Persist List

Friday, 31st Jan, 2014 by Krilling for Company in Base Behavior, Mathematics, Narcissistic numbers and tagged Arithmetic, arithmetical, base 10, base 108, base 11, base 12, base 27, base 9, Base Behavior, base behaviour, integer, integers, math, mathematics, maths, multiplicative persistence, narcissistic multiplicative persistence sums, narcissistic number, narcissistic numbers, recreational math, recreational mathematics, recreational maths, sum of integers, sum of products | 1 Comment

Multiplicative persistence is a complex term but a simple concept. Take a number, multiply its digits, repeat. Sooner or later the result is a single digit:

25 → 2 x 5 = 10 → 1 x 0 = 0 (mp=2)
39 → 3 x 9 = 27 → 2 x 7 = 14 → 1 x 4 = 4 (mp=3)

So 25 has a multiplicative persistence of 2 and 39 a multiplicative persistence of 3. Each is the smallest number with that m.p. in base-10. Further records are set by these numbers:

77 → 49 → 36 → 18 → 8 (mp=4)
679 → 378 → 168 → 48 → 32 → 6 (mp=5)
6788 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=6)
68889 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=7)
2677889 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=8)
26888999 → 4478976 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=9)
3778888999 → 438939648 → 4478976 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=10)

Now here’s base-9:

25_[b=9] → 11 → 1 (mp=2)
38_[b=9] → 26 → 13 → 3 (mp=3)
57_[b=9] → 38 → 26 → 13 → 3 (mp=4)
477_[b=9] → 237 → 46 → 26 → 13 → 3 (mp=5)
45788_[b=9] → 13255 → 176 → 46 → 26 → 13 → 3 (mp=6)
2577777_[b=9] → 275484 → 13255 → 176 → 46 → 26 → 13 → 3 (mp=7)

And base-11:

26_[b=11] → 11 → 1 (mp=2)
3A_[b=11] → 28 → 15 → 5 (mp=3)
69_[b=11] → 4A → 37 → 1A → A (=10_b=10) (mp=4)
269_[b=11] → 99 → 74 → 26 → 11 → 1 (mp=5)
3579_[b=11] → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=6)
26778_[b=11] → 3597 → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=7)
47788A_[b=11] → 86277 → 3597 → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=8)
67899AAA_[b=11] → 143A9869 → 299596 → 2A954 → 2783 → 286 → 88 → 59 → 41 → 4 (mp=9)
77777889999_[b=11] → 2AA174996A → 143A9869 → 299596 → 2A954 → 2783 → 286 → 88 → 59 → 41 → 4 (mp=10)

I was also interested in the narcissism of multiplicative persistence. That is, are any numbers equal to the sum of the numbers created while calculating their multiplicative persistence? Yes:

86 = (8 x 6 = 48) + (4 x 8 = 32) + (3 x 2 = 6)

I haven’t found any more in base-10 (apart from the trivial 0 to 9) and can’t prove that this is the only one. Base-9 offers this:

78_[b=9] = 62 + 13 + 3

I can’t find any at all in base-11, but here are base-12 and base-27:

57_[b=12] = 2B + 1A + A
A8_[b=12] = 68 + 40 + 0

4[23]_[b=27] = 3B + 16 + 6
7[24]_[b=27] = 66 + 19 + 9
A[18]_[b=27] = 6[18] + 40 + 0
[26][24]_[b=27] = [23]3 + 2F + 13 + 3
[26][23][26]_[b=27] = [21]8[23] + 583 + 4C + 1[21] + [21]

But the richest base I’ve found so far is base-108, with fourteen narcissistic multiplicative-persistence sums:

