Auld Lang Syne
Image: Wikipedia

The yacht club, which was directly across from the island, would always have a big New Year’s party. If the wind was blowing from that direction to the Rock, you could actually hear people laughing, you could hear music, you could hear girls laughing. You know, you could hear all the sounds coming from the free world, at the Rock. And New Year’s was always the night we heard it.

There was never a day you didn’t see what the hell you were losing, and what you were missing, you know. It was all there for you to see. There’s life. There’s everything I want in my life, and it’s there. It’s a mile or a mile and half away. And yet I can’t get to it.

— Jim Quillen, Alcatraz Inmate 586, in narration recorded for the self-guiding tour


There was once a man who married a sweet little wife; but when he set out with her from her father’s house, he found that she had never been taught to walk. They had a long way to go, and there was nothing for him to do but to carry her; and as he carried her she grew heavier and heavier.

Then they came to a wide, deep river, and he found that she had never been taught to swim. So he told her to hold fast to his shoulder, and started to swim with her across the river. And as he swam she grew frightened, and dragged him down in her struggles. And the river was deep and wide, and the current ran fast; and once or twice she nearly had him under. But he fought his way through, and landed her safely on the other side; and behold, he found himself in a strange country, beyond all imagining delightful. And as he looked about him and gave thanks, he said to himself:

‘Perhaps if I hadn’t had to carry her over, I shouldn’t have kept up long enough to get here myself.’

— Edith Wharton, The Valley of Childish Things, and Other Emblems, 1896

One to Go

33 = 88661289752875283 + (-8778405442862239)3 + (-2736111468807040)3

That result was discovered by Andrew Booker of the University of Bristol just this year.

It leaves 42 as the only positive integer less than 100 that has not been represented as the sum of three cubes.

(We can omit numbers that give a remainder of 4 or 5 when divided by 9, since those are known to be ineligible. But can every other integer be expressed in this way? It’s an open problem.)

(Thanks, Kate.)

09/06/2019 UPDATE: The case of 42 has now been solved, by Andrew Booker at Bristol and Andrew Sutherland at MIT:

42 = (-80538738812075974)3 + 804357581458175153 + 126021232973356313

The lowest unsolved case is now 114.

In a Word

n. a detailed imaginary world, especially one created by a child

When English curate Patrick Brontë brought home a box of wooden soldiers in June 1829, his 12-year-old son Branwell shared them with his sisters. “This is the Duke of Wellington! It shall be mine!” cried 13-year-old Charlotte, and 11-year-old Emily and 9-year-old Anne took up heroes of their own. In the children’s shared imagination, the “Young Men” traveled to the west coast of Africa; settled there after a war with the indigenous Ashantee tribes; elected Arthur Wellesley, the Duke of Wellington, as their leader; and founded the Great Glass Town at the delta of the River Niger.

After 1831 Emily and Ann “seceded” to create a separate imaginary country, Gondal, and after 1834 Charlotte and Branwell developed Glass Town into yet another imaginary nation, Angria. In various combinations the four edited magazines, wrote histories, and composed stories, poems, and plays about these shared fantasy worlds, with alliances, feuds, and love affairs that play out across Africa and the Pacific.

These writings eventually filled 484 pages before maturing interests inevitably sent the Brontës in different directions, but this early work helped to shape the themes and styles of their later poems and novels.

Another Puzzling Commute

A few weeks after his first confusing journey home from the train station, Smith again finishes work ahead of schedule and takes an early train home. This time he arrives at his suburban station half an hour early. Again, rather than wait for the chauffeur, he starts walking home. And as before, he meets his chauffeur on the road, who picks him up promptly and takes him home. How many minutes early do they reach the house this time?

Click for Answer

Image: Wikimedia Commons

Canadian puzzle designer Jeff Widderich invented this game in 2007. The goal is to place a digit 1-9 in each white cell so that each crossword-style “word” contains a straight, that is, a set of consecutive numbers in some order. For example, the top row of five digits might contain 62534, but not 91548.

One other constraint: Each full row or column must contain no repeated digits. That means, for example, that each of the two long vertical “words” will contain all nine digits. The digits in black cells count toward this constraint — the 9 in the black cell near the center means that no 9 appears elsewhere in its row or column. Can you complete the rest of the diagram?

Click for Answer

An Unsolved Puzzle

Sam Loyd’s 1903 Eighth Book of Tan explores the world of tangrams, the pastime of constructing specified shapes from a given set of seven pieces:

The book includes a few “paradoxes,” two of which I’ve mentioned here before. But here’s another:

“The seventh and eighth figures represent the mysterious square, built with seven pieces; then with one corner clipped off, and still the same seven pieces employed.”

The book includes no solution. The square on the left is just the regular “block” formation above. But if anyone has discovered how Loyd produced a “clipped” square using the same seven pieces, I haven’t been able to find it.

12/22/2020 UPDATE: Reader Alon Shaham came up with this solution:

shaham solution

Here the seven pieces are used to make both figures, rather than each figure separately. Arguably Loyd’s description is artfully ambiguous. (Thanks also to reader Andrew Davison.)