Lat’s That

In a magic square of numbers, all rows, columns and diagonals have the same sum, or magic total. Here is an example:

1*5*9
8*3*4
6*7*2

(mt=15)

Here’s another:

06*07*11*10
15*02*14*03
04*13*01*16
09*12*08*05

(mt=34)

And another:

04*25*20*10*06
01*13*11*21*19
23*09*07*08*18
15*16*03*14*17
22*02*24*12*05

(mt=65)

And another:

35*15*10*18*11*22
05*25*33*12*07*29
34*30*04*14*21*08
02*16*27*17*23*26
03*24*09*19*36*20
32*01*28*31*13*06

(mt=111)

In all those magic squares, the magic total is fixed: the sum of all numbers from 1 to 36 is 666, so any individual line in a 6×6 magic square has to equal 666 / 6 or 111. In other kinds of magic figure, this rule doesn’t apply:

2*7*3
4***8
6*5*1

(mt=12)

6*3*4
2***8
5*7*1

(mt=13)

8*5*1
2***6
4*3*7

(mt=14)

8*1*6
4***2
3*5*7

(mt=15)

Continue reading Lat’s That

Advertisements

Who Guards the Guardianistas?

“…We’re not so much a reaction against what’s going on – it’s more down to the music that we’re into – but in terms of guitar music there hasn’t been much in terms of louder groups.” – Bored of cookie-cutter conformity in music?, The Guardian, 6/iii/ 2014.


Elsewhere other-posted:

Ex-term-in-ate!

Priamonds and Pearls

Interesting patterns emerge when primes are represented as white blocks in a series of n-width left-right lines laid vertically, one atop the other. When the line is five blocks wide, the patterns look like this (the first green block is 1, followed by primes 2, 3 and 5, then 7 in the next line):
5line

(Click for larger version)

Right at the bottom of the first column is an isolated prime diamond, or priamond (marked with a green block). It consists of the four primes 307-311-313-317, where the three latter primes equal 307 + 4 and 6 and 10, or 307 + 5-1, 5+1 and 5×2 (the last prime in the first column is 331 and the first prime in the second is 337). About a third of the way down the first column is a double priamond, consisting of 97, 101, 103, 107, 109 and 113. For a given n, then, a priamond is a set of primes, p1, p2, p3 and p4, such that p2 = p1 + n-1, p3 = p + n+1 and p4 = p1 + 2n.

There are also fragments of pearl-necklace in the columns. One is above the isolated priamond. It consists of four prime-blocks slanting from left to right: 251-257-263-269, or 251 + 6, 12 and 18. A prearl-necklace, then, is a set of primes, p1, p2, p3…, such that p2 = p1 + n+i, p3 = p + 2(n+i)…, where i = +/-1. Now here are the 7-line and 9-line:

7line

Above: 7-line for primes

9line

Above: 9-line for primes

In the 9-line, you can see a prime-ladder marked with a red block. It consists of the primes 43-53-61-71-79-89-97-107, in alternate increments of 10 and 8, or 9+1 and 9-1. A prime-ladder, then, is a set of primes, p1, p2, p3, p4…, such that p2 = p1 + n+1, p3 = p + 2n, p3 = p + 3n+1…

And here is an animated gif of lines 5 through 51:

lines5to51

(Click or open in new window for larger version or if file fails to animate)

No Plaice Like Olm

European Reptile and Amphibian Guide by Axel KwetEuropean Reptile and Amphibian Guide, Axel Kwet (New Holland 2009)

An attractive book about animals that are mostly attractive, sometimes strange, always interesting. It devotes photographs and descriptive text to all the reptiles and amphibians found in Europe, from tree-frogs to terrapins, from skinks to slow-worms. Some of the salamanders look like heraldic mascots, some of the lizards like enamel jewellery, and some of the toads like sumo wrestlers with exotic skin-diseases. When you leaf through the book, you’ve moving through several kinds of space: geographic and evolutionary, aesthetic and psychological. Europe is a big place and has a lot of reptilian and amphibian variety, including one species of turtle, the loggerhead, Caretta caretta, and one species of chameleon, the Mediterranean, Chamaeleo chamaeleon.