4[92]_[b=108] = 3[44] + 1[24] + [24]
5[63]_[b=108] = 2[99] + 1[90] + [90]
7[96]_[b=108] = 6[24] + 1[36] + [36]
A[72]_[b=108] = 6[72] + 40 + 0
[19][81]_[b=108] = E[27] + 3[54] + 1[54] + [54]
[26][96]_[b=108] = [23]C + 2[60] + 1C + C
[35][81]_[b=108] = [26][27] + 6[54] + 30 + 0
[37][55]_[b=108] = [18][91] + F[18] + 2[54] + 10 + 0
[73][60]_[b=108] = [40][60] + [22][24] + 4[96] + 3[60] + 1[72] + [72]
[107][66]_[b=108] = [65][42] + [25][30] + 6[102] + 5[72] + 3[36] + 10 + 0
[71][84]_[b=108] = [55][24] + C[24] + 2[72] + 1[36] + [36]
[107][99]_[b=108] = [98]9 + 8[18] + 1[36] + [36]
5[92][96]_[b=108] = 3[84][96] + 280 + 0
8[107][100]_[b=108] = 7[36][64] + 1[41][36] + D[72] + 8[72] + 5[36] + 1[72] + [72]

Update (10/ii/14): The best now is base-180 with eighteen multiplicative-persistence sums.

5[105]_[b=180] = 2[165] + 1[150] + [150]
7[118]_[b=180] = 4[106] + 2[64] + [128]
7[160]_[b=180] = 6[40] + 1[60] + [60]
8[108]_[b=180] = 4[144] + 3[36] + [108]
A[120]_[b=180] = 6[120] + 40 + 0 (s=5)
[19][135]_[b=180] = E[45] + 3[90] + 1[90] + [90]
[21][108]_[b=180] = C[108] + 7[36] + 1[72] + [72]
[26][160]_[b=180] = [23][20] + 2[100] + 1[20] + [20]
[31][98]_[b=180] = [16][158] + E8 + [112]
[35][135]_[b=180] = [26][45] + 6[90] + 30 + 0 (s=10)
[44][96]_[b=180] = [23][84] + A[132] + 7[60] + 2[60] + [120]
[71][140]_[b=180] = [55][40] + C[40] + 2[120] + 1[60] + [60]
[73][100]_[b=180] = [40][100] + [22][40] + 4[160] + 3[100] + 1[120] + [120]
[107][110]_[b=180] = [65][70] + [25][50] + 6[170] + 5[120] + 3[60] + 10 + 0
[107][165]_[b=180] = [98]F + 8[30] + 1[60] + [60] (s=15)
[172][132]_[b=180] = [126][24] + [16][144] + C[144] + 9[108] + 5[72] + 20 + 0
5[173][145]_[b=180] = 3[156][145] + 2[17]0 + 0
E[170][120]_[b=180] = 8[146][120] + 4[58][120] + [154][120] + [102][120] + [68]0 + 0

She Say, She Sigh, She Sow

Thursday, 23rd Jan, 2014 by Krilling for Company in Mathematics, Philosophy, Philosophy of mathematics, Quotations and tagged Ada Lovelace, Book of Nature, Byron's daughter, math, mathematical truth, mathematics, maths, quotation, quotation about math, quotation about mathematics, quotation about maths, quotations, Victorian mathematics | Leave a comment

“Those who view mathematical science, not merely as a vast body of abstract and immutable truths, whose intrinsic beauty, symmetry and logical completeness, when regarded in their connection together as a whole, entitle them to a prominent place in the interest of all profound and logical minds, but as possessing a yet deeper interest for the human race, when it is remembered that this science constitutes the language through which alone we can adequately express the great facts of the natural world, and those unceasing changes of mutual relationship which, visibly or invisibly, consciously or unconsciously to our immediate physical perceptions, are interminably going on in the agencies of the creation we live amidst: those who thus think on mathematical truth as the instrument through which the weak mind of man can most effectively read his Creator’s works, will regard with especial interest all that can tend to facilitate the translation of its principles into explicit practical forms.” — Ada Lovelace (née Byron) (1815-52).

