# Sprime Time

All fans of recreational math love palindromic numbers. It’s mandatory, man. 101, 727, 532235, 8810188, 1367755971795577631 — I love ’em! But where can you go after palindromes? Well, you can go to palindromes in a higher dimension. Numbers like 101, 727, 532235 and 8810188 are 1-d palindromes. That is, they’re palindromic in one dimension: backwards and forwards. But numbers like 181818189 and 646464640 aren’t palindromic in one dimension. They’re palindromic in two dimensions:

1 8 1
8 9 8
1 8 1

n=181818189

6 4 6
4 0 4
6 4 6

n=646464640

They’re 2-d palindromes or spiral numbers, that is, numbers that are symmetrical when written as a spiral. You start with the first digit on the top left, then spiral inwards to the center, like this for a 9-digit spiral (9 = 3×3): And this for a 36-digit spiral (36 = 6×6): Spiral numbers are easy to construct, because you can reflect and rotate the numbers in one triangular slice of the spiral to find all the others:   You could say that the seed for the spiral number above is 7591310652, because you can write that number in descending lines, left-to-right, as a triangle.

Here are some palindromic numbers with nine digits in base 3 — as you can see, some are both palindromic numbers and spiral numbers. That is, some are palindromic in both one and two dimensions:

1  0  1

0  1  0

1  0  1

n=101010101

1  0  1

0  2  0

1  0  1

n=101010102

1  1  1

1  0  1

1  1  1

n=111111110

1  1  1

1  1  1

1  1  1

n=111111111

2  0  2

0  1  0

2  0  2

n=202020201

2  0  2

0  2  0

2  0  2

n=202020202

2  2  2

2  1  2

2  2  2

n=222222221

2  2  2

2  2  2

2  2  2

n=222222222

But palindromic primes are even better than ordinary palindromes. Here are a few 1-d palindromic primes in base 10:

101
151
73037
7935397
97356765379
1091544334334451901
1367755971795577631
70707270707
39859395893
9212129
7436347
166000661
313
929

And after 1-d palindromic primes, you can go to 2-d palindromic primes. That is, to spiral primes or sprimes — primes that are symmetrical when written as a spiral:

3 6 3
6 7 6
3 6 3

n=363636367 (prime)
seed=367 (see definition above)

9 1 9
1 3 1
9 1 9

n=919191913 (prime)
seed=913

3 7 8 6 3 6 8 7 3
7 9 1 8 9 8 1 9 7
8 1 9 0 9 0 9 1 8
6 8 0 5 5 5 0 8 6
3 9 9 5 7 5 9 9 3
6 8 0 5 5 5 0 8 6
8 1 9 0 9 0 9 1 8
7 9 1 8 9 8 1 9 7
3 7 8 6 3 6 8 7 3

n=378636873786368737863687378636879189819189819189819189819090909090909090555555557 (prime)
seed=378639189909557 (l=15)

And why stop with spiral numbers — and sprimes — in two dimensions? 363636367 is a 2-sprime, being palindromic in two dimensions. But the digits of a number could be written to form a symmetrical cube in three, four, five and more dimensions. So I assume that there are 3-sprimes, 4-sprimes, 5-sprimes and more out there. Watch this space.

# The Power of Powder

• Racine carrée de 2, c’est 1,414 et des poussières… Et quelles poussières ! Des grains de sable qui empêchent d’écrire racine de 2 comme une fraction. Autrement dit, cette racine n’est pas dans Q. — Rationnel mon Q: 65 exercices de styles, Ludmilla Duchêne et Agnès Leblanc (2010)

• The square root of 2 is 1·414 and dust… And what dust! Grains of sand that stop you writing the root of 2 as a fraction. Put another way, this root isn’t in Q [the set of rational numbers].

