They Say, They Sigh, They Sow #2

“Epitaxial mismatches in the lattices of nickelate ultra-thin films can be used to tune the energetic landscape of Mott materials and thereby control conductor/insulator transitions.” — On the road to Mottronics, ScienceDaily, 24/ii/2014.

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Reverssum

Here’s a simple sequence. What’s the next number?

1, 2, 4, 8, 16, 68, 100, ?

The rule I’m using is this: Reverse the number, then add the sum of the digits. So 1 doubles till it becomes 16. Then 16 becomes 61 + 6 + 1 = 68. Then 68 becomes 86 + 8 + 6 = 100. Then 100 becomes 001 + 1 = 2. And the sequence falls into a loop.

Reversing the number means that small numbers can get big and big numbers can get small, but the second tendency is stronger for the first few seeds:

• 1 → 2 → 4 → 8 → 16 → 68 → 100 → 2
• 2 → 4 → 8 → 16 → 68 → 100 → 2
• 3 → 6 → 12 → 24 → 48 → 96 → 84 → 60 → 12
• 4 → 8 → 16 → 68 → 100 → 2 → 4
• 5 → 10 → 2 → 4 → 8 → 16 → 68 → 100 → 2
• 6 → 12 → 24 → 48 → 96 → 84 → 60 → 12
• 7 → 14 → 46 → 74 → 58 → 98 → 106 → 608 → 820 → 38 → 94 → 62 → 34 → 50 → 10 → 2 → 4 → 8 → 16 → 68 → 100 → 2
• 8 → 16 → 68 → 100 → 2 → 4 → 8
• 9 → 18 → 90 → 18
• 10 → 2 → 4 → 8 → 16 → 68 → 100 → 2

An 11-seed is a little more interesting:

11 → 13 → 35 → 61 → 23 → 37 → 83 → 49 → 107 → 709 → 923 → 343 → 353 → 364 → 476 → 691 → 212 → 217 → 722 → 238 → 845 → 565 → 581 → 199 → 1010 → 103 → 305 → 511 → 122 → 226 → 632 → 247 → 755 → 574 → 491 → 208 → 812 → 229 → 935 → 556 → 671 → 190 → 101 → 103 (11 leads to an 18-loop from 103 at step 26; total steps = 44)

Now try some higher bases:

• 1 → 2 → 4 → 8 → 15 → 57 → 86 → 80 → 15 (base=11)
• 1 → 2 → 4 → 8 → 14 → 46 → 72 → 34 → 4A → B6 → 84 → 58 → 96 → 80 → 14 (base=12)
• 1 → 2 → 4 → 8 → 13 → 35 → 5B → C8 → A6 → 80 → 13 (base=13)
• 1 → 2 → 4 → 8 → 12 → 24 → 48 → 92 → 36 → 6C → DA → C8 → A4 → 5A → B6 → 80 → 12 (base=14)
• 1 → 2 → 4 → 8 → 11 → 13 → 35 → 5B → C6 → 80 → 11 (base=15)
• 1 → 2 → 4 → 8 → 10 → 2 (base=16)

Does the 1-seed always create a short sequence? No, it gets pretty long in base-19 and base-20:

• 1 → 2 → 4 → 8 → [16] → 1D → DF → [17]3 → 4[18] → 107 → 709 → 914 → 424 → 42E → E35 → 54[17] → [17]5C → C7D → D96 → 6B3 → 3C7 → 7D6 → 6EE → E[16]2 → 2[18]8 → 90B → B1A → A2E → E3[17] → [17]5A → A7B → B90 → AC→ DD → F1 → 2C → C[16] → [18]2 → 40 → 8 (base=19)
• 1 → 2 → 4 → 8 → [16] → 1C → CE → F[18] → 108 → 80A → A16 → 627 → 731 → 13[18] → [18]43 → 363 → 36F → F77 → 794 → 4A7 → 7B5 → 5CA → ADC → CF5 → 5[17]4 → 4[18]B → B[19][17] → [18]1[18] → [18]3F → F5E → E79 → 994 → 4AB → BB9 → 9D2 → 2ED → DFB → B[17]C → C[19]B → C1E → E2[19] → [19]49 → 96B → B7F → F94 → 4B3 → 3C2 → 2D0 → D[17] → [19]3 → 51 → 1B → BD → EF → [17]3 → 4[17] → [18]5 → 71 → 1F → F[17] → [19]7 → 95 → 63 → 3F → [16]1 → 2D → D[17] (base=20)