Six Six Nix

Saturday, 18th Jan, 2014 by Krilling for Company in Base Behavior, Digit-sums, Mathematics, Narcissistic numbers and tagged 1088, Arithmetic, base 10, base 11, base 6, base 7, base 9, base-dependent integer sequences, digit sum, digit sums, integer bases, Integer Sequences, math, mathematics, maths, number sums, recreational math, recreational mathematics, recreational maths, sum-product number, sum-product numbers | Leave a comment

4 x 3 = 13. A mistake? Not in base-9, where 13 = 1×9^1 + 3 = 12 in base-10. This means that 13 is a sum-product number in base-9: first add its digits, then multiply them, then multiply the digit-sum by the digit-product: (1+3) x (1×3) = 13_[9]. There are four more sum-product numbers in this base:

2086_[9] = 17 x 116 = (2 + 8 + 6) x (2 x 8 x 6) = 1536_[10] = 16 x 96
281876_[9] = 35 x 7333 = (2 + 8 + 1 + 8 + 7 + 6) x (2 x 8 x 1 x 8 x 7 x 6) = 172032_[10] = 32 x 5376
724856_[9] = 35 x 20383 = (7 + 2 + 4 + 8 + 5 + 6) x (7 x 2 x 4 x 8 x 5 x 6) = 430080_[10] = 32 x 13440
7487248_[9] = 44 x 162582 = (7 + 4 + 8 + 7 + 2 + 4 + 8) x (7 x 4 x 8 x 7 x 2 x 4 x 8) = 4014080_[10] = 40 x 100352

And that’s the lot, apart from the trivial 0 = (0) x (0) and 1 = (1) x (1), which are true in all bases.

What about base-10?

135 = 9 x 15 = (1 + 3 + 5) x (1 x 3 x 5)
144 = 9 x 16 = (1 + 4 + 4) x (1 x 4 x 4)
1088 = 17 x 64 = (1 + 8 + 8) x (1 x 8 x 8)

1088 is missing from the list at Wikipedia and the Encyclopedia of Integer Sequences, but I like the look of it, so I’m including it here. Base-11 has five sum-product numbers:

419_[11] = 13 x 33 = (4 + 1 + 9) x (4 x 1 x 9) = 504_[10] = 14 x 36
253_[11] = [10] x 28 = (2 + 5 + 3) x (2 x 5 x 3) = 300_[10] = 10 x 30
2189_[11] = 19 x 121 = (2 + 1 + 8 + 9) x (2 x 1 x 8 x 9) = 2880_[10] = 20 x 144
7634_[11] = 19 x 419 = (7 + 6 + 3 + 4) x (7 x 6 x 3 x 4) = 10080_[10] = 20 x 504
82974_[11] = 28 x 3036 = (8 + 2 + 9 + 7 + 4) x (8 x 2 x 9 x 7 x 4) = 120960_[10] = 30 x 4032

But the record for bases below 50 is set by 7:

22_[7] = 4 x 4 = (2 + 2) x (2 x 2) = 16_[10] = 4 x 4
505_[7] = 13 x 34 = (5 + 5) x (5 x 5) = 250_[10] = 10 x 25
242_[7] = 11 x 22 = (2 + 4 + 2) x (2 x 4 x 2) = 128_[10] = 8 x 16
1254_[7] = 15 x 55 = (1 + 2 + 5 + 4) x (1 x 2 x 5 x 4) = 480_[10] = 12 x 40
2343_[7] = 15 x 132 = (2 + 3 + 4 + 3) x (2 x 3 x 4 x 3) = 864_[10] = 12 x 72
116655_[7] = 33 x 2424 = (1 + 1 + 6 + 6 + 5 + 5) x (1 x 1 x 6 x 6 x 5 x 5) = 21600_[10] = 24 x 900
346236_[7] = 33 x 10362 = (3 + 4 + 6 + 2 + 3 + 6) x (3 x 4 x 6 x 2 x 3 x 6) = 62208_[10] = 24 x 2592
424644_[7] = 33 x 11646 = (4 + 2 + 4 + 6 + 4 + 4) x (4 x 2 x 4 x 6 x 4 x 4) = 73728_[10] = 24 x 3072

And base-6? Six Nix. There are no sum-product numbers unique to that base (to the best of my far-from-infallible knowledge). Here is the full list for base-3 to base-50 (not counting 0 and 1 as sum-product numbers):