# Thrice Dice Twice

A once very difficult but now very simple problem in probability from Ian Stewart’s Do Dice Play God? (2019):

For three dice [Girolamo] Cardano solved a long-standing conundrum [in the sixteenth century]. Gamblers had long known from experience that when throwing three dice, a total of 10 is more likely than 9. This puzzled them, however, because there are six ways to get a total of 10:

1+4+5; 1+3+6; 2+4+4; 2+2+6; 2+3+5; 3+3+4

But also six ways to get a total of 9:

1+2+6; 1+3+5; 1+4+4; 2+2+5; 2+3+4; 3+3+3

So why does 10 occur more often?

To see the answer, imagine throwing three dice of different colors: red, blue and yellow. How many ways can you get 9 and how many ways can you get 10?

 Roll Total=9 Dice #1 (Red) Dice #2 (Blue) Dice #3 (Yellow) 01 9 = 1 2 6 02 9 = 1 3 5 03 9 = 1 4 4 04 9 = 1 5 3 05 9 = 1 6 2 06 9 = 2 1 6 07 9 = 2 2 5 08 9 = 2 3 4 09 9 = 2 4 3 10 9 = 2 5 2 11 9 = 2 6 1 12 9 = 3 1 5 13 9 = 3 2 4 14 9 = 3 3 3 15 9 = 3 4 2 16 9 = 3 5 1 17 9 = 4 1 4 18 9 = 4 2 3 19 9 = 4 3 2 20 9 = 4 4 1 21 9 = 5 1 3 22 9 = 5 2 2 23 9 = 5 3 1 24 9 = 6 1 2 25 9 = 6 2 1 Roll Total=10 Dice #1 (Red) Dice #2 (Blue) Dice #3 (Yellow) 01 10 = 1 3 6 02 10 = 1 4 5 03 10 = 1 5 4 04 10 = 1 6 3 05 10 = 2 2 6 06 10 = 2 3 5 07 10 = 2 4 4 08 10 = 2 5 3 09 10 = 2 6 2 10 10 = 3 1 6 11 10 = 3 2 5 12 10 = 3 3 4 13 10 = 3 4 3 14 10 = 3 5 2 15 10 = 3 6 1 16 10 = 4 1 5 17 10 = 4 2 4 18 10 = 4 3 3 19 10 = 4 4 2 20 10 = 4 5 1 21 10 = 5 1 4 22 10 = 5 2 3 23 10 = 5 3 2 24 10 = 5 4 1 25 10 = 6 1 3 26 10 = 6 2 2 27 10 = 6 3 1

# Toxic Turntable #22

Currently listening…

• Dźmutia Zirih, Plz Yrslf (1976)
• Far Beyond Xanadu, Dionysus’ Holy Name (1992)
• Yolanda Grovedrew, Not for Duke War (1997)
• Egzotiq, Vous N’Êtes Que (1984)
• Doctor Yacht, Invoke the Geigar (2009)
• Forschung-239, Jisirlo (1995)
• Gary Jophe, Silver Sands (1992)
• მზის მგელი, მგლისთვალება (2008)
• Helios Epoch, Nahtloser Neuntöter (2009)
• WihlhiW, Gaze Fix (1996)
• Ossafracht, Lokomotiv Zinken (2002)
• Vora xMqa, Future Is An Asylum (2015)
• հաց և գինի, Պետրիկոր (2020)
• Floris Nox, God is Caffeinated (1988)
• Phonophoro L.G., El Coro del Abismo (1988)
• Oscar’s Vital Glove, We Hate Tweeve (2003)
• Ecofoxes, When the Hen (1994)
• ბვემწა, ფვიტი ჰმრე (2017)
• Aoatt Leit, Trey Drake (1993)
• Audiosun, Lucus (Non Lucendo) (1995)
• Hildegard von Bingen, Hortus Deliciarum (2018)
• Ikexon, H.M.T. (2014)

Previously pre-posted

Toxic Turntable #1#2#3#4#5#6#7#8#9#10#11#12#13#14#15#16#17#18#19#20#21

# Potent Pencivity

“A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence.” — Howard Whitley Eves (1911-2004)