Assignment 12

 

The unresolved problem

 

 

 

Consider 4 real numbers A, B, C, and D arranged in a row and their successive differences in absolute terms as shown in the array below:

 

                                                                                                           

 

                                                                           

 

                        

 

 

If the procedure is continued for the fourth row, fifth row and so on, we ultimately end with a row having 4 zeros. The aim is to have a maximum number of non-zero zeros.

For example, if A = 1, B = 4, C = 5, and D = 9, we have the following structure.

Example 1:

 

Difference

A

B

C

D

 

1

4

5

9

1st

3

1

4

8

2nd

2

3

4

5

3rd

1

1

1

3

4th

0

0

2

2

5th

0

2

0

2

6th

2

2

2

2

 

0

0

0

0

 

Table 12.1

 

 

 

 

We observe that after the 6th difference all the terms are zero. The problem is to find a set of four numbers such that we have a maximum of non-zero rows.

 

Example 2:

 

 

Difference

A

B

C

D

1st

900

4

500

260

2nd

896

496

240

640

3rd

400

256

400

256

4th

144

144

144

144

 

0

0

0

0

 

Table 12.2

 

 

 

Observe that the maximum difference in each row decreases as we move down the rows. Further, the row just above the final zero row has all even terms or are divisible by 2. 

 

Example 3:

In this example, we select four real numbers.

 

Difference

A

B

C

D

1st

0.9

0.4

0.7

1.5

2nd

0.5

0.3

0.8

0.6

3rd

0.2

0.5

0.2

0.1

4th

0.3

0.3

0.1

0.1

5th

0

0.2

0

0.2

6th

0.2

0.2

0.2

0.2

 

0

0

0

0

 

Table 12.3

 

 

 

The problem has a ÔcyclicÕ nature since the difference between the first and fourth numbers in each row are being considered. The four numbers in example 1 have been represented on a circle to give another representation of the problem.

 

 

 

 

 

 

 

 

Figure 12.1

 

 

 

A number of possibilities for A, B, C, and D have been tried out in attempt to maximize the number of non-zero rows. If we choose A, B, C, and D such that , , and , then this increases the number of non-zero rows substantially for certain values of A. For instance, if we choose A = 1.83928675521416, we have 61 non-zero rows as shown in Table 12.4. 

 

 

 

