# 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

# The Art Grows Onda

Anyone interested in recreational mathematics should seek out three compendiums by Ian Stewart: Professor Stewart’s Cabinet of Mathematical Curiosities (2008), Professor Stewart’s Hoard of Mathematical Treasures (2009) and Professor Stewart’s Casebook of Mathematical Mysteries (2014). They’re full of ideas and puzzles and are excellent introductions to the scope and subtlety of maths. I first came across Alexander’s Horned Sphere in one of them. I also came across this simpler shape that packs infinity into a finite area:

I call it a horned triangle or unicorn triangle and it reminds me of a wave curling over, like Katsushika Hokusai’s The Great Wave off Kanagawa (c. 1830) (“wave” is unda in Latin and onda in Spanish).

The Great Wave off Kanagawa by Katsushika Hokusai (1760–1849)

To construct the unicorn triangle, you take an equilateral triangle with sides of length 1 and erect a triangle with sides of length 0.5 on one of its corners. Then on the corresponding corner of the new triangle you erect a triangle with sides of length 0.25. And so on, for ever.

When you double the sides of a polygon, you quadruple the area: a 1×1 square has an area of 1, a 2×2 square has an area of 4. Accordingly, when you halve the sides of a polygon, you quarter the area: a 1×1 square has an area of 1, a 0.5 x 0.5 square has an area of 0.25 or 1/4. So if the original triangle of the unicorn triangle above has an area of 1 rather than sides of 1, the first triangle added has an area of 0.25 = 1/4, the next an area of 0.0625 = 1/16, and so on. The infinite sum is this:

1/4 + 1/16 + 1/256 + 1/1024 + 1/4096 + 1/16384…

Which equals 1/3. This becomes important when you see the use made of the shape in Stewart’s book. The unicorn triangle is a rep-tile, or a shape that can be divided into smaller copies of the same shape:

An equilateral triangle can be divided into four copies of itself, each 1/4 of the original area. If an equilateral triangle with an area of 4 is divided into three unicorn triangles, each unicorn has an area of 1 + 1/3 and 3 * (1 + 1/3) = 4.

Because it’s a rep-tile, a unicorn triangle is also a fractal, a shape that is self-similar at smaller and smaller scales. When one of the sub-unicorns is dropped, the fractals become more obvious:

Elsewhere other-posted:

# Performativizing Papyrocentricity #37

Papyrocentric Performativity Presents:

Maths and Marmosets – The Great Mathematical Problems: Marvels and Mysteries of Mathematics, Ian Stewart (Profile Books 2013)

Be Ear Now – Sonic Wonderland: A Scientific Odyssey of Sound, Trevor Cox (Vintage 2015)

Exquisite Bulgarity – The Future of Architecture in 100 Buildings, Mark Kushner (Simon & Schuster 2015)

Stellar StoryDiscovering the Universe: The Story of Astronomy, Paul Murdin (Andre Deutsch 2014)

Terms of EndrearmentShe Literally Exploded: The Daily Telegraph Infuriating Phrasebook, Christopher Howse and Richard Preston (Constable 2007)

Or Read a Review at Random: RaRaR

# He Say, He Sigh, He Sow #20

“In 1997, Fabrice Bellard announced that the trillionth digit of π, in binary notation, is 1.” — Ian Stewart, The Great Mathematical Problems (2013).