A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number.
Consider a number divided into one, like 1/3 or 1/7. In base ten, the remainder, and so the digits, of 1/3 repeats at once: 0·3333... However, the remainders of 1/7 repeat over six, or 7-1, digits: 1/7 = 0·142857142857142857... If you examine the multiples of 1/7, you can see that each is a cyclic permutation of these six digits:
1/7 = 0·1 4 2 8 5 7... 2/7 = 0·2 8 5 7 1 4... 3/7 = 0·4 2 8 5 7 1... 4/7 = 0·5 7 1 4 2 8... 5/7 = 0·7 1 4 2 8 5... 6/7 = 0·8 5 7 1 4 2...If the digits are laid out as a square, it is obvious that each row will sum to 1+4+2+8+5+7, or 27, and only slightly less obvious that each column will also do so, and consequently we have a magic square:
1 4 2 8 5 7 2 8 5 7 1 4 4 2 8 5 7 1 5 7 1 4 2 8 7 1 4 2 8 5 8 5 7 1 4 2However, neither diagonal sums to 27, but all other prime reciprocals in base ten with maximum period of p-1 produce squares in which all rows and columns sum to the same total.
Other properties of Prime Reciprocals: Midy's theorem
The repeating pattern of an even number of digits in the quotients when broken in half are the nines-complement of each half:
1/7 = 0.142,857,142,857 ... +0.857,142 --------- 0.999,999 1/11 = 0.09090,90909 ... +0.90909,09090 ----- 0.99999,99999 1/13 = 0.076,923 076,923 ... +0.923,076 --------- 0.999,999 1/17 = 0.05882352,94117647 +0.94117647,05882352 ------------------- 0.99999999,99999999 1/19 = 0.052631578,947368421 ... +0.947368421,052631578 ---------------------- 0.999999999,999999999Ekidhikena Purvena From: Bharati Krishna Tirtha's Vedic mathematics#By one more than the one before
Concerning the number of decimal places shifted in the quotient per multiple of 1/19:
01/19 = 0.052631578,947368421 02/19 = 0.1052631578,94736842 04/19 = 0.21052631578,9473684 08/19 = 0.421052631578,947368 16/19 = 0.8421052631578,94736A factor of 2 in the numerator produces a shift of one decimal place to the right in the quotient.
In the square from 1/19, with maximum period 18 and row-and-column total of 81, both diagonals also sum to 81, and this square is therefore fully magic: 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...The same phenomenon occurs with other primes in other bases, and the following table lists some of them, giving the prime, base, and magic total (derived from the formula base-1 x prime-1 / 2):
| Prime | Base | 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 |
| 1,877 | 19 | 16,884 |
| 1,933 | 146 | 140,070 |
| 2,011 | 26 | 25,125 |
| 2,027 | 2 | 1,013 |
| 2,141 | 63 | 66,340 |
| 2,539 | 2 | 1,269 |
| 3,187 | 97 | 152,928 |
| 3,373 | 11 | 16,860 |
| 3,659 | 126 | 228,625 |
| 3,947 | 35 | 67,082 |
| 4,261 | 2 | 2,130 |
| 4,813 | 2 | 2,406 |
| 5,647 | 75 | 208,902 |
| 6,113 | 3 | 6,112 |
| 6,277 | 2 | 3,138 |
| 7,283 | 2 | 3,641 |
| 8,387 | 2 | 4,193 |
Famous quotes containing the words prime, reciprocal, magic and/or square:
“In time, after a dozen years of centering their lives around the games boys play with one another, the boys’ bodies change and that changes everything else. But the memories are not erased of that safest time in the lives of men, when their prime concern was playing games with guys who just wanted to be their friendly competitors. Life never again gets so simple.”
—Frank Pittman (20th century)
“I had no place in any coterie, or in any reciprocal self-advertising. I stood alone. I stood outside. I wanted only to learn. I wanted only to write better.”
—Ellen Glasgow (1873–1945)
“The middle years of parenthood are characterized by ambiguity. Our kids are no longer helpless, but neither are they independent. We are still active parents but we have more time now to concentrate on our personal needs. Our children’s world has expanded. It is not enclosed within a kind of magic dotted line drawn by us. Although we are still the most important adults in their lives, we are no longer the only significant adults.”
—Ruth Davidson Bell. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 3 (1978)
“O for a man who is a man, and, as my neighbor says, has a bone in his back which you cannot pass your hand through! Our statistics are at fault: the population has been returned too large. How many men are there to a square thousand miles in this country? Hardly one.”
—Henry David Thoreau (1817–1862)