But every species, no matter how dissimilar in size and appearance, has a common ancestor: the tiny crested newt (Triturus cristatus) to the far north-west in Scotland and the two-and-a-half metre whip snake (Dolichophis caspius) to the far south-east in Greece; the sun-loving Hermann’s tortoise (Testudo hermanni), with its sturdy shell, and the pallid and worm-like olm (Proteus anguinus), which lives in “underground streams in limestone karst country along the coast from north-east Italy to Montenegro” (pg. 55). Long-limbed or limbless, sun-loving or sun-shunning, soft-skinned or scaly – they’re all variations on a common theme.

Sample page

Sample page from European Reptile and Amphibian Guide


And that’s where aesthetic and psychological space comes in, because different species and families evoke different impressions and emotions. Why do snakes look sinister and skinks look charming? But snakes are sinuous too and in a way it’s a shame that a photograph can capture their endlessly varying loops and curves as easily as it can capture the ridigity of a tortoise. At one time a book like this would have had paintings or drawings. Nowadays, it has photographs. The images are more realistic but less enchanted: the images are no longer mediated by the hand, eye and brain of an artist. But some enchantment remains: the glass lizard, Pseudopus apodus, peering from a holly bush on page 199 reminds me of Robert E. Howard’s “The God in the Bowl”, because there’s an alien intelligence in its gaze. Glass lizards are like snakes but can blink and retain “tiny, barely visible vestiges of the hind legs” (pg. 198).

Other snake-like reptiles retain vestiges of the fore-limbs too, like the Italian three-toed skink (Chalcides chalcides). The slow-worm, Anguis fragilis, has lost its limbs entirely, but doesn’t look sinister like a snake and can still blink. Elsewhere, some salamanders have lost not limbs but lungs: the Italian cave salamander, Speleomantes italicus, breathes through its skin and the lining of its mouth. So does Gené’s cave salamander, Speleomantes genei, which is found only on the island of Sardinia. It “emits an aromatic scent when touched” (pg. 54). Toads can emit toxins and snakes can inject venoms: movement in evolutionary space means movement in chemical space, because every alteration in an animal’s appearance and anatomy involves an alteration in the chemicals created by its body. But chemical space is two-fold: genotypic and phenotypic. The genes change and so the products of the genes change. The external appearance of every species is like a bookmark sticking out of the Book of Life, fixing its position in gene-space. You have to open that book to see the full recipe for the animal’s anatomy, physiology and behaviour, though not everything is specified by the genes.

Pleuronectes platessa on the sea-floor

Pleuronectes platessa on the sea-floor


The force of gravity is one ingredient in an animal’s development, for example. So is sunlight or its absence. Or water, sand, warmth, cold. The descendants of that common ancestor occupy many ecological niches. And in fact one of those descendants wrote this book: humans and all other mammals share an ancestor with frogs, skinks and vipers. Before that, we were fish. So a plaice is a distant cousin of an olm, despite the huge difference in their appearance and habitat. One is flat, one is tubular. One lives in the sea, one lives in caves. But step by step, moving through genomic and topological space, you can turn a plaice into an olm. Or into anything else in this book. Just step back through time to the common ancestor, then take another evolutionary turning. One ancestor, many descendants. That ancestor was itself one descendant among many of something even earlier.
Olm in a Slovenian cave

Olm in a Slovenian cave


But there’s another important point: once variety appeared, it began to interact with itself. Evolutionary environment includes much more than the inanimate and inorganic. We mammals share more than an ancestor with reptiles and amphibians: we’ve also shared the earth. So we’re written into their genes and some of them are probably written into ours. Mammalian predators have influenced the evolution of skin-colour and psychology, making some animals camouflaged and cautious, others obtrusive and aggressive. But it works both ways: perhaps snakes seem sinister because we’re born with snake-sensitive instincts. If it’s got no limbs and it doesn’t blink, it might have a dangerous bite. That’s why the snake section of this book seems so different to the salamander section or the frog section. But all are interesting and all are important. This is a small book with some big themes.

