maps as mazes -...

57
Izidor Hafner Maps as Mazes Connect points: A875 S, 30 W<,B875 S, 90 E< and points C845 N, 30 W<,D845 N, 30 E< 60 30 0 30 60 180 120 60 0 60 120 A B C D 60 30 0 30 60 180 120 60 0 60 120

Upload: phungkien

Post on 29-Aug-2019

219 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

Izidor Hafner

Maps as Mazes

Connect points: A875 S, 30 W<,B875 S, 90 E<

and points C845 N, 30 W<,D845 N, 30 E<

60

30

0

30

60

180 120 60 0 60 120

A B

C D

60

30

0

30

60

180 120 60 0 60 120

Page 2: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

Problems

I. Hafner, Maps as Mazes 2

Page 3: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

1.

Connect points: A875 S, 30 E<,B815 N, 90 W<

and points C815 N, 30 E<,D815 S, 90 E<

60

30

0

30

60

180 120 60 0 60120

I. Hafner, Maps as Mazes 3

Page 4: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

2.

Connect points: A875 S, 150 W<,B875 S, 30 E<

and points C845 N, 30 E<,D845 S, 30 W<

60 30 0 30 60

180120

60

0

60

120

I. Hafner, Maps as Mazes 4

Page 5: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

3.

Connect points: A875 S, 90 W<,B875 S, 30 W<

and points C815 S, 30 E<,D875 N, 30 W<

60

30

0

30

60

180 120 60 0 60120

I. Hafner, Maps as Mazes 5

Page 6: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

4.

Connect points: A875 N, 30 E<,B875 S, 30 E<

and points C845 N, 30 E<,D845 N, 30 E<

60

30

0

30

60

180 120 60 0 60120

I. Hafner, Maps as Mazes 6

Page 7: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

5.

Connect points: A875 N, 150 W<,B875 S, 150 E<

and points C845 S, 30 E<,D815 S, 30 E<

6030

0

3060

180 120 600 60 120

I. Hafner, Maps as Mazes 7

Page 8: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

6.

Connect points: A815 S, 90 W<,B875 N, 30 W<

and points C845 S, 30 W<,D845 N, 30 W<

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 8

Page 9: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

7.

Connect points: A875 S, 150 E<,B875 N, 150 W<

and points C815 N, 30 E<,D815 N, 90 W<

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 9

Page 10: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

8.

Connect points: A875 N, 90 W<,B875 S, 150 E<

and points C815 N, 30 E<,D815 S, 150 E<

60

30

0

30

60

180 120 600 60 120

I. Hafner, Maps as Mazes 10

Page 11: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

9.

Connect points: A875 S, 30 W<,B875 N, 150 W<

and points C815 N, 30 E<,D815 S, 30 E<

60

30

0

30

60180

120

60

0

60

120

I. Hafner, Maps as Mazes 11

Page 12: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

10.

Connect points: A875 S, 30 W<,B875 S, 150 W<

and points C845 N, 150 E<,D815 N, 90 W<

60

30

0

30

60

180

120

60

0

60

120

I. Hafner, Maps as Mazes 12

Page 13: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

11.

Connect points: A875 S, 30 E<,B875 S, 150 W<

and points C845 N, 30 W<,D845 N, 30 E<

60

30

0

30

60

180 120 60 060 120

I. Hafner, Maps as Mazes 13

Page 14: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

12.

Connect points: A875 S, 90 W<,B845 N, 90 W<

and points C815 S, 30 E<,D845 N, 30 E<

60

30

0

30

60

180 120 60 0 60120

I. Hafner, Maps as Mazes 14

Page 15: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

13.

Connect points: A875 S, 90 E<,B875 S, 30 W<

and points C815 N, 30 E<,D845 N, 30 E<

60

30

0

30

60

180120 60 0 60 120

I. Hafner, Maps as Mazes 15

Page 16: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

14.

Connect points: A875 N, 90 W<,B875 N, 150 E<

and points C815 N, 30 E<,D845 N, 30 W<

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 16

Page 17: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

15.

Connect points: A875 S, 30 W<,B875 N, 30 W<

and points C845 N, 90 E<,D845 N, 30 E<

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 17

Page 18: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

17.

Connect points: A875 N, 30 W<,B875 S, 150 W<

and points C845 N, 30 W<,D815 N, 90 E<

60

30

0

30

60

180 120 60 0 60120

I. Hafner, Maps as Mazes 18

Page 19: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

18.

Connect points: A875 N, 30 W<,B875 S, 90 E<

and points C845 S, 30 W<,D815 N, 90 W<

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 19

Page 20: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

19.

Connect points: A815 S, 150 W<,B875 N, 90 E<

and points C875 N, 30 W<,D815 S, 150 E<

60

30

0

30

60

120 60 060 120

I. Hafner, Maps as Mazes 20

Page 21: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

21.

Connect points: A845 S, 150 W<,B875 S, 30 E<

and points C815 S, 30 E<,D815 S, 150 E<

60

30

0

30

60

180 120 60 060 120

I. Hafner, Maps as Mazes 21

Page 22: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

22.

