The Numbers in Dice Stacking and Balancing

Behold the 120-sided dice, the d120:
http://thedicelab.com/d120.html

Of course, the d120 is available on Maths Gear!
https://mathsgear.co.uk/products/d120-dice

Here is the numerically balanced d20 as well.
https://mathsgear.co.uk/products/numerically-balanced-d20

If you’re in the USA, it might be easier to buy direct from the Dice Lab guys via MathArtFun. All of their unique dice are available here:
http://www.mathartfun.com/DiceLabDice.html

If you’re quick, get one of the free dice from the video!
UPDATE: They are all gone!

Read the mathematics behind finding the numerically balanced d20:
http://oberlin.edu/math/faculty/bosch/nbd_abridged.pdf

This is a spreadsheet with the numbers for all 15 different d6 dice. Or the numbers are pasted below.
https://www.dropbox.com/s/t03on5cwwmjvysf/d6-layouts.xlsx?dl=0

Here are my previous spherical videos with Henry Segerman:
https://www.youtube.com/watch?v=x0gRcRd_rHs
https://www.youtube.com/watch?v=Yp12c3-IL-I

A NOTE ON NOTATION:
I deliberately say “dice” instead of “die”. That’s my personal preference. However, I accidentally flip between “vertex” and “vertices” for the singular of “vertices”. Sorry about that.

CORRECTIONS:
- The top middle dice in the diagram accidentally has two 5s and no 4. The opposite facing face pairs should be (2,1) (3,6) (4,5). First spotted by Haxxx5, followed by Sam.
- Let me know if you spot any more mistakes!

THANKS:
Thanks to Jared Ro Bear who originally emailed me about spin-down dice.

Huge thanks to my Patreon supporters, a subset of which are:
Patrick Stover
Malcolm Rowe
Caitlin Mosca
Doug Jones
Philippe von Bergen

Music by Howard Carter
Design by Simon Wright

MATT PARKER: Stand-up Mathematician
Website: http://standupmaths.com/
Maths book: http://makeanddo4D.com/
Nerdy maths toys: http://mathsgear.co.uk/

Numbers on dice Sum around vertices Vertex variance Opp faces sum Opp variance
(2,1) (3,4) (5,6) [9,10,10,10,11,11,11,12] 0.75 [3,7,11] 10.67
(2,1) (3,5) (4,6) [8,10,10,9,12,11,11,13] 2.25 [3,8,10] 8.67
(2,1) (3,6) (4,5) [8,11,9,9,12,12,10,13] 2.75 [3,9,9] 8
(3,1) (2,4) (5,6) [8,10,9,10,11,12,11,13] 2.25 [4,6,11] 8.67
(3,1) (2,5) (4,6) [7,10,9,9,12,12,11,14] 4.25 [4,7,10] 6
(3,1) (2,6) (4,5) [7,11,8,9,12,13,10,14] 5.25 [4,8,9] 4.67
(4,1) (2,3) (5,6) [8,9,9,11,10,12,12,13] 2.75 [5,5,11] 8
(4,1) (2,5) (3,6) [6,9,9,9,12,12,12,15] 6.75 [5,7,9] 2.67
(4,1) (2,6) (3,5) [6,10,8,9,12,13,11,15] 7.25 [5,8,8] 2
(5,1) (2,3) (4,6) [7,8,9,11,10,12,13,14] 5.25 [6,5,10] 4.67
(5,1) (2,4) (3,6) [6,8,9,10,11,12,13,15] 7.25 [6,6,9] 2
(5,1) (2,6) (3,4) [6,10,7,10,11,14,11,15] 8.25 [6,8,7] 0.67
(6,1) (2,3) (4,5) [7,8,8,12,9,13,13,14] 6.75 [7,5,9] 2.67
(6,1) (2,4) (3,5) [6,8,8,11,10,13,13,15] 8.25 [7,6,8] 0.67
(6,1) (2,5) (3,4) [6,9,7,11,10,14,12,15] 8.75 [7,7,7] 0