Then it settles down again:

• 1 → 2 → 4 → 8 → [16] → 1B → BD → EE → [16]0 → 1B (base=21)
• 1 → 2 → 4 → 8 → [16] → 1A → AC → DA → BE → FE → [16]0 → 1A (base=22)
• 1 → 2 → 4 → 8 → [16] → 19 → 9B → C6 → 77 → 7[21] → [22]C → EA → BF → [16]E → [16]0 → 19 (base=23)

Base-33 is also short:

1 → 2 → 4 → 8 → [16] → [32] → 1[31] → [32]0 → 1[31] (base=33)

And so is base-35:

1 → 2 → 4 → 8 → [16] → [32] → 1[29] → [29][31] → [33][19] → [21]F → [16][22] → [23][19] → [20][30] → [32]0 → 1[29] (base=35)

So what about base-34?

1 → 2 → 4 → 8 → [16] → [32] → 1[30] → [30][32] → 10[24] → [24]0[26] → [26]26 → 63[26] → [26]47 → 75[29] → [29]6E → E8A → A9C → CA7 → 7B7 → 7B[32] → [32]C[23] → [23]E[31] → [31][16][23] → [23][18][33] → [33][20][29] → [29][23]D → D[25][26] → [26][27]9 → 9[29][20] → [20][30][33] → [33][33]1 → 21[32] → [32]23 → 341 → 14B → B4[17] → [17]59 → 96E → E74 → 485 → 58[21] → [21]95 → 5A[22] → [22]B8 → 8C[29] → [29]D[23] → [23]F[26] → [26][17][19] → [19][19][20] → [20][21]9 → 9[23]2 → 2[24]9 → 9[25]3 → 3[26]C → C[27]A → A[28][27] → [27][30]7 → 7[32][23] → [24]01 → 11F → F1[18] → [18]2F → F3[19] → [19]4[18] → [18]5[26] → [26]6[33] → [33]8[23] → [23]A[29] → [29]C[17] → [17]E[19] → [19]F[33] → [33][17][18] → [18][19][33] → [33][21][20] → [20][24]5 → 5[26]1 → 1[27]3 → 3[27][32] → [32][28][31] → [31][31][21] → [22]0C → C1[22] → [22]2D → D3[25] → [25]4[20] → [20]66 → 67[18] → [18]83 → 39D → D9[28] → [28]A[29] → [29]C[27] → [27]E[29] → [29][16][29] → [29][19]1 → 1[21]A → A[21][33] → [33][23]6 → 6[25][27] → [27][26][30] → [30][29]8 → 8[31][29] → [29][33]8 → 91[31] → [31]2[16] → [16]4C → C5E → E69 → 979 → 980 → 8[26] → [27]8 → 9[28] → [29]C → E2 → 2[30] → [31]0 → 1[28] → [28][30] → [32][18] → [20]E → F[20] → [21][16] → [17][24] → [25][24] → [26]6 → 7[24] → [25]4 → 5[20] → [20][30] → [32]2 → 3[32] → [33]4 → 62 → 2E → E[18] → [19]C → D[16] → [17]8 → 98 → 8[26] (1 leads to a 30-loop from 8[26] / 298 in base-34 at step 111; total steps = 141)