A

B

C

D

1

1.839286755

3.382975768

6.222262523

11.44452505

2

1.543689013

2.839286755

5.222262523

9.605238291

3

1.295597743

2.382975768

4.382975768

8.061549278

4

1.087378025

2

3.67857351

6.765951536

5

0.912621975

1.67857351

3.087378025

5.67857351

6

0.765951536

1.408804515

2.591195485

4.765951536

7

0.642852979

1.18239097

2.174756051

4

8

0.539537991

0.992365081

1.825243949

3.357147021

9

0.45282709

0.832878869

1.531903072

2.81760903

10

0.380051779

0.699024203

1.285705958

2.36478194

11

0.318972424

0.586681755

1.079075982

1.984730161

12

0.267709331

0.492394227

0.905654179

1.665757737

13

0.224684896

0.413259953

0.760103558

1.398048406

14

0.188575057

0.346843605

0.637944849

1.17336351

15

0.158268548

0.291101244

0.535418662

0.984788453

16

0.132832696

0.244317418

0.449369791

0.826519906

17

0.111484722

0.205052373

0.377150114

0.69368721

18

0.093567651

0.172097741

0.316537096

0.582202487

19

0.07853009

0.144439355

0.265665392

0.488634836

20

0.065909264

0.121226037

0.222969445

0.410104746

21

0.055316773

0.101743407

0.187135302

0.344195482

22

0.046426635

0.085391894

0.15706018

0.288878709

23

0.03896526

0.071668286

0.131818529

0.242452074

24

0.032703026

0.060150243

0.110633545

0.203486815

25

0.027447217

0.050483302

0.092853269

0.170783789

26

0.023036086

0.042369967

0.077930519

0.143336572

27

0.019333881

0.035560552

0.065406052

0.120300486

28

0.016226671

0.0298455

0.054894434

0.100966605

29

0.013618829

0.025048934

0.046072171

0.084739934

30

0.011430104

0.021023238

0.038667763

0.071121105

31

0.009593133

0.017644525

0.032453342

0.059691

32

0.008051392

0.014808816

0.027237659

0.050097867

33

0.006757424

0.012428843

0.022860208

0.042046475

34

0.005671418

0.010431366

0.019186267

0.035289051

35

0.004759947

0.008754901

0.016102784

0.029617632

36

0.003994954

0.007347883

0.013514849

0.024857685

37

0.003352929

0.006166966

0.011342837

0.020862731

38

0.002814037

0.005175871

0.009519895

0.017509802

39

0.002361834

0.004344024

0.007989908

0.014695765

40

0.00198219

0.003645884

0.006705858

0.012333932

41

0.001663694

0.003059974

0.005628074

0.010351742

42

0.00139628

0.0025681

0.004723667

0.008688048

43

0.00117182

0.002155567

0.00396438

0.007291768

44

0.000983747

0.001808813

0.003327388

0.006119947

45

0.000825067

0.001518574

0.00279256

0.005136201

46

0.000693507

0.001273986

0.002343641

0.004311134

47

0.000580478

0.001069655

0.001967493

0.003617627

48

0.000489177

0.000897838

0.001650133

0.003037148

49

0.000408661

0.000752296

0.001387015

0.002547971

50

0.000343635

0.000634719

0.001160956

0.002139311

51

0.000291084

0.000526237

0.000978354

0.001795676

52

0.000235153

0.000452117

0.000817321

0.001504591

53

0.000216964

0.000365204

0.00068727

0.001269438

54

0.00014824

0.000322066

0.000582168

0.001052474

55

0.000173826

0.000260103

0.000470306

0.000904234

56

8.62769E-05

0.000210203

0.000433928

0.000730408

57

0.000123926

0.000223725

0.00029648

0.000644132

58

9.97991E-05

7.27547E-05

0.000347652

0.000520205

59

2.70444E-05

0.000274897

0.000172554

0.000420406

60

0.000247852

0.000102343

0.000247852

0.000393362

61

0.000145509

0.000145509

0.000145509

0.000145509

 

0

0

0

0

 

 

Table 12.4

 

 

 

If we choose A = 0.543689012692077, we equally have 61 rows as shown in Table 12.5. 

 

 

 