Connect points: A875 S, 30 W<,B875 N, 150 E<

and points C815 N, 90 W<,D845 N, 90 W<

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 22

Page 23: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

23.

Connect points: A875 S, 30 E<,B875 N, 150 E<

and points C815 N, 30 W<,D845 N, 90 W<

60

30

0

30

60

180120 60 0 60 120

I. Hafner, Maps as Mazes 23

Page 24: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

24.

Connect points: A875 S, 30 E<,B875 N, 30 W<

and points C845 N, 90 W<,D815 N, 150 W<

60

30

0

30 60

180

120

60

0

60

120

I. Hafner, Maps as Mazes 24

Page 25: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

25.

Connect points: A845 S, 90 W<,B875 S, 150 E<

and points C845 N, 30 E<,D845 N, 30 E<

60

30

0

30

60

180 120 60 060 120

I. Hafner, Maps as Mazes 25

Page 26: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

26.

Connect points: A875 S, 150 E<,B875 N, 30 E<

and points C875 N, 30 W<,D845 N, 30 E<

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 26

Page 27: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

27.

Connect points: A875 S, 30 E<,B875 S, 30 W<

and points C815 N, 30 W<,D845 N, 150 E<

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 27

Page 28: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

28.

Connect points: A875 S, 90 W<,B815 N, 150 E<

and points C815 N, 90 E<,D845 N, 30 E<

60

30

0

30

60

180 12060 0 60 120

I. Hafner, Maps as Mazes 28

Page 29: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

29.

Connect points: A875 N, 150 E<,B875 N, 90 W<

and points C815 N, 30 W<,D815 N, 90 W<

60

30

0

30

60

180 120 60 060 120

I. Hafner, Maps as Mazes 29

Page 30: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

Solutions

I. Hafner, Maps as Mazes 30

Page 31: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

1.

A

BC

D

60

30

0

30

60

180 120 60 0 60120

I. Hafner, Maps as Mazes 31

Page 32: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

2.

AB

CD

60 30 0 30 60

180120

60

0

60

120

I. Hafner, Maps as Mazes 32

Page 33: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

3.

A B

C

D

60

30

0

30

60

180 120 60 0 60120

I. Hafner, Maps as Mazes 33

Page 34: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

4.

A

B

CD

60

30

0

30

60

180 120 60 0 60120

I. Hafner, Maps as Mazes 34

Page 35: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

5.

A

B

C

D

6030

0

3060

180 120 600 60 120

I. Hafner, Maps as Mazes 35

Page 36: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

6.

A

B

C

D

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 36

Page 37: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

7.

A

B

CD

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 37

Page 38: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

8.

A

B

C

D

60

30

0

30

60

180 120 600 60 120

I. Hafner, Maps as Mazes 38

Page 39: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

9.

A

B

C

D

60

30

0

30

60180

120

60

0

60

120

I. Hafner, Maps as Mazes 39

Page 40: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

10.

A

B

C

D

60

30

0

30

60

180

120

60

0

60

120

I. Hafner, Maps as Mazes 40

Page 41: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

11.

A B

CD

60

30

0

30

60

180 120 60 060 120

I. Hafner, Maps as Mazes 41

Page 42: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

12.

A

B

C

D

60

30

0

30

60

180 120 60 0 60120

I. Hafner, Maps as Mazes 42

Page 43: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

13.

A

B

C

D

60

30

0

30

60

180120 60 0 60 120

I. Hafner, Maps as Mazes 43

Page 44: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

14.

AB

C

D

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 44

Page 45: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

15.

A

B

CD

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 45

Page 46: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

17.

A

B

C

D

60

30

0

30

60

180 120 60 0 60120

I. Hafner, Maps as Mazes 46

Page 47: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

18.

A

B

C

D

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 47

Page 48: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

19.

A

BC

D

60

30

0

30

60

120 60 060 120

I. Hafner, Maps as Mazes 48

Page 49: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

21.

A

B

CD

60

30

0

30

60

180 120 60 060 120

I. Hafner, Maps as Mazes 49

Page 50: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

22.

A

B

C

D

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 50

Page 51: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

23.

A

B

C

D

60

30

0

30

60

180120 60 0 60 120

I. Hafner, Maps as Mazes 51

Page 52: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

24.

A

B

C

D

60

30

0

30 60

180

120

60

0

60

120

I. Hafner, Maps as Mazes 52

Page 53: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

25.

A

B

CD

60

30

0

30

60

180 120 60 060 120

I. Hafner, Maps as Mazes 53

Page 54: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

26.

A

BC

D

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 54

Page 55: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

27.

AB

C

D

60

30

0

30

60

180 120 60 0 60 120

I. Hafner, Maps as Mazes 55

Page 56: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

28.

A

BC

D

60

30

0

30

60

180 12060 0 60 120

I. Hafner, Maps as Mazes 56

Page 57: Maps as Mazes - matematika.fe.uni-lj.simatematika.fe.uni-lj.si/html/people/izidor/homepage/MapsPDF/ToPoints.pdf2. Connect points: A875 S, 150 W

29.

A B

CD

60

30

0

30

60

180 120 60 060 120

I. Hafner, Maps as Mazes 57