## Description

Michael Engel is an Austrian mathematician and magician. He researches and writes books on games, mathematics and magic. Michael discovered this particular dice series through trial and error – he listed all possibilities of 21 dots and compared them by pairs!

A set of non-transitive dice has the peculiar property that no matter which of them is chosen, one of the remaining ones has a strong statistical advantage over it. Many varieties of non-transitive dice have been discovered over the years, but Michael’s set has special properties which make them superior in several ways:

- Each of the 4 dice has exactly 21 spots, same as a normal die.
- They use only the numbers 1 to 6, same as a normal die.
- They have a winning factor of at least 1.5, higher than most non-transitive dice sets.

The children’s game of “rock, scissors, paper” is an example of a non-transitive relationship – rock beats scissors beats paper beats rock. In this set of Michael Engel Dice, red beats black beats green beats blue beats red!

**Numbering:**

Red {1, 1, 4, 5, 5, 5};

Black