Performativizing Papyrocentricity #20

Papyrocentric Performativity Presents:

Clive AliveC.S. Lewis: A Life, Alister McGrath (Hodder & Staughton 2013)

Ink TuneNick Drake: Dreaming England, Nathan Wiseman-Trowse (Reaktion Books 2013)

Stan’s FansAwaydays, Kevin Sampson (Vintage 1998)

Words at WarPoetry of the First World War: An Anthology, ed. Tim Kendall (Oxford University Press 2013) (posted @ Overlord of the Über-Feral)


Or Read a Review at Random: RaRaR

Miss This

1,729,404 is seven digits long. If you drop one digit at a time, you can create seven more numbers from it, each six digits long. If you add these numbers, something special happens:

1,729,404 → 729404 (missing 1) + 129404 (missing 7) + 179404 (missing 2) + 172404 + 172904 + 172944 + 172940 = 1,729,404

So 1,729,404 is narcissistic, or equal to some manipulation of its own digits. Searching for numbers like this might seem like a big task, but you can cut the search-time considerably by noting that the final two digits determine whether a number is a suitable candidate for testing. For example, what if a seven-digit number ends in …38? Then the final digit of the missing-digit sum will equal (3 x 1 + 8 x 6) modulo 10 = (3 + 48) mod 10 = 51 mod 10 = 1. This means that you don’t need to check any seven-digit number ending in …38.

But what about seven-digit numbers ending in …57? Now the final digit of the sum will equal (5 x 1 + 7 x 6) modulo 10 = (5 + 42) mod 10 = 47 mod 10 = 7. So seven-digit numbers ending in …57 are possible missing-digit narcissistic sums. Then you can test numbers ending …157, …257, …357 and so on, to determine the last-but-one digit of the sum. Using this method, one quickly finds the only two seven-digit numbers of this form in base-10:

1,729,404 → 729404 + 129404 + 179404 + 172404 + 172904 + 172944 + 172940 = 1,729,404

1,800,000 → 800000 + 100000 + 180000 + 180000 + 180000 + 180000 + 180000 = 1,800,000

What about eight-digit numbers? Only those ending in these two digits need to be checked: …00, …23, …28, …41, …46, …64, …69, …82, …87. Here are the results:

• 13,758,846 → 3758846 + 1758846 + 1358846 + 1378846 + 1375846 + 1375846 + 1375886 + 1375884 = 13,758,846
• 13,800,000 → 3800000 + 1800000 + 1300000 + 1380000 + 1380000 + 1380000 + 1380000 + 1380000 = 13,800,000
• 14,358,846 → 4358846 + 1358846 + 1458846 + 1438846 + 1435846 + 1435846 + 1435886 + 1435884 = 14,358,846
• 14,400,000 → 4400000 + 1400000 + 1400000 + 1440000 + 1440000 + 1440000 + 1440000 + 1440000 = 14,400,000
• 15,000,000 → 5000000 + 1000000 + 1500000 + 1500000 + 1500000 + 1500000 + 1500000 + 1500000 = 15,000,000
• 28,758,846 → 8758846 + 2758846 + 2858846 + 2878846 + 2875846 + 2875846 + 2875886 + 2875884 = 28,758,846
• 28,800,000 → 8800000 + 2800000 + 2800000 + 2880000 + 2880000 + 2880000 + 2880000 + 2880000 = 28,800,000
• 29,358,846 → 9358846 + 2358846 + 2958846 + 2938846 + 2935846 + 2935846 + 2935886 + 2935884 = 29,358,846
• 29,400,000 → 9400000 + 2400000 + 2900000 + 2940000 + 2940000 + 2940000 + 2940000 + 2940000 = 29,400,000

But there are no nine-digit sumbers, or nine-digit numbers that supply missing-digit narcissistic sums. What about ten-digit sumbers? There are twenty-one:

