex. What is p?	
	A)  3    B)  4    C)  5    D)  6	

60.	A semiregular tiling has two equilateral triangles and two regular p-gons at each vertex. What is p?	
	A)  3    B)  4    C)  5    D)  6	

Answer Key

1.	B	
2.	B	
3.	C	
4.	C	
5.	D	
6.	D	
7.	C	
8.	B	
9.	B	
10.	B	
11.	A	
12.	A	
13.	A	
14.	B	
15.	B	
16.	B	
17.	B	
18.	D	
19.	D	
20.	B	
21.	D	
22.	C	
23.	A	
24.	A	
25.	B	
26.	B	
27.	A	
28.	C	
29.	A	
30.	B	
31.	C	
32.	A	
33.	C	
34.	B	
35.	B	
36.	B	
37.	A	
38.	C	
39.	C	
40.	C	
41.	A	
42.	A	
43.	B	
44.	A	
45.	A	
46.	C	
47.	B	
48.	B	
49.	B	
50.	C	
51.	C	
52.	B	
53.	D	
54.	A	
55.	A	
56.	B	
57.	C	
58.	A	
59.	B	
60.	D