Pangrammatic Loops

A marvelous variation on self-inventorying lists, from the inimitable Lee Sallows:

Recalling that a self-enumerating pangram corresponds to a closed loop of length 1, here follows a loop of length 2, which is to say, a pair of pangrams that enumerate each other. The pangrams are both minimal in the sense of containing none but essential letters with no “and”s or other devices openly or surreptitously added.

ONE A, ONE B, ONE C, ONE D, THIRTYONE E, FOUR F, ONE G, FIVE H, FIVE I, ONE J, ONE K, ONE L, ONE M, TWENTYTWO N, SEVENTEEN O, ONE P, ONE Q, SEVEN R, FOUR S, ELEVEN T, THREE U, FIVE V, FOUR W, ONE X, THREE Y, ONE Z.

ONE A, ONE B, ONE C, ONE D, THIRTYTWO E, SEVEN F, ONE G, FOUR H, FIVE I, ONE J, ONE K, TWO L, ONE M, TWENTY N, NINETEEN O, ONE P, ONE Q, SEVEN R, THREE S, NINE T, FOUR U, SEVEN V, THREE W, ONE X, THREE Y, ONE Z.

An alternative (non-minimal) pair includes plural s’s:

ONE A, ONE B, ONE C, ONE D, TWENTYSEVEN E’S, SIX F’S, ONE G, THREE H’S, SIX I’S, ONE L, TWENTY N’S, SIXTEEN O’S, ONE P, ONE Q, SIX R’S, NINETEEN S’S, TWELVE T’S, FOUR U’S, FOUR V’S, FIVE W’S, THREE X’S, FOUR Y’S, ONE Z.

ONE A, ONE B, ONE C, ONE D, TWENTYNINE E’S, FIVE F’S, ONE G, THREE H’S, SEVEN I’S, ONE J, ONE K, TWO L’S, ONE M, TWENTY N’S, SIXTEEN O’S, ONE P, ONE Q, SIX R’S, TWENTY S’S, TEN T’S, FOUR U’S, THREE V’S, FOUR W’S, FIVE X’S, THREE Y’S, ONE Z.

In similar vein, pangrammatic loops of length 3 follow, but now in shorthand, using arabic numerals to stand for number words, i.e. 1 = one, 2 = two, etc. The first list is enumerated by the second, the second by the third and the third by the first. The 1st loop contains minimal pangrams, the 2nd, pangrams with plural s’s:

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z
1  1  1  1 31  5  1  5  9  1  1  1  1 20 16  1  1  5  5 11  1  4  3  4  2  1
1  1  1  1 28  7  1  3  8  1  1  2  1 20 18  1  1  5  2  8  3  6  3  2  3  1
1  1  1  1 31  2  5  9  7  1  1  1  1 16 15  1  1  5  3 16  1  3  6  2  3  1

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z
1  1  1  1 32  5  2  3  7  1  1  1  1 22 18  1  1  3 19 14  2  6  7  2  3  1
1  1  1  1 32  3  2  6  6  1  1  1  1 20 18  1  1  6 19 16  2  4  7  2  3  1
1  1  1  1 27  2  2  5  8  1  1  1  1 19 17  1  1  5 21 14  2  2  6  5  3  1

Here also a minimal pangrammatic loop of length 4 (no equivalent using plural s’s exists):

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z
1  1  1  1 25  4  2  4  7  1  1  2  1 16 18  1  1  5  5 11  3  4  5  4  2  1
1  1  1  1 28  9  2  3  7  1  1  2  1 16 18  1  1  6  3  9  5  7  5  2  2  1
1  1  1  1 30  3  3  5  9  1  1  1  1 20 15  1  1  3  5 12  1  5  6  3  2  1
1  1  1  1 30  6  1  6  8  1  1  2  1 17 14  1  1  6  2 12  1  5  4  2  3  1

“There exist no minimal pangrammatic loops of length 5 or longer until we reach lengths 10, 33, and 55 (no plural s’s) and lengths 15, 22, 23, 207 and 312 (with plural s’s),” he adds. “This completes what I believe to be an exhaustive survey of all self-enumerating minimal pangrammatic loops.”

(Thanks, Lee.)

A New Find

From Lee Sallows, a remarkable new self-inventorying list:

ONE A, ONE B, ONE C, ONE D, TWENTYEIGHT E, SEVEN F, FIVE G, FIVE H, EIGHT I, ONE J, ONE K, ONE L, ONE M, EIGHTEEN N, EIGHTEEN O, ONE P, ONE Q, FOUR R, TWO S, TEN T, FOUR U, FIVE V, FOUR W, ONE X, TWO Y, ONE Z

“I may be wrong, but I believe it to be the most concise self-descriptive (or ‘self-enumerating’) English pangram yet discovered, with as many as 12 of its 26 letters occurring just once.”

(Thanks, Lee!)

12/18/2019 UPDATE: We’ve learned that the same self-descriptive pangram had already been found in 1998 by Gilles Esposito-Farese, in collaboration with Éric Angelini and Nicolas Graner.

Sums and Sums

lee sallows self-descriptive magic square

Something new from Lee Sallows: a self-descriptive magic square. Each row, column, and long diagonal adds up to 20, and every letter used is correctly counted.

“You may notice that the square includes a fox. But don’t be foxed by the fox. Just enjoy him. For this is not merely any old fox. No, it is our old friend the quick brown fox that jumped over that lazy dog!”

(Thanks, Lee!)

The Egyptian Lo Shu

https://commons.wikimedia.org/wiki/File:Magic_Square_Lo_Shu.svg

Another contribution from Lee Sallows:

“The smallest, oldest and most famous magic square of all is the specimen of Chinese origin known as the Lo shu. In this, the numbers from 1 to 9 are so placed that their sum taken in any row, column or diagonal is 15. This is another way of saying that the sum of any three of them lying in a straight line is 15. Less well known is the ‘Egyptian’ Lo shu (seen below) in which the same numbers are rearranged in a triangular formation that exhibits the same property.”

(From his book Geometric Magic Squares, 2013.) (Thanks, Lee.)

sallows egyptian lo shu

Podcast Episode 219: The Greenbrier Ghost

https://commons.wikimedia.org/wiki/File:ZonaHeasterShue.jpg

In 1897, shortly after Zona Shue was found dead in her West Virginia home, her mother went to the county prosecutor with a bizarre story. She said that her daughter had been murdered — and that her ghost had revealed the killer’s identity. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Greenbrier Ghost, one of the strangest courtroom dramas of the 19th century.

We’ll also consider whether cats are controlling us and puzzle over a delightful oblivion.

See full show notes …