25x25x25 bimagic cube of John-R. Hendricks analyzed

 

John-R. Hendricks constructed in 2000 the first known bimagic cube (see website http://www.multimagie.com/English/Cube.htm). I have analyzed the bimagic 25x25x25 cube. It is possible to use Hendricks' method and to construct 15625 different bimagic 25x25x25 cubes. One result is even symmetric.

The first grid is based on the following 5x5 square:

 

   

65

65

65

65

65

     
 

65

         

65

   

55

 

1

6

11

16

21

     

75

 

15

20

25

5

10

 

65

65

70

 

24

4

9

14

19

 

65

65

65

 

8

13

18

23

3

 

65

65

60

 

17

22

2

7

12

 

65

65

 


The 5x5 square vierkant is not fully magic, because not all rows give thje magic sum of 65.

The first level of the first grid consists of the 25 shifted versions of the 5x5 squares on a 2x2 carpet. It has the following row-column coordinates (e.g. square 5 - 3 has 2 in the top left corner):


1 - 1
  5 - 3   4 - 5   3 - 2   
2 - 4
2 - 5   1 - 2   5 - 4   4 - 1   
3 - 3
3 - 4   2 - 1   1 - 3   5 - 5   
4 - 2
4 - 3   3 - 5   2 - 2   1 - 4   
5 - 1
5 - 2   4 - 4   3 - 1   2 - 3   
1 - 5
 

Level 1 up to 25 of the first grid are the shifted versions of the first level with the following row/column-coordinates on a 2x2 carpet of the first level:

1-1, 4-2, 2-3, 5-4, 3-5, 4-5, 2-1, 5-2, 3-3, 1-4, 2-4, 5-5, 3-1, 1-2, 4-3, 5-3, 3-4, 1-5, 4-1, 2-2, 3-2, 1-3, 4-4, 2-5 and 5-1.

The second grid is based on the diagonal shifted version of the 5x5 square which is used for the construction of the first grid:

 
5x5 square first grid      --> diagonal shift second grid

1

6

11

16

21

   

1

20

9

23

12

15

20

25

5

10

   

19

8

22

11

5

24

4

9

14

19

   

7

21

15

4

18

8

13

18

23

3

   

25

14

3

17

6

17

22

2

7

12

   

13

2

16

10

24

 

 

The 5x5 square of the second grid is a shifted  version of a symmetric and panmagic (= ultra magic) 5x5 square! The first level of the second grid consists of the 25 shifted versions of the 5x5 square on a 2x2 carpet. It has the following row-column coordinates (e.g. square 3 - 1 has 7 in the top left corner):


1 - 1
  3 - 1   5 - 1   2 - 1   
4 - 1
1 - 5   3 - 5   5 - 5   2 - 5   
4 - 5
1 - 4   3 - 4   5 - 4   2 - 4   
4 - 4
1 - 3   3 - 3   5 - 3   2 - 3   
4 - 3
1 - 2   3 - 2   5 - 2   2 - 2   
4 - 2
 


Level 1 up to 25 of the second grid are the shifted versions of the first level with the following row/column-coordinates on a 2x2 carpet of the first level:


1-1, 3-2, 5-3, 2-4, 4-5, 4-3, 1-4, 3-5, 5-1, 2-2, 2-5, 4-1, 1-2, 3-3, 5-4, 5-2, 2-3, 4-4, 1-5, 3-1, 3-4, 5-5, 2-1, 4-2 en 1-3.

The third grid is based on the following 5x5 square:

 

   

15

40

65

90

115

     
 

55

         

65

   

65

 

1

8

15

17

24

     

65

 

4

6

13

20

22

 

65

65

65

 

2

9

11

18

25

 

65

75

65

 

5

7

14

16

23

 

65

60

65

 

3

10

12

19

21

 

65

70


  
The square vierkant is not fully magic, not all rows and not all (pan)diagonals give the magic sum of 65.

The first level of the third grid consists of the 25 shifted versions of the 5x5 square on a 2x2 carpet. It has the following row-column coordinates (e.g. square 4 - 2 has 7 in the top left corner):


1 - 1
   4 - 2   2 - 3   5 - 4  
3 - 5
4 - 4   2 - 5   5 - 1   3 - 2  
1 - 3
2 - 2   5 - 3   3 - 4   1 - 5  
4 - 1
5 - 5   3 - 1   1 - 2   4 - 3  
2 - 4
3 - 3   1 - 4   4 - 5   2 - 1  
5 - 2

 

 

Level 1 up to 25 of the third grid are the shifted versions of the first level with the following row/column-coordinates on a 2x2 carpet of the first level:

1-1, 1-3, 1-5, 1-2, 1-4, 2-4, 2-1, 2-3, 2-5, 2-2, 3-2, 3-4, 3-1, 3-3, 3-5, 4-5, 4-2, 4-4, 4-1, 4-3, 5-3, 5-5, 5-2, 5-4 and 5-1.