	5 in base-11	4 in base-21	3 in base-31	2 in base-41
	4 in base-12	5 in base-22	1 in base-32	3 in base-42
0 in base-3	3 in base-13	4 in base-23	3 in base-33	4 in base-43
2 in base-4	3 in base-14	2 in base-24	4 in base-34	5 in base-44
1 in base-5	2 in base-15	3 in base-25	2 in base-35	6 in base-45
0 in base-6	2 in base-16	6 in base-26	2 in base-36	7 in base-46
8 in base-7	6 in base-17	0 in base-27	3 in base-37	3 in base-47
1 in base-8	5 in base-18	1 in base-28	3 in base-38	7 in base-48
5 in base-9	7 in base-19	0 in base-29	1 in base-39	5 in base-49
3 in base-10	3 in base-20	2 in base-30	2 in base-40	3 in base-50

Hex Appeal

Tuesday, 7th Jan, 2014 by Krilling for Company in Geometry, Mathematics, Rep-Tiles and tagged animated gif, animated gifs, double-triangle, equilateral triangle, equilateral triangles, fish rep-tile, fish rep-tiles, five sides, five vertices, four right triangles, geometric, geometric dissection, geometric dissections, geometry, Great Sphinx at Giza, hexiamonds, isosceles right triangle, isosceles right triangles, math, mathematics, maths, pentagon, pentagonal rep-tile, pentagonal rep-tiles, Pentagons, polyiamond, polyiamonds, polyshape, polyshapes, recreational math, recreational mathematics, recreational maths, rep-4 sphinx, rep-9 sphinx, rep-tile, Rep-Tiles, right triangle, right triangles, self-replication, sphinx, sphinx hexiamond, tessellation, tessellations, three equilateral triangles, tiling, tilings, triangle, Triangles, vertex | Leave a comment

A polyiamond is a shape consisting of equilateral triangles joined edge-to-edge. There is one moniamond, consisting of one equilateral triangle, and one diamond, consisting of two. After that, there are one triamond, three tetriamonds, four pentiamonds and twelve hexiamonds. The most famous hexiamond is known as the sphinx, because it’s reminiscent of the Great Sphinx of Giza:

The Great Sphinx of Giza

It’s famous because it is the only known pentagonal rep-tile, or shape that can be divided completely into smaller copies of itself. You can divide a sphinx into either four copies of itself or nine copies, like this (please open images in a new window if they fail to animate):

So far, no other pentagonal rep-tile has been discovered. Unless you count this double-triangle as a pentagon:

It has five sides, five vertices and is divisible into sixteen copies of itself. But one of the vertices sits on one of the sides, so it’s not a normal pentagon. Some might argue that this vertex divides the side into two, making the shape a hexagon. I would appeal to these ancient definitions: a point is “that which has no part” and a line is “a length without breadth” (see Neuclid on the Block). The vertex is a partless point on the breadthless line of the side, which isn’t altered by it.

But, unlike the sphinx, the double-triangle has two internal areas, not one. It can be completely drawn with five continuous lines uniting five unique points, but it definitely isn’t a normal pentagon. Even less normal are two more rep-tiles that can be drawn with five continuous lines uniting five unique points: the fish that can be created from three equilateral triangles and the fish that can be created from four isosceles right triangles:

Think Ink

Tuesday, 7th Jan, 2014 by Krilling for Company in Biology, Book Reviews, Mathematics, Physics, Psychology, Science and tagged 50 Quantum Physics Ideas You Really Need to Know, action at a distance, advanced math, advanced mathematics, advanced maths, advanced physics, agriculture, Albert Einstein, Asia, biology, Bohr, brain evolution, chess, Danish physicist Niels Bohr, energy, Everett, evolution, evolutionary biology, Fermi, Feynman, Fifty Quantum Physics Ideas, HBD, Higgs, high finance, human bio-diversity, human biology, human evolution, Joanne Baker, linguistics, literacy, manipulating symbols, math, mathematics, maths, matter, matter and energy, Meso-America, Mesopotamia, particle-wave duality, physics, pure mathematics, quantum physics, Quercus, Schrödinger, spoken language, Steven Weinberg, sub-Saharan Africa, symbols, the brain, the human brain, ultra-violet catastrophe, What Is Life?, written language | Leave a comment

50 Quantum Physics Ideas You Really Need to Know, Joanne Baker (Quercus 2013)

A very good introduction to a very difficult subject. A very superficial introduction too, because it doesn’t use proper mathematics. If it did, I’d be lost: like most people’s, my maths is far too weak for me to understand quantum physics. Here’s one of the side-quotes that help make this book such an interesting read: “We must be clear that when it comes to atoms, language can be used only as in poetry.”

