From Lee Sallows:
From Lee Sallows:
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.
From Lee Sallows, “A novel geometric proof that 16 is a square number.” :)
Lee Sallows sent this self-descriptive rectangle tiling: The grid catalogs its own contents by arranging its 70 letters and 14 spaces into 14 itemizing phrases.
Bonus: The rectangle measures 7 × 12, which is commemorated by the two strips that meet in the top left-hand corner. And “The author’s signature is also incorporated.”
Belle Gunness was one of America’s most prolific female serial killers, luring lonely men to her Indiana farm with promises of marriage, only to rob and kill them. In this week’s episode of the Futility Closet podcast we’ll tell the story of The LaPorte Black Widow and learn about some of her unfortunate victims.
We’ll also break back into Buckingham Palace and puzzle over a bet with the devil.
In 1986 British electronics engineer Lee Sallows invented the alphamagic square:
As in an ordinary magic square, each row, column, and long diagonal produces the same sum. But when the number in each cell is replaced by the length of its English name (25 -> TWENTY-FIVE -> 10), a second magic square is produced:
Now British computer scientist Chris Patuzzo, who found the percentage-reckoned pangram that we covered here in November 2015, has created a double alphamagic square:
Each row, column, and long diagonal here totals 303370120164. If the number in each cell is replaced by the letter count of its English name (using “and” after “hundred,” e.g. ONE HUNDRED AND FORTY-EIGHT BILLION SEVEN HUNDRED AND TWENTY-EIGHT MILLION THREE HUNDRED AND SEVENTY-EIGHT THOUSAND THREE HUNDRED AND SEVENTY-EIGHT), then we get a new magic square, with a common sum of 345:
And this is itself an alphamagic square! Replace each number with the length of its name and you get a third magic square, this one with a sum of 60:
Chris has found 50 distinct doubly alphamagic squares, listed here. I suppose there must be some limit to this — is a triple alphamagic square even possible?
(Thanks, Chris and Lee.)
Lee Sallows has been working on a new experiment in self-reference that he calls self-descriptive squares, arrays of numbers that inventory their own contents. Here’s an example of a 4×4 square:
The sums of the rows and columns are listed to the right and below the square. These sums also tally the number of times that each row’s rightmost entry, or each column’s lowermost entry, appears in the square. So, for example, the sum of the top row is 3, and that row’s rightmost entry is 1; correspondingly, the number 1 appears three times in the square. Likewise, the sum of the rightmost column is 2, and the lowermost entry in that column, 4, appears twice in the square.
In this example this property extends to the diagonals — and, pleasingly, each sum applies to both ends of its diagonal. The northwest-southeast diagonal totals 2, and both -2 and 4 appear twice in the square. And the southwest-northeast diagonal totals 3, and both 1 and 0 appear three times.
“Easy to understand, but not so easy to produce!” he writes. “I’m still in the throes of figuring out the surprisingly complicated theory of such squares. It turns out there are just two basic squares of 3×3. One of them can be found at the centre of this 5×5 example, which is therefore a concentric self-descriptive square:”