**1:**
Explain the Pigeonhole principle

79% |

**2a:**
Assume 1200 people are in the same room. How many people can you guaranteed find
that celebrate their birthday on the same day?
**2b:**
Assume 1500 people are in the same room. How many people can you guaranteed find
that celebrate their birthday on the same day?

73% |

**3a:**
List the first ten Fibonacci numbers

100% |

**3b:**
What has the Golden ratio to do with Fibonacci numbers?

75% |

**4a:**
Compute the value of the following compound fraction:
**4b:**
Compute the value of the following compound fraction:

63% |

B: 2/3

**5a:**
Write the number 60 as a sum of distinct nonconsecutive Fibonacci numbers.
**5b:**
Write the number 70 as a sum of distinct nonconsecutive Fibonacci numbers.

90% |

B: 70 = 55 + 13 + 2.

**6a:**
Write the number 130 as a product of prime numbers.
**6b:**
Write the number 126 as a product of prime numbers.

67% |

B: 126 = 2 · 3 · 3 · 7.

**7a:**
Who gave the first proof that there are infinitely many prime numbers?.

0% |

**7b:**
How many prime numbers are there?.

83% |

**8:**
Is the number 123 + 123^{11} prime? Why or why not?

18% |

**9a:**
Reduce 3·5·7·11·13+1 modulo 11.
(Meaning, find that number among 0,1,2,...10 that is equivalent to
3·5·7·11·13+1 (mod 11).)
**9b:**
Reduce 3·5·7·11·13+1 modulo 7.
(Meaning, find that number among 0,1,2,...6 that is equivalent to
3·5·7·11·13+1 (mod 7).)

38% |

B: In the same way 3·5·7·11·13+1 ≡ 3·5·0·11·13+1 = 1 (mod 7).

**10a:**
It is Monday today. What day will there be in 789 days?
**10b:**
It is Monday today. What day will there be in 779 days?

50% |

For (b) it`s Wednesday, since 779 = 777+2 = 7 · 111 + 2 ≡ 2 (mod 7).

Here I was also disappointed. That is the reason for introducing the concept of modulo n. Questions like "It is 3 o clock in the afternoon---what times is it in 800 hours?" Everybody in this class should know how to do this.

**11:**
When are two numbers "equivalent modulo 11"?

30% |

This is the main concept of this section. Everybody should understand it, and should also be able to explain it.

**12:**
What does the Art Gallery Theorem say?

40% |

13a:
Find a 3-coloring of the vertices (such that every triangle has all three colors)
of the following triangulation of an art gallery. |

13b:
Find a 3-coloring of the vertices (such that every triangle has all three colors)
of the following triangulation of an art gallery. |

88% |

B: ....

**14:**
Explain a simple procedure how to produce another golden rectangle of different size from
a given golden rectangle.

75% |

**15a:**
How many edges does a convex polyhedron with 20 vertices and 11 faces have?
**15b:**
How many edges does a convex polyhedron with 15 vertices and 18 faces have?

69% |

B: Since V-E+F=2, and V=15 and F=18, we get E=31.

**16a:**
You have a polyhedron with 14 faces, all of them triangles. How many edges does the polyhedron have?
**16b:**
You have a polyhedron with 16 faces, all of them triangles. How many edges does the polyhedron have?

46% |

A: 3 · 14 = 2 · E, hence E=21.

B: 3 · 16 = 2 · E, hence E=24.

**17a:**
Which one of the following three graphs is a plane graph?

**17b:**
Which one of the following three graphs is a plane graph?

79% |

B: The first one.

**18:**
For the plane graph in the previous question, draw its dual.

54% |

**19:**
Explain what Euler's polyhedron formula states about polyhedra.
Who proved this result?

45% |

**20:**
Does the graph to the right have an open or closed Eulerian tour?
If it doesn't, why not, if it has, write it down (as a sequence of the vertices).

35% |