Reciprocal Arrangements

by Simon Whitechapel

Some magic squares are born, not made. Take, for example, the reciprocal 1/19 in base 10. Unlike 1/3 in the same base, which repeats after one digit (0·333...), it repeats after 18 digits, like this:

0·0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0...

The remainders at each step are:

1 10 5 12 6 3 11 15 17 18 9 14 7 13 16 8 4 2 1 10 5 12 6 3 11 15 17 18 9 14 7 13 16 8 4 2 1 10 5 12 6 3 11 15 17 18 9 14 7 13 16 8 4 2 1...

Because all the numbers 1 through 18 are represented in the remainders, the digits of the multiples of 1/19 (2/19, 3/19, and so on) are cyclic permutations of the digits of 1/19, like this:

01/19 = 0·0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1... 02/19 = 0·1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2... 03/19 = 0·1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3... 04/19 = 0·2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4... 05/19 = 0·2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5... 06/19 = 0·3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6... 07/19 = 0·3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7... 08/19 = 0·4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8... 09/19 = 0·4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9... 10/19 = 0·5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0... 11/19 = 0·5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1... 12/19 = 0·6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2... 13/19 = 0·6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3... 14/19 = 0·7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4... 15/19 = 0·7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5... 16/19 = 0·8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6... 17/19 = 0·8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7... 18/19 = 0·9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8...

Each row sums to 81, and because each row is a cyclic permutation of the digits of 1/19, so does each column. And, in the particular case of 1/19, so do the digits of the left-right and right-left diagonal:

Note: If the digits of one diagonal sum to the magic total of each row and column, so must the digits of the other diagonal: each digit, d, in one diagonal has a mirrored digit in the other that equals 9-d (see above). The sum of both diagonals therefore equals 9 x 18, or 162. If the sum of one diagonal is 81, the sum of the other must be 162 - 81 = 81. Similar reasoning applies to other prime reciprocal magic squares.

1/19 therefore forms a true magic square, or rather, because it only uses the digits 0-9, a true pseudo-magic square.

Pseudo-magic squares like this are rare: the next in base 10 is formed by the multiples of 1/383.^* When you add other bases, however, they become slightly more common, and the table below lists a selection of them, giving the prime, base, and magic total (derived from the formula base-1 x num-1 / 2).