That’s by the Jewish-Danish physicist Niels Bohr (1885-1962). It applies to quantum physics in general. Without the full maths, you’re peering through a frost-covered window into a sweetshop, you’re not inside sampling the wares. But even without the full maths, the concepts and ideas in this book are still difficult and challenging, from the early puzzles thrown up by the ultra-violet catastrophe to the ingenious experiments that have proved particle-wave duality and action at a distance.

But there’s a paradox here.

Continue reading: Think Ink…

Rep It Up

Tuesday, 17th Dec, 2013 by Krilling for Company in Geometry, Mathematics, Rep-Tiles and tagged animated gifs, equilateral triangle, equilateral triangles, geometric, geometric dissection, geometric dissections, geometry, isosceles right triangle, isosceles right triangles, math, mathematics, maths, polyiamond, polyiamonds, polyshape, polyshapes, recreational math, recreational mathematics, recreational maths, rep-tile, Rep-Tiles, right triangle, right triangles, self-replication, tessellation, tessellations, tiling, tilings, triangle, Triangles, twelve equilateral triangles | Leave a comment

When I started to look at rep-tiles, or shapes that can be divided completely into smaller copies of themselves, I wanted to find some of my own. It turns out that it’s easy to automate a search for the simpler kinds, like those based on equilateral triangles and right triangles.

(Please open the following images in a new window if they fail to animate)

Previously pre-posted (please peruse):

• Rep-Tile Reflections

Hextra Texture

Thursday, 5th Dec, 2013 by Krilling for Company in Fractals, Geometry, Mathematics and tagged animated fractal, animated fractals, animated gif, animated gifs, equilateral triangle, equilateral triangles, fractal, fractal method, fractal methods, fractals, geometric, geometry, hexagon, Hexagons, math, mathematics, maths, recreational math, recreational mathematics, recreational maths, triangle, Triangles | Leave a comment

A hexagon can be divided into six equilateral triangles. An equilateral triangle can be divided into a hexagon and three more equilateral triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.

Previously pre-posted (please peruse):

• Fractal Fourmulas

Prime Time

Friday, 29th Nov, 2013 by Krilling for Company in Mathematics, Prime numbers, Reciprocals and tagged 29, math, mathematics, maths, maximum period prime reciprocal, maximum period reciprocal, maximum period reciprocals, prime number, prime numbers, primes, reciprocal, reciprocals, recreational math, recreational mathematics, recreational maths, twenty-nine | Leave a comment