1,107,488,889; 1,107,489,042; 1,111,088,889; 1,111,089,042; 3,277,800,000; 3,281,400,000; 4,388,888,889; 4,388,889,042; 4,392,488,889; 4,392,489,042; 4,500,000,000; 5,607,488,889; 5,607,489,042; 5,611,088,889; 5,611,089,042; 7,777,800,000; 7,781,400,000; 8,888,888,889; 8,888,889,042; 8,892,488,889; 8,892,489,042 (21 numbers)

Finally, the nine eleven-digit sumbers all take this form:

30,000,000,000 → 0000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 = 30,000,000,000

So that’s forty-one narcissistic sumbers in base-10. Not all of them are listed in Sequence A131639 at the Encyclopedia of Integer Sequences, but I think I’ve got my program working right. Other bases show similar patterns. Here are some missing-digit narcissistic sumbers in base-5:

• 1,243 → 243 + 143 + 123 + 124 = 1,243 (b=5) = 198 (b=10)
• 1,324 → 324 + 124 + 134 + 132 = 1,324 (b=5) = 214 (b=10)
• 1,331 → 331 + 131 + 131 + 133 = 1,331 (b=5) = 216 (b=10)
• 1,412 → 412 + 112 + 142 + 141 = 1,412 (b=5) = 232 (b=10)

• 100,000 → 00000 + 10000 + 10000 + 10000 + 10000 + 10000 = 100,000 (b=5) = 3,125 (b=10)
• 200,000 → 00000 + 20000 + 20000 + 20000 + 20000 + 20000 = 200,000 (b=5) = 6,250 (b=10)
• 300,000 → 00000 + 30000 + 30000 + 30000 + 30000 + 30000 = 300,000 (b=5) = 9,375 (b=10)
• 400,000 → 00000 + 40000 + 40000 + 40000 + 40000 + 40000 = 400,000 (b=5) = 12,500 (b=10)

And here are some sumbers in base-16:

5,4CD,111,0EE,EF0,542 = 4CD1110EEEF0542 + 5CD1110EEEF0542 + 54D1110EEEF0542 + 54C1110EEEF0542 + 54CD110EEEF0542 + 54CD110EEEF0542 + 54CD110EEEF0542 + 54CD111EEEF0542 + 54CD1110EEF0542 + 54CD1110EEF0542 + 54CD1110EEF0542 + 54CD1110EEE0542 + 54CD1110EEEF542 + 54CD1110EEEF042 + 54CD1110EEEF052 + 54CD1110EEEF054 (b=16) = 6,110,559,033,837,421,890 (b=10)

6,5DD,E13,CEE,EF0,542 = 5DDE13CEEEF0542 + 6DDE13CEEEF0542 + 65DE13CEEEF0542 + 65DE13CEEEF0542 + 65DD13CEEEF0542 + 65DDE3CEEEF0542 + 65DDE1CEEEF0542 + 65DDE13EEEF0542 + 65DDE13CEEF0542 + 65DDE13CEEF0542 + 65DDE13CEEF0542 + 65DDE13CEEE0542 + 65DDE13CEEEF542 + 65DDE13CEEEF042 + 65DDE13CEEEF052 + 65DDE13CEEEF054 (b=16) = 7,340,270,619,506,705,730 (b=10)

10,000,000,000,000,000 → 0000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 = 10,000,000,000,000,000 (b=16) = 18,446,744,073,709,551,616 (b=10)

F0,000,000,000,000,000 → 0000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 = F0,000,000,000,000,000 (b=16) = 276,701,161,105,643,274,240 (b=10)

Next I’d like to investigate sumbers created by missing two, three and more digits at a time. Here’s a taster:

1,043,101 → 43101 (missing 1 and 0) + 03101 (missing 1 and 4) + 04101 (missing 1 and 3) + 04301 + 04311 + 04310 + 13101 + 14101 + 14301 + 14311 + 14310 + 10101 + 10301 + 10311 + 10310 + 10401 + 10411 + 10410 + 10431 + 10430 + 10431 = 1,043,101 (b=5) = 18,526 (b=10)