Non-Transitive Dice Are Weird

Non-transitive dice are beyond weird. As seen in the image below of Bradley Efron’s four-dice nontransitive set, these dice are not like normal six-sided dice in that each has a unique pattern of numbers.

Imagine setting up a game where each person picks a die and then rolls that die 50 times. Whichever die rolls higher, the most times wins.

The purple die will roll a higher number than the yellow die 2/3rds of the time.

The yellow die, in turn, will roll a higher number than the red die 2/3rds of the time.

Finally, the red die will roll a higher number than the green die 2/3rds of the time.

Here’s where the weirdness of the non-transitive nature of the dice comes into play. The green die will roll a higher number than the purple die 2/3rds of the time.

Wikipedia has a nice entry on non-transitive dice, including a detailed look at the probabilities of the Efron dice. The set of Efron’s dice in the image is available for $5 plus shipping from the National Museum of Mathematics.

Non-Transitive Dice
Non-Transitive Dice

EFF Dice-Generated Passphrases

The Electronic Frontier Foundation has its own version of DiceWare, complete with several different wordlists for randomly generating passphrases with dice.

EFF’s new long list, referenced in the directions above, is designed for memorability and passphrase strength. We recommend selecting a minimum of six words from our long wordlist, or when using any other list of this size. The more words you use, the stronger the passphrase. Different wordlists may produce passphrases with different degrees of memorability, but you don’t get a significantly different passphrase strength by using one wordlist over another, if the lists are the same length.

Stormcrow’s D2020

Stormcrow decided to do its version of a D2020, this one with sides printed with actual 2020 horrors, such as Murder Hornets, and the ones likely left to still surprise us, such as World War III.


The sides include:

  • Locust Swarm
  • Online Education
  • Zoom Happy Hour
  • Meth Gators
  • Market Crash
  • Bleach Injection
  • Lockdown Hair
  • Donald Trump
  • Quarantine
  • Australian Brushfires 
  • Riots
  • Hand Sanitizer
  • Murder Hornets
  • Autonomous Zone
  • Recession
  • World War 3
  • Brexit
  • X Æ A-12
  • Tiger King
  • Covid-19


DiceKeys reminds me a lot of Diceware. The user gets a set of special dice, in this case, which they roll into a box.


The user then closes the box, preserving their DiceKeys pattern indefinitely. The result of the dice roll is then used to seed a hash algorithm that, in turn, generates application-specific passwords and even U2F tokens.


DiceKeys also comes with an app that can assemble the master password automatically by scanning the dice (including their orientation, which the app uses to generate further entropy), and the QR code-like symbols on the top and bottom of the dice.

DiceKeys are backup security keys with 196 bits of security made of 25 custom dice and a rugged holder, built to last a lifetime. . . . As password managers add support for DiceKeys, you’ll also be able to use your DiceKey in place of a `master’ password. . . .

Use the open source DiceKeys app to quickly read your DiceKey from a device. Our API allows apps and services to derive their own private secrets from your DiceKey without those apps seeing the key itself.

Our reference implementation runs in most modern web browsers, allowing it to work on an incredibly diverse range of devices. While built with web-based technologies (TypeScript & WebAssembly), it runs entirely locally on your device.

We are also developing Android and iOS versions to provide a richer experience on those devices.

The cost for a set of DiceKeys looks to run about US$25.