An alternative rule is to add the digit-sum first and then reverse the result. Now 8 becomes 8 + 8 = 16 and 16 becomes 61. Then 61 becomes 61 + 6 + 1 = 68 and 68 becomes 86. Then 86 becomes 86 + 8 + 6 = 100 and 100 becomes 001 = 1:

• 1 → 2 → 4 → 8 → 61 → 86 → 1
• 2 → 4 → 8 → 61 → 86 → 1 → 2
• 3 → 6 → 21 → 42 → 84 → 69 → 48 → 6
• 4 → 8 → 61 → 86 → 1 → 2 → 4
• 5 → 1 → 2 → 4 → 8 → 62 → 7 → 48 → 6 → 27 → 63 → 27
• 6 → 21 → 42 → 84 → 69 → 48 → 6
• 7 → 41 → 64 → 47 → 85 → 89 → 601 → 806 → 28 → 83 → 49 → 26 → 43 → 5 → 6 → 27 → 63 → 27
• 8 → 61 → 86 → 1 → 2 → 4 → 8
• 9 → 81 → 9
• 10 → 11 → 31 → 53 → 16 → 32 → 73 → 38 → 94 → 701 → 907 → 329 → 343 → 353 → 463 → 674 → 196 → 212 → 712 → 227 → 832 → 548 → 565 → 185 → 991 → 101 → 301 → 503 → 115 → 221 → 622 → 236 → 742 → 557 → 475 → 194→ 802 → 218 → 922 → 539 → 655 → 176 → 91 → 102 → 501 → 705 → 717 → 237 → 942 → 759 → 87 → 208 → 812 → 328 → 143 → 151 → 851 → 568 → 785 → 508 → 125 → 331 → 833 → 748 → 767 → 787 → 908 → 529 → 545 → 955 → 479 → 994 → 6102 → 1116 → 5211 → 225 → 432 → 144 → 351 → 63 → 27 → 63

Words at War

Front cover of Poetry of the First World War edited by Tim KendallPoetry of the First World War: An Anthology, ed. Tim Kendall (Oxford University Press 2013)

J.R.R. Tolkien and C.S. Lewis are famous names today, but both might have died young in the First World War. If so, they would now be long forgotten. Generally speaking, novelists, essayists and scholars take time to mature and need time to create. Poets are different: they can create something of permanent value in a few minutes. This helps explain why nearly half the men chosen for this book did not reach their thirties:

• Rupert Brooke (1887-1915)
• Julian Grenfell (1888-1915)
• Charles Sorley (1895-1915)
• Patrick Shaw Stewart (1888-1917)
• Arthur Graeme West (1891-1917)
• Isaac Rosenberg (1890-1918)
• Wilfred Owen (1893-1918)

And none of them left substantial bodies of work. Indeed, “except for some schoolboy verse”, Patrick Shaw Stewart is known for only one poem, which “was found written on the back flyleaf of his copy of A.E. Housman’s A Shropshire Lad after his death” (pg. 116). It begins like this:

I saw a man this morning
  Who did not wish to die:
I ask and cannot answer,
  If otherwise wish I.

(From I saw a man this morning)

Housman is here too, with Epitaph on an Army of Mercenaries, which Kipling, also here, is said to have called “the finest poem of the First World War” (pg. 14). I don’t agree and I would prefer less Kipling and no Thomas Hardy. That would have left space for something I wish had been included: translations from French and German. The First World War was fought by speakers of Europe’s three major languages and this book makes me realize that I know nothing about war poetry in French and German.