Prime
Base
Magic Total

19 10 81

53 12 286

53 34 858

59 2 29

67 2 33

83 2 41

89 19 792

167 68 5,561

199 41 3,960

199 150 14,751

211 2 105

223 3 222

293 147 21,316

307 5 612

383 10 1,719

389 360 69,646

397 5 792

421 338 70,770

487 6 1,215

503 420 105,169

587 368 107,531

593 3 592

631 87 27,090

677 407 137,228

757 759 286,524

787 13 4,716

811 3 810

977 1,222 595,848

1,033 11 5,160

1,187 135 79,462

1,307 5 2,612

1,499 11 7,490

1,637 509 415,544

1,877 19 16,884

1,877 553 517,776

1,931 2,898 2,795,605

1,933 146 140,070

2,011 26 25,125

2,027 2 1,013

2,141 63 66,340

2,179 448 486,783

2,417 4,029 4,865,824

2,539 2 1,269

2,579 1,720 2,215,791

2,657 3,322 4,410,288

2,843 2,686 3,815,385

3,187 97 152,928

3,373 11 16,860

3,631 420 760,485

3,659 126 228,625

3,947 35 67,082

4,001 421 840,000

4,019 2,011 4,038,090

4,127 322 662,223

4,261 2 2,130

4,337 239 515,984

4,357 2,764 6,017,814

4,813 2 2,406

5,101 539 1,371,900

5,527 67 182,358

5,647 75 208,902

5,861 3,664 10,732,590

5,867 85 246,372

6,067 542 1,640,853

6,113 3 6,112

6,197 75 229,252

6,277 2 3,138

6,389 133 421,608

6,857 1,696 5,810,460

6,869 1,273 4,368,048

7,019 114 396,517

7,283 2 3,641

7,351 15 51,450

8,117 31 121,740

8,387 2 4,193

8,609 38 159,248

9,029 110 492,026

10,061 12 55,330

10,133 14 65,858

10,259 181 923,220

10,289 585 3,004,096

10,457 29 146,384

10,457 61 313,680

10,529 78 405,328

10,667 8 37,331

10,831 7 32,490

10,859 8 38,003

11,503 55 310,554

11,831 94 550,095

11,813 436 2,569,110

13,463 45 296,164

13,513 14 87,828

13,627 5 27,252

13,913 193 1,335,552

14,543 91 654,390

15,319 3 15,318

15,373 2 7,686

15,749 17 125,984

16,349 2 8,174

16,573 15 116,004

17,327 131 1,126,190

22,123 3 22,122

22,259 6 55,645

22,409 3 22,408

22,817 3 22,816

24,229 6 60,570

24,859 3 24,858

30,707 2 15,353

31,357 60 925,002

Magic prime reciprocals

*The list of magic prime reciprocals in base 10 begins 19, 383, 32327, 34061, 45341, 61967, 65699, 117541, 158771... (see On-Line Encyclopedia of Integer Sequences).

Maths Index

Main Index

Prime	Base	Magic Total
19	10	81
53	12	286
53	34	858
59	2	29
67	2	33
83	2	41
89	19	792
167	68	5,561
199	41	3,960
199	150	14,751
211	2	105
223	3	222
293	147	21,316
307	5	612
383	10	1,719
389	360	69,646
397	5	792
421	338	70,770
487	6	1,215
503	420	105,169
587	368	107,531
593	3	592
631	87	27,090
677	407	137,228
757	759	286,524
787	13	4,716
811	3	810
977	1,222	595,848
1,033	11	5,160
1,187	135	79,462
1,307	5	2,612
1,499	11	7,490
1,637	509	415,544
1,877	19	16,884
1,877	553	517,776
1,931	2,898	2,795,605
1,933	146	140,070
2,011	26	25,125
2,027	2	1,013
2,141	63	66,340
2,179	448	486,783
2,417	4,029	4,865,824
2,539	2	1,269
2,579	1,720	2,215,791
2,657	3,322	4,410,288
2,843	2,686	3,815,385
3,187	97	152,928
3,373	11	16,860
3,631	420	760,485
3,659	126	228,625
3,947	35	67,082
4,001	421	840,000
4,019	2,011	4,038,090
4,127	322	662,223
4,261	2	2,130
4,337	239	515,984
4,357	2,764	6,017,814
4,813	2	2,406
5,101	539	1,371,900
5,527	67	182,358
5,647	75	208,902
5,861	3,664	10,732,590
5,867	85	246,372
6,067	542	1,640,853
6,113	3	6,112
6,197	75	229,252
6,277	2	3,138
6,389	133	421,608
6,857	1,696	5,810,460
6,869	1,273	4,368,048
7,019	114	396,517
7,283	2	3,641
7,351	15	51,450
8,117	31	121,740
8,387	2	4,193
8,609	38	159,248
9,029	110	492,026
10,061	12	55,330
10,133	14	65,858
10,259	181	923,220
10,289	585	3,004,096
10,457	29	146,384
10,457	61	313,680
10,529	78	405,328
10,667	8	37,331
10,831	7	32,490
10,859	8	38,003
11,503	55	310,554
11,831	94	550,095
11,813	436	2,569,110
13,463	45	296,164
13,513	14	87,828
13,627	5	27,252
13,913	193	1,335,552
14,543	91	654,390
15,319	3	15,318
15,373	2	7,686
15,749	17	125,984
16,349	2	8,174
16,573	15	116,004
17,327	131	1,126,190
22,123	3	22,122
22,259	6	55,645
22,409	3	22,408
22,817	3	22,816
24,229	6	60,570
24,859	3	24,858
30,707	2	15,353
31,357	60	925,002