2-15-2011

 

This section mines data produced from the Zimmermann contest.   The examples for maximum retention below order 9 lend themselves to a attempt to enumerate all solutions.

 

 

6x6  Magic Square – data from Zimmermann’s contest.  Below - the top two submitted retention patterns.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

17

19

20

24

23

8

 

 

9

25

27

22

8

20

 

13

31

34

3

5

25

 

 

26

6

4

33

23

19

 

28

7

10

36

4

26

 

 

24

1

35

5

30

16

 

27

1

16

12

33

22

 

 

21

32

12

15

3

28

 

15

35

2

6

32

21

 

 

14

34

2

7

36

18

 

11

18

29

30

14

9

 

 

17

13

31

29

11

10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

192 units retained

 

 

 

185 units retained

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Find all examples for the  6x6 magic square retaining 192 units of water (only one pattern known)

 

Case Split:

 

              C4 = 36 ; cs = 45 ;  solutions:  8855

              C4 = 36 ; cs = 44 ;  solutions:  2495

              C4 = 36 ; cs = 43 ;  solutions:  1695

              C4 = 35 ; cs = 45 ;  solutions:  1232

                       Total =   6 x 6 MS  192 u      14277

               CS = sum of 4 corner cells

              To avoid symmetrical solutions  B2 < E5.

               ( Walter Trump’s data)

 

 

 

7x7 Magic Square – data from Zimmermann’s contest.  Below - the top two submitted retention patterns.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

24

38

39

40

19

12

 

 

13

30

22

41

39

20

10

 

29

45

13

11

4

46

27

 

 

31

15

48

7

4

45

25

 

41

18

20

22

25

7

42

 

 

27

44

17

18

21

8

40

 

23

44

6

17

14

43

28

 

 

23

42

5

29

9

43

24

 

31

1

49

9

48

2

35

 

 

34

1

47

6

49

2

36

 

32

10

15

47

8

37

26

 

 

35

11

3

46

16

38

26

 

16

33

34

30

36

21

5

 

 

12

32

33

28

37

37

19

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  418 units retained

 

 

 

 

  417 units retained

 

 

 

 

 

Find all examples for the  7 x 7 magic square retaining 418 units of water  (only one pattern known)

 

 

1

24

38

40

39

18

15

3

23

38

39

41

19

12

26

45

8

13

10

44

29

26

45

11

13

5

46

29

41

17

21

20

25

9

42

43

18

21

20

25

8

40

28

46

11

19

5

43

23

24

44

15

17

6

42

27

32

3

49

6

48

2

35

31

2

49

9

48

1

35

31

7

14

47

12

37

27

32

10

7

47

14

37

28

16

33

34

30

36

22

4

16

33

34

30

36

22

4

F6 = 37    E7 = 35      CS = 36     

8351 total

F6 = 37    E7 = 35     CS =  35             

1583 total

3

22

38

39

40

20

13

4

23

39

40

38

19

12

24

46

12

14

6

44

29

26

46

13

11

6

45

28

41

18

21

19

27

7

42

42

17

20

21

25

9

41

28

45

11

17

5

43

26

24

44

5

18

14

43

27

32

2

49

8

47

1

36

32

2

49

8

48

1

35

31

9

10

48

15

37

25

31

10

15

47

7

36

29

16

33

34

30

35

23

4

16

33

34

30

37

22

3

F6 = 37   E7 = 36  CS = 35,36              320 total

F6 = 36  E7 = 35   CS = 35      178 total

 

 

 

1-5

21 - 25

38-41

38-41

38-41

17-21

11-15

22-29

43-48

3-17

5-16

3-16

43-47

23-30

40-43

15-20

18-22

18-22

24-28

7-10

40-43

22-29

43-47

3-17

16-20

3-16

42-45

23-30

31-33

1-5

47-49

6-12

46-49

1-3

E7 35-36

31-33

7-10

4-17

46-49

4-17

F6 36-37

23-30

14-18

31-34

33-34

29-30

35-37

20-24

3-6

 

Table above -   known number range for each individual cell …. 

 

***  the F1 compiler can actually solve for all solutions in a reasonable amount of time given the following constraints  #1.  Of course the normal magic square constraints  #2. The individual cell constraints noted above  #3.  The corner sum constraint  #4. The sum of the lake border  #5. The sum of the retained water > 417 units. 

 

 

Case Split:

 

 

              F6 = 37 ;  E7 = 35;   cs = 36 ;        solutions:     8351

              F6 = 37;  E7 = 35;    cs = 35 ;        solutions:     1583

              F6 = 37;  E7 = 36;    cs = 35,36 ;   solutions        320

              F6 = 36 ;  E7 = 35;    cs = 35;        solutions:       178

              Total     7 x 7 MS          418 u                          10,432

 

                    ( Walter Trump’s data)

              

 

 

8x8  Magic Square – data from Zimmermann’s contest.  Below - the top two submitted retention patterns.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

25

41

38

50

47

40

16

 

1

28

49

39

40

44

42

17

36

43

22

62

8

6

51

32

 

29

50

4

63

56

2

13

43

42

1

63

18

56

26

5

49

 

48

5

62

23

11

64

6

41

29

59

13

20

27

57

7

48

 

30

57

22

15

25

18

58

35

54

14

21

33

23

15

61

39

 

52

16

24

31

21

19

59

38

52

17

34

24

10

60

19

44

 

53

10

36

26

20

61

8

46

35

64

11

12

58

4

46

30

 

33

60

12

9

55

7

47

37

9

37

55

53

28

45

31

2

 

14

34

51

54

32

45

27

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

797 units retained

 

 

 

 

795 units retained

 

 

 

 

 

 

Find all examples for the  8x8 magic square retaining 797 units of water (only one pattern known)

 

 

***  F1 compiler – using the same logic / constraints that worked for the 7x7 MS max retention above produce miniscule results given the scope of the 8x8 problem.  I added alot of individual cell constraints and got 254 solutions in 48 hours.  Walter Trump’s skill set produced > 50k solutions in a hour and gave him the following estimation on the number of 8x8 magic squares noted below.

 

 

Estimating the number of magic 8x8-squares with maximal water retention

 

( Walter Trump’s notes 2-14-2011)

 

The following mask was used for a first count.

The shown entries were not changed during this count.

Additionally limitations: A4 > 33  and  E8 > 33.

The program found  206,422  solutions.

Possible permutations of numbers and expected factor
that increases the total number of squares

Numbers

Factor

1, 2

2

12, 15

2

44, 45

2

41, 42

2

60, 61

2

62, 63

2

52, 53

2

54, 55

2

47, 48, 49, 50

24

56, 57, 58, 59

24

The total factor is nearly 150,000

But there are some more possible permutations: 61, 62  or  60, 63  or 53, 54  and so on.

They probably do not bring the same number of solutions. (factor: unknown > 1)

We do not flip 43 and 46 in order to avoid symmetrical solutions.

Changing certain numbers

We can change the values of several numbers:

Larger pair of corner values: 10, 17 or 11, 16 or 12, 15 or 13, 14

            All these values should bring us nearly the same number of squares (factor: 4)

H8 = 3, 4  instead of 2 and smaller numbers for the other corners
            Surely less successful (factor: unknown > 1)

B7 = 64, 63, 62  - but 63 and 62 are less successful than 64 (factor: unknown > 1)

Solutions with a smaller corner sum

We can also choose 29 or 28 instead of 30 for the sum of the numbers in the corner.

But we will get not as many squares for these smaller sums (factor: unknown > 1)

 

All in all we can assume that the factor is much higher than  1,000,000.

Thus the number of squares is at least  200,000 x 1,000,000 = 200,000,000,000

Considering all the unknown factors we may assume that

the total amount of squares is about 1,000,000,000,000

 

( Walter Trump’s notes 2-14-2011)

 

 

1 

37

57

47

66

50

53

55

 3 

43

59

11

77

25

75

10

15

54

58

 7 

78

21

34

22

81

12

56

45

73

14

23

32

30

27

74

51

69

17

39

28

52

36

35

26

67

44

72

18

38

42

13

20

76

46

64

 6 

80

16

31

24

79

 8 

61

40

65

 9 

70

19

71

 4 

62

29

 5 

33

63

49

68

48

60

41

 2 

1408  Units Retained   Walter Trump  12 June 2010

 

 

1 

37

57

47

66

50

53

55

 3 

43

59

11

77

25

75

10

15

54

58

 7 

78

21

34

22

81

12

56

45

73

14

23

32

30

27

74

51

69

17

39

28

52

36

35

26

67

44

72

18

38

42

13

20

76

46

64

 6 

80

16

31

24

79

 8 

61

40

65

 9 

70

19

71

 4 

62

29

 5 

33

63

49

68

48

60

41

 2 

Walter Trump found the soution for the 9x9 magic square that retained the most water.  Lets examine this square and follow the logic of its design.  All the largest numbers are used in the lake border ( 65-81). Notice that the largest numbers (78,79,80,81)  are placed as close to the center of the square as possible.  The smallest numbers in the lake border ( 66,67,68,69) are placed as peripherally as possible

 

1 

37

57

47

66

50

53

55

 3 

43

59

11

77

25

75

10

15

54

58

 7 

78

21

34

22

81

12

56

45

73

14

23

32

30

27

74

51

69

17

39

28

52

36

35

26

67

44

72

18

38

42

13

20

76

46

64

 6 

80

16

31

24

79

 8 

61

40

65

 9 

70

19

71

 4 

62

29

 5 

33

63

49

68

48

60

41

 2 

 

 

 

Below ... the next largest numbers (53-65) are used to construct the pond borders.  Note that the largest numbers in the pond borders (59,62,65) are placed as close to the center of the square as possible.

 

 

1 

37

57

47

66

50

53

55

 3 

43

59

11

77

25

75

10

15

54

58

 7 

78

21

34

22

81

12

56

45

73

14

23

32

30

27

74

51

69

17

39

28

52

36

35

26

67

44

72

18

38

42

13

20

76

46

64

 6 

80

16

31

24

79

 8 

61

40

65

 9 

70

19

71

 4 

62

29

 5 

33

63

49

68

48

60

41

 2 

 

 

 

 

Notice that the next largest number (52) is placed in the c enter of the square

 

 

1 

37

57

47

66

50

53

55

 3 

43

59

11

77

25

75

10

15

54

58

 7 

78

21

34

22

81

12

56

45

73

14

23

32

30

27

74

51

69

17

39

28

52

36

35

26

67

44

72

18

38

42

13

20

76

46

64

 6 

80

16

31

24

79

 8 

61

40

65

 9 

70

19

71

 4 

62

29

 5 

33

63

49

68

48

60

41

 2 

 

 

 

The next sequence of largest numbers ( 44-51) is shown below.  If you have the largest numbers placed peripherally, they are not going to be in the water retaining areas displacing water.

 

1 

37

57

47

66

50

53

55

 3 

43

59

11

77

25

75

10

15

54

58

 7 

78

21

34

22

81

12

56

45

73

14

23

32

30

27

74

51

69

17

39

28

52

36

35

26

67

44

72

18

38

42

13

20

76

46

64

 6 

80

16

31

24

79

 8 

61

40

65

 9 

70

19

71

 4 

62

29

 5 

33

63

49

68

48

60

41

 2 

 

 

 

A 3d viewer is now available for the water retention concept

 

http://users.eastlink.ca/~sharrywhite/Download.html

 

 

Order 13 max retention

Order 13

 

 

 

Maximum Retention 28 x 28 Magic Square ... Jarek Wroblewski

 

 

 

http://users.eastlink.ca/~sharrywhite/Download.html

 

Harry White has done a superb job in making the 3d viewer available for the water retention concept.  ( see link above)