It would be interesting to compare it with the poetry in English. Were traditional forms mingling with modernism in the same way? I assume so. Wilfred Owen looked back to Keats and the assonance of Anglo-Saxon verse:

Our brains ache, in the merciless iced winds that knive us…
Wearied we keep awake because the night is silent…
Low, drooping flares confuse our memory of the salient…
Worried by silence, sentries whisper, curious, nervous,
  But nothing happens. (Exposure)

David Jones (1895-1974) looked forward:

   You can hear the silence of it:
You can hear the rat of no-man’s-land
rut-out intricacies,
weasel-out his patient workings,
scrut, scrut, sscrut,
harrow-out earthly, trowel his cunning paw;
redeem the time of our uncharity, to sap his own amphibi-
ous paradise.
   You can hear his carrying-parties rustle our corruptions
through the night-weeds – contest the choicest morsels in his
tiny conduits, bead-eyed feast on us; by a rule of his nature,
at night-feast on the broken of us. (In Parenthesis)

But is there Gerard Manley Hopkins in that? And in fact In Parenthesis was begun “in 1927 or 1928” and published in 1937. T.S. Eliot called it “a work of genius” (pg. 200). I’d prefer to disagree, but I can’t: you can feel the power in the extract given here. Isaac Rosenberg had a briefer life and left briefer work, but was someone else who could work magic with words:

A worm fed on the heart of Corinth,
Babylon and Rome.
Not Paris raped tall Helen,
But this incestuous worm
Who lured her vivid beauty
To his amorphous sleep.
England! famous as Helen
Is thy betrothal sung.
To him the shadowless,
More amorous than Solomon.

A beautiful poem about an ugly thing: death. A mysterious poem too. And a sardonic one. Rosenberg says much with little and I think he was a much better poet than the more famous Siegfried Sassoon and Robert Graves. They survived the war and wrote more during it, which helps explain their greater fame. But the flawed poetry of Graves was sometimes appropriate to its ugly theme:

To-day I found in Mametz Wood
A certain cure for lust of blood:

Where, propped against a shattered trunk,
  In a great mess of things unclean,
Sat a dead Boche; he scowled and stunk
  With clothes and face a sodden green,
Big-bellied, spectacled, crop-haired,
Dribbling black blood from nose and beard.

A poem like that is a cure for romanticism, but that’s part of what makes Wilfred Owen a better and more interesting poet than Graves. Owen’s romanticism wasn’t cured: there’s conflict in his poems about conflict:

I saw his round mouth’s crimson deepen as it fell,
  Like a sun, in his last deep hour;
Watched the magnificent recession of farewell,
  Clouding, half gleam, half glower,
And a last splendour burn the heavens of his cheek.
  And in his eyes
The cold stars lighting, very old and bleak,
  In different skies.

But how good is Owen’s work? He was a Kurt Cobain of his day: good-looking, tormented and dying young. You can’t escape the knowledge of early death when you read the poetry of one or listen to the music of other. That interferes with objective appraisal. But the flaws in Owen’s poetry add to its power, increasing the sense of someone writing against time and struggling for greatness in a bad place. The First World War destroyed a lot of poets and perhaps helped destroy poetry too, raising questions about tradition that some answered with nihilism. As Owen asks in Futility:

Was it for this the clay grew tall?
—O what made fatuous sun-beams toil
To break earth’s sleep at all?

Some of the poets here were happy to go to war, but it wasn’t the Homeric adventure anticipated by Patrick Shaw Stewart. He learnt that high explosive is impersonal, bullets kill at great distance and machines don’t need rest. Poetry of the First World War is about a confrontation: between flesh and metal, brains and machinery. It’s an interesting anthology that deserves much more time than I have devoted to it. The notes aren’t intrusive, the biographies are brief but illuminating, and although Tim Kendall is a Professor of English Literature he has let his profession down by writing clear prose and eschewing jargon. He’s also included some “Music-Hall and Trench Songs” and they speak for the ordinary and sometimes illiterate soldier. The First World War may be the most important war in European history and this is a good introduction to some of the words it inspired.

Bill Self

I would be disturbed and dismayed if Will Self ever wrote an essay on Evelyn Waugh or Clark Ashton Smith. In fact, I hope he has never even heard of CAS. But I’m happy to see Self writing in the Guardian on William Burroughs. It’s a perfect setting for a perfect pairing. And Self, like Christopher Hitchens, raises a very interesting question. What is his mother-tongue? Quechua? Tagalog? Sumerian? Whatever it is, it’s not even remotely related to English.