1/29_[b=2] = 0·0000100011010011110111001011… (l=28)
1/29_[b=3] = 0·0002210102011122200121202111… (l=28)
1/29_[b=5] = 0·00412334403211… (l=14)
1/29_[b=7] = 0·0145536… (l=7)
1/29_[b=11] = 0·04199534608387[10]69115764[10]2723… (l=28)
1/29_[b=13] = 0·05[10]9[11]28[12]7231[10]4… (l=14)
1/29_[b=17] = 0·09[16]7… (l=4)
1/29_[b=19] = 0·0[12]89[15][13][14]7[16]73[17][13]1[18]6[10]9354[11]2[11][15]15[17]… (l=28)
1/29_[b=23] = 0·0[18]5[12][15][19][19]… (l=7)
1/29_[b=29] = 0·1 (l=1)
1/29_[b=31] = 0·1248[17]36[12][25][20]9[19]7[14][29][28][26][22][13][27][24][18]5[10][21][11][23][16]… (l=28)
1/29_[b=37] = 0·1[10]7[24]8[34][16][21][25][19]53[30][22][35][26][29][12][28]2[20][15][11][17][31][33]6[14]… (l=28)
1/29_[b=41] = 0·1[16][39][24]… (l=4)
1/29_[b=43] = 0·1[20][32][26][29][28]7[17][34]4[19][11][37]2[41][22][10][16][13][14][35][25]8[38][23][31]5[40]… (l=28)
1/29_[b=47] = 0·1[29]84[40][24][14][27][25][43][35][30][37][12][45][17][38][42]6[22][32][19][21]3[11][16]9[34]… (l=28)
1/29_[b=53] = 0·1[43][45][36][29][12][42]… (l=7)
1/29_[b=59] = 0·2… (l=1)
1/29_[b=61] = 0·26[18][56][48][23]8[25][14][44][10][31][33][39][58][54][42]4[12][37][52][35][46][16][50][29][27][21]… (l=28)
1/29_[b=67] = 0·2[20][53]9[16][11][36][64][46][13][57][50][55][30]… (l=14)
1/29_[b=71] = 0·2[31][58][53][61][14][48][68][39][12][17]9[56][22]… (l=14)
1/29_[b=73] = 0·2[37][55][27][50][25][12][42][57][65][32][52][62][67][70][35][17][45][22][47][60][30][15]7[40][20][10]5… (l=28)
1/29_[b=79] = 0·2[57][16][27][19]5[35][32][54][38][10][70][65][29][76][21][62][51][59][73][43][46][24][40][68]8[13][49]… (l=28)
1/29_[b=83] = 0·2[71][45][65][68][57][20]… (l=7)
1/29_[b=89] = 0·36[12][24][49]9[18][36][73][58][27][55][21][42][85][82][76][64][39][79][70][52][15][30][61][33][67][46]… (l=28)
1/29_[b=97] = 0·3[33][43][46][80][26][73][56][83][60][20]6[66][86][93][63][53][50][16][70][23][40][13][36][76][90][30][10]… (l=28)

Performativizing Papyrocentricity #16

Wednesday, 27th Nov, 2013 by Krilling for Company in Book Reviews, Notices and tagged botanical illustration, botanical illustrations, botany, Britain, British botany, British Isles, British landscapes, British wild flowers, cartoon, cartoons, Futurama, Geology, hard rock, heavy metal, irrational number, irrational numbers, φ, J.R. Press, Leonora Box, Mario Livio, math, mathematics, maths, phi, rock music, Ronald Turnbull, Simon Singh, Spinal Tap, Status Quo, the golden ratio, The Simpsons | Leave a comment

Papyrocentric Performativity Presents:

• Brit Grit – Granite and Grit: A Walker’s Guide to the Geology of British Mountains, Ronald Turnbull (Francis Lincoln 2011)

• Singh Summing Simpsons – The Simpsons and Their Mathematical Secrets, Simon Singh (Bloomsbury 2013)

• Go with the Quo – Status Quo: Still Doin’ It – The Official Updated Edition, compiled by Bob Young, edited by Francis Rossi and Rick Parfitt (Omnibus Press 2013)

• Breeding Bunnies – The Golden Ratio: The Story of Phi, the Extraordinary Number of Nature, Art and Beauty, Mario Livio (Headline Review 2003) (posted @ Overlord of the Über-Feral)

• Brit Bot Book – Reader’s Digest Field Guide to the Wild Flowers of Britain, J.R. Press et al, illustrated Leonora Box et al (1981) (@ O.o.t.Ü.-F.)

Or Read a Review at Random: RaRaR

The Brain in Train

Monday, 25th Nov, 2013 by Krilling for Company in Mathematics, Philosophy and tagged brain, determinism, deterministic, epistemology, Fermat's last theorem, Free Will, infinitude of the primes, logic, math, mathematical proof, mathematics, maths, mind, philosophy, proof, reason, reasoning, recognising the truth, recognizing the truth, sense-data, sense-organs, strict determinism, strictly determined, truth | Leave a comment

I feel odd when I consider this possibility: that all my thoughts are strictly determined, no more under my control than a straw in a gale or a stone in an avalanche. It seems paradoxical to have strictly determined thoughts about strictly determined thoughts. But is it? And is strict determinism fatal for finding the truth? I don’t think so. In fact, I think that strict determinism is essential for truth. But irrelevant associations get in the way of our understanding this. If our thoughts are determined, they seem like automatic trains running on rigid tracks. We might want to go to the station marked “Truth”, but if the switches are set wrong, the train will never get there. Or it will thunder through and never stop.

Continue reading The Brain in Train…

Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral

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