A

B

C

D

1

0.543689013

0.295597743

0.160713245

0.087378025

2

0.24809127

0.134884498

0.073335219

0.456310987

3

0.113206772

0.061549278

0.382975768

0.208219717

4

0.051657494

0.32142649

0.174756051

0.095012945

5

0.269768995

0.146670439

0.079743106

0.043355451

6

0.123098557

0.066927333

0.036387655

0.226413545

7

0.056171224

0.030539677

0.190025889

0.103314988

8

0.025631547

0.159486212

0.086710901

0.047143764

9

0.133854665

0.072775311

0.039567137

0.021512218

10

0.061079355

0.033208174

0.018054919

0.112342448

11

0.027871181

0.015153255

0.094287529

0.051263093

12

0.012717926

0.079134274

0.043024435

0.023391913

13

0.066416348

0.036109839

0.019632523

0.010673987

14

0.030306509

0.016477316

0.008958536

0.055742361

15

0.013829193

0.00751878

0.046783825

0.025435852

16

0.006310413

0.039265045

0.021347974

0.011606659

17

0.032954632

0.017917071

0.009741315

0.005296246

18

0.015037561

0.008175757

0.004445069

0.027658386

19

0.006861804

0.003730688

0.023213317

0.012620826

20

0.003131117

0.01948263

0.010592492

0.005759021

21

0.016351513

0.008890138

0.00483347

0.002627905

22

0.007461375

0.004056668

0.002205566

0.013723608

23

0.003404707

0.001851102

0.011518043

0.006262233

24

0.001553605

0.009666941

0.005255809

0.002857526

25

0.008113335

0.004411131

0.002398284

0.00130392

26

0.003702204

0.002012848

0.001094363

0.006809415

27

0.001689356

0.000918485

0.005715052

0.003107211

28

0.000770872

0.004796567

0.002607841

0.001417854

29

0.004025695

0.002188726

0.001189986

0.000646983

30

0.001836969

0.00099874

0.000543004

0.003378713

31

0.000838229

0.000455736

0.002835709

0.001541744

32

0.000382493

0.002379973

0.001293965

0.000703515

33

0.00199748

0.001086008

0.00059045

0.000321021

34

0.000911472

0.000495557

0.000269429

0.001676458

35

0.000415914

0.000226128

0.001407029

0.000764986

36

0.000189786

0.001180901

0.000642043

0.000349072

37

0.000991115

0.000538858

0.000292971

0.000159286

38

0.000452257

0.000245887

0.000133685

0.000831829

39

0.000206369

0.000112202

0.000698144

0.000379572

40

9.4167E-05

0.000585942

0.000318571

0.000173203

41

0.000491775

0.00026737

0.000145368

7.90361E-05

42

0.000224404

0.000122002

6.63322E-05

0.000412739

43

0.000102403

5.56697E-05

0.000346406

0.000188334

44

4.67329E-05

0.000290737

0.000158072

8.59315E-05

45

0.000244004

0.000132664

7.21408E-05

3.91986E-05

46

0.000111339

6.05236E-05

3.29421E-05

0.000204805

47

5.08159E-05

2.75814E-05

0.000171863

9.34657E-05

48

2.32345E-05

0.000144282

7.83973E-05

4.26498E-05

49

0.000121047

6.58843E-05

3.57474E-05

1.94154E-05

50

5.51628E-05

3.01369E-05

1.63321E-05

0.000101632

51

2.5026E-05

1.38048E-05

8.52997E-05

4.64689E-05

52

1.12212E-05

7.14949E-05

3.88308E-05

2.14429E-05

53

6.02737E-05

3.26641E-05

1.73878E-05

1.02218E-05

54

2.76096E-05

1.52763E-05

7.16604E-06

5.00519E-05

55

1.23332E-05

8.11028E-06

4.28859E-05

2.24424E-05

56

4.22296E-06

3.47756E-05

2.04435E-05

1.01091E-05

57

3.05527E-05

1.43321E-05

1.03344E-05

5.88618E-06

58

1.62206E-05

3.9977E-06

4.44821E-06

2.46665E-05

59

1.22229E-05

4.50514E-07

2.02183E-05

8.44591E-06

60

1.17724E-05

1.97678E-05

1.17724E-05

3.77695E-06

61

7.9954E-06

7.9954E-06

7.9954E-06

7.9954E-06

 

0

0

0

0

 

 

Table 12.5

 

 

 

 

Interestingly, the two numbers 0.543689012692077 and 1.83928675521416 are related,

 

 

i.e., they are reciprocals. The large number of decimal places for the above two values of A shows that accuracy is also a determining factor in the number of non-zero rows.

 

Table 12.6 shows the number of non-zero rows as the accuracy of  A increases from 1.839 (3 d.p) to 1.83928675521416 (14 d.p).

 

 

Value of A

No. of non-zero rows

1.839

19

1.8392

21

1.83928

26

1.839286

30

1.8392867

33

1.83928675

37

1.839286755

42

1.8392867552

46

1.83928675521

49

1.839286755214

54

1.8392867552141

54

1.83928675521416

61

 

 

Table 12.6

 

 

 

The values in Table 12.4 and 12.5 are plotted in Figure 12.1 and Figure 12.2, respectively to show how the numbers decrease along the four columns.

 

 

Figure 12.1

 

 

 

 

 

Figure 12.2

 

 

 

We observe that the values decrease monotically in Figure 12.1 in contrast to Figure 12.2. The values in Table 12.4 have also been plotted along the rows. Once again, the values display some pattern as shown in Figure 12.3.

 

 

Figure 12.3

 

 

The topmost curve is for row 1. The second curve is for row two and so on. The decreasing pattern in Figures 12.1 and 12.2 may suggest a clue to the problem of finding the maximum number of non-zero rows.  

 

 

Ajay Ramful

3 December 2006

 

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