William Burroughs — the original Junkie — Will Self, The Guardian, 1/ii/2014.

Entitled Junkie: Confessions of an Unredeemed Drug Addict and authored pseudonymously by “William Lee” (Burroughs’ mother’s maiden name – he didn’t look too far for a nom de plume) …

[Self missed his chance there: nom de guerre would have been much better.]

The two-books-in-one format was not uncommon in 1950s America …

Despite its subhead, Wyn did think the book had a redemptive capability …

Both Junkie and Narcotic Agent have covers of beautiful garishness, featuring 1950s damsels in distress. On the cover of Junkie a craggy-browed man is grabbing a blond lovely from behind; one of his arms is around her neck, while the other grasps her hand, within which is a paper package. The table beside them has been knocked in the fray, propelling a spoon, a hypodermic, and even a gas ring, into inner space.

This cover illustration is, in fact, just that: an illustration of a scene described by Burroughs in the book. “When my wife saw I was getting the habit again, she did something she had never done before. I was cooking up a shot two days after I’d connected with Old Ike. My wife grabbed the spoon and threw the junk on the floor. I slapped her twice across the face and she threw herself on the bed, sobbing …” That this uncredited and now forgotten hack artist should have chosen one of the few episodes featuring the protagonist’s wife to use for the cover illustration represents one of those nastily serendipitous ironies that Burroughs himself almost always chose to view as evidence of the magical universe. …

… if you turn to his glossary of junk lingo and jive talk – you will see how many arcane drug terms have metastasised into the vigorous language. …

Burroughs viewed the postwar era as a Götterdämmerung and a convulsive re-evaluation of values. …

An open homosexual and a drug addict, his quintessentially Midwestern libertarianism led him to eschew any command economy of ethics …

For Burroughs, the re-evaluation was both discount and markup …

… and perhaps it was this that made him such a great avatar of the emergent counterculture. …

Janus-faced, and like some terminally cadaverous butler, Burroughs ushers in the new society of kicks for insight as well as kicks’ sake. …

Let’s return to that cover illustration with its portrayal of “William Lee” as Rock Hudson and his common-law wife, Joan Vollmer, as Kim Novak.

When I say Burroughs himself must have regarded the illustration – if he thought of it at all – as evidence of the magical universe he conceived of as underpinning and interpenetrating our own …

Much has been written and even more conjectured about the killing. Burroughs himself described it as “the accidental shooting death”; and although he jumped bail, he was only convicted – in absentia by the Mexican court – of homicide. …

When Burroughs was off heroin he was a bad, blackout drunk (for evidence you need look no further than his own confirmation in Junky). …

By the time Burroughs was living in Tangier in the late 1950s, his sense of being little more than a cipher, or a fictional construct, had become so plangent …

Burroughs was the perfect incarnation of late 20th-century western angst precisely because he was an addict. Self-deluding, vain, narcissistic, self-obsessed, and yet curiously perceptive about the sickness of the world if not his own malaise, Burroughs both offered up and was compelled to provide his psyche as a form of Petri dish, within which were cultured the obsessive and compulsive viruses of modernity. …

In a thin-as-a-rake’s progress …

… a deceptively thin, Pandora’s portfolio of an idea …

It is Burroughs’ own denial of the nature of his addiction that makes this book capable of being read as a fiendish parable of modern alienation. …

For, in describing addiction as “a way of life”, Burroughs makes of the hypodermic a microscope, through which he can examine the soul of man under late 20th-century capitalism.

William Burroughs – the original Junkie, The Guardian, 1/ii/2014.


The big disappointment is that he didn’t use in terms of.