It is possible to use in each grid one of the 25 shifted versions of the 5x5 squares. So there are 25x25x25 is 15625 different possibilities to construct a Hendricks' bimagic 25x25x25 cube. It is even possible to construct a symmetric version; see below the 13th (= middle) level:
  

 

13th (= middle) level of the symmetric version of Hendricks' bimagic 25x25x25 cube

6875

10955

15060

1040

5145

10021

14851

206

4936

9041

13922

3002

4107

8212

9817

2198

3153

7883

12113

13718

6099

7054

11784

12764

1369

8434

9414

14144

2749

4329

12335

13315

2420

3400

7605

13106

1586

5691

7296

11376

632

5487

6467

11197

15277

4533

9263

10368

14473

428

6893

11748

12703

1808

5913

10794

15524

979

5084

6689

14695

50

4755

8985

10590

2966

3946

8651

9631

13861

3742

7847

11927

13532

2012

9202

10182

14912

267

4497

9978

14083

2563

4168

8273

13129

2359

3339

8069

12174

1405

6135

7240

11345

12950

5301

6281

11011

15241

1221

7661

12391

13496

1951

3556

11562

12542

1647

5852

7457

15463

818

5548

6503

10733

614

4719

8824

10404

14509

3765

8620

9600

14305

2785

129

4984

9089

10069

14799

4030

8135

9865

13970

3075

7926

12031

13636

2241

3221

11827

12807

1287

6017

7122

15103

1083

5188

6793

10898

2463

3443

7548

12253

13358

5739

7344

11449

13029

1509

6390

11245

15350

680

5410

10286

14391

496

4576

9306

14187

2667

4272

8477

9457

922

5002

6732

10837

15567

4823

8903

10508

14738

93

8724

9679

13784

2889

3994

12000

13580

2060

3665

7770

12646

1851

5956

6936

11666

2606

4211

8316

9921

14001

3257

8112

12217

13197

2277

7158

11263

12993

1473

6178

11059

15164

1144

5374

6329

14960

315

4420

9150

10230

1695

5800

7380

11610

12590

5591

6571

10651

15381

861

8867

10472

14552

532

4637

9518

14373

2828

3808

8538

13419

1899

3604

7709

12439

9783

13888

3118

4098

8178

13684

2164

3144

7999

12079

1335

6065

7045

11775

12855

5231

6836

10941

15046

1001

9007

10112

14842

197

4902

11492

13097

1552

5657

7262

15268

748

5453

6433

11163

419

4524

9354

10334

14439

4320

8425

9380

14235

2715

7591

12321

13276

2381

3486

10551

14656

11

4866

8971

13827

2932

3912

8642

9747

2103

3708

7813

11918

13523

5879

6984

11714

12694

1799

6655

10760

15615

970

5075

12140

13245

2350

3305

8035

12911

1391

6246

7201

11306

1187

5292

6272

11102

15207

4463

9193

10173

14878

358

8364

9969

14074

2529

4134

10724

15429

784

5514

6619

14625

580

4685

8790

10395

2771

3851

8581

9561

14291

3547

7627

12482

13462

1942

7448

11528

12508

1738

5843

3187

7917

12022

13727

2207

7088

11818

12798

1253

6108

10989

15094

1074

5154

6759

14765

245

4975

9055

10035

3036

4016

8246

9826

13931

5396

6476

11206

15311

666

9297

10252

14482

462

4567

9448

14153

2633

4363

8468

13349

2429

3409

7514

12369

1625

5705

7310

11415

13020

3960

8690

9670

13775

2980

7856

11961

13566

2046

3626

11632

12737

1842

5947

6902

15533

888

5118

6723

10803

59

4789

8894

10624

14704

6169

7149

11354

12959

1439

6320

11050

15130

1235

5340

10216

14946

276

4381

9236

14117

2597

4177

8282

9887

2268

3373

8078

12183

13163

4728

8833

10438

14543

523

8504

9609

14339

2819

3799

12405

13385

1990

3595

7700

12551

1656

5761

7491

11596

827

5557

6537

10642

15497

12841

1321

6026

7006

11861

1117

5222

6802

10907

15012

4893

9123

10078

14808

163

8169

9774

13979

3084

4064

12070

13675

2130

3235

7965

14405

385

4615

9345

10325

2676

4281

8386

9491

14221

3452

7557

12287

13267

2497

7353

11458

13063

1543

5648

11129

15359

714

5444

6424

13614

2094

3699

7779

11884

1765

5995

6975

11680

12660

5036

6641

10871

15576

931

8937

10542

14647

102

4832

9713

13818

2923

3878

8733

15198

1153

5258

6363

11093

349

4429

9159

10139

14994

4250

8330

9935

14040

2520

8021

12226

13206

2311

3291

11297

12877

1482

6212

7192

14257

2862

3842

8572

9527

1908

3513

7743

12473

13428

5809

7414

11519

12624

1704

6585

10690

15420

775

5605

10481

14586

566

4671

8751

 

 
See all grids and all levels of the 25x25x25 bimagic cube in the download below:

 

 

Download
25x25x25, Bimagic 25x25x25 cube, grids.x
Microsoft Excel werkblad 1.8 MB
Download
25x25x25, Bimagic 25x25x25 cube, result.
Microsoft Excel werkblad 3.6 MB