Performativizing Papyrocentricity #19

Papyrocentric Performativity Presents:

Book in BlackBlack Sabbath: Symptom of the Universe, Mick Wall (Orion Books 2013)

Critical Math – A Mathematician Reads the Newspaper, John Allen Paulos (Penguin 1996)

Rude BoysRuthless: The Global Rise of the Yardies, Geoff Small (Warner 1995)

K-9 KonundrumDog, Peter Sotos (TransVisceral Books 2014)

Ghosts in the CathedralThe Neutrino Hunters: The Chase for the Ghost Particle and the Secrets of the Universe, Ray Jayawardhana (Oneworld 2013) (posted @ Overlord of the Über-Feral)


Or Read a Review at Random: RaRaR

Block and Goal

123456789. How many ways are there to insert + and – between the numbers and create a formula for 100? With pen and ink it takes a long time to answer. With programming, the answer will flash up in an instant:

01. 1 + 2 + 3 - 4 + 5 + 6 + 78 + 9 = 100
02. 1 + 2 + 34 - 5 + 67 - 8 + 9 = 100
03. 1 + 23 - 4 + 5 + 6 + 78 - 9 = 100
04. 1 + 23 - 4 + 56 + 7 + 8 + 9 = 100
05. 12 - 3 - 4 + 5 - 6 + 7 + 89 = 100
06. 12 + 3 + 4 + 5 - 6 - 7 + 89 = 100
07. 12 + 3 - 4 + 5 + 67 + 8 + 9 = 100
08. 123 - 4 - 5 - 6 - 7 + 8 - 9 = 100
09. 123 + 4 - 5 + 67 - 89 = 100
10. 123 + 45 - 67 + 8 - 9 = 100
11. 123 - 45 - 67 + 89 = 100

And the beauty of programming is that you can easily generalize the problem to other bases. In base b, how many ways are there to insert + and – in the block [12345…b-1] to create a formula for b^2? When b = 10, the answer is 11. When b = 11, it’s 42. Here are two of those formulae in base-11:

123 - 45 + 6 + 7 - 8 + 9 + A = 100[b=11]
146 - 49 + 6 + 7 - 8 + 9 + 10 = 121

123 + 45 + 6 + 7 - 89 + A = 100[b=11]
146 + 49 + 6 + 7 - 97 + 10 = 121

When b = 12, it’s 51. Here are two of the formulae:

123 + 4 + 5 + 67 - 8 - 9A + B = 100[b=12]
171 + 4 + 5 + 79 - 8 - 118 + 11 = 144

123 + 4 + 56 + 7 - 89 - A + B = 100[b=12]
171 + 4 + 66 + 7 - 105 - 10 + 11 = 144

So that’s 11 formulae in base-10, 42 in base-11 and 51 in base-12. So what about base-13? The answer may be surprising: in base-13, there are no +/- formulae for 13^2 = 169 using the numbers 1 to 12. Nor are there any formulae in base-9 for 9^2 = 81 using the numbers 1 to 8. If you reverse the block, 987654321, the same thing happens. Base-10 has 15 formulae, base-11 has 54 and base-12 has 42. Here are some examples:

9 - 8 + 7 + 65 - 4 + 32 - 1 = 100
98 - 76 + 54 + 3 + 21 = 100

A9 + 87 - 65 + 4 - 3 - 21 = 100[b=11]
119 + 95 - 71 + 4 - 3 - 23 = 121

BA - 98 + 76 - 5 - 4 + 32 - 1 = 100[b=12]
142 - 116 + 90 - 5 - 4 + 38 - 1 = 144

But base-9 and base-13 again have no formulae. What’s going on? Is it a coincidence that 9 and 13 are each one more than a multiple of 4? No. Base-17 also has no formulae for b^2 = 13^2 = 169. Here is the list of formulae for bases-7 thru 17:

1, 2, 0, 11, 42, 51, 0, 292, 1344, 1571, 0 (block = 12345...)
3, 2, 0, 15, 54, 42, 0, 317, 1430, 1499, 0 (block = ...54321)

To understand what’s going on, consider any sequence of consecutive integers starting at 1. The number of odd integers in the sequence must always be greater than or equal to the number of even integers:

1, 2 (1 odd : 1 even)
1, 2, 3 (2 odds : 1 even)
1, 2, 3, 4 (2 : 2)
1, 2, 3, 4, 5 (3 : 2)
1, 2, 3, 4, 5, 6 (3 : 3)
1, 2, 3, 4, 5, 6, 7 (4 : 3)
1, 2, 3, 4, 5, 6, 7, 8 (4 : 4)

The odd numbers in a sequence determine the parity of the sum, that is, whether it is odd or even. For example:

1 + 2 = 3 (1 odd number)
1 + 2 + 3 = 6 (2 odd numbers)
1 + 2 + 3 + 4 = 10 (2 odd numbers)
1 + 2 + 3 + 4 + 5 = 15 (3 odd numbers)
1 + 2 + 3 + 4 + 5 + 6 = 21 (3 odd numbers)
1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 (4 odd numbers)

If there is an even number of odd numbers, the sum will be even; if there is an odd number, the sum will be odd. Consider sequences that end in a multiple of 4:

1, 2, 3, 4 → 2 odds : 2 evens
1, 2, 3, 4, 5, 6, 7, 8 → 4 : 4
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 → 6 : 6

Such sequences always contain an even number of odd numbers. Now, consider these formulae in base-10:

1. 12 + 3 + 4 + 56 + 7 + 8 + 9 = 99
2. 123 - 45 - 67 + 89 = 100
3. 123 + 4 + 56 + 7 - 89 = 101

They can be re-written like this:

1. 1×10^1 + 2×10^0 + 3×10^0 + 4×10^0 + 5×10^1 + 6×10^0 + 7×10^0 + 8×10^0 + 9×10^0 = 99

2. 1×10^2 + 2×10^1 + 3×10^0 – 4×10^1 – 5×10^0 – 6×10^1 – 7×10^0 + 8×10^1 + 9×10^0 = 100

3. 1×10^2 + 2×10^1 + 3×10^0 + 4×10^0 + 5×10^1 + 6×10^1 + 7×10^0 – 8×10^1 – 9×10^0 = 101

In general, the base-10 formulae will take this form:

1×10^a +/- 2×10^b +/- 3×10^c +/– 4×10^d +/– 5×10^e +/– 6×10^f +/– 7×10^g +/– 8×10^h +/– 9×10^i = 100

It’s important to note that the exponent of 10, or the power to which it is raised, determines whether an odd number remains odd or becomes even. For example, 3×10^0 = 3×1 = 3, whereas 3×10^1 = 3×10 = 30 and 3×10^2 = 3×100 = 300. Therefore the number of odd numbers in a base-10 formula can vary and so can the parity of the sum. Now consider base-9. When you’re trying to find a block-formula for 9^2 = 81, the formula will have to take this form:

1×9^a +/- 2×9^b +/- 3×9^c +/- 4×9^d +/- 5×9^e +/- 6×9^f +/- 7×9^g +/- 8×9^h = 81

But no such formula exists for 81 (with standard exponents). It’s now possible to see why this is so. Unlike base-10, the odd numbers in the formula will remain odd what the power of 9. For example, 3×9^0 = 3×1 = 3, 3×9^1 = 3×9 = 27 and 3×9^2 = 3×81 = 243. Therefore base-9 formulae will always contain four odd numbers and will always produce an even number. Odd numbers in base-2 always end in 1, even numbers always end in 0. Therefore, to determine the parity of a sum of integers, convert the integers to base-2, discard all but the final digit of each integer, then sum the 1s. In a base-9 formula, these are the four possible results:

1 + 1 + 1 + 1 = 4
1 + 1 + 1 - 1 = 2
1 + 1 - 1 - 1 = 0
1 - 1 - 1 - 1 = -2

The sum represents the parity of the answer, which is always even. Similar reasoning applies to base-13, base-17 and all other base-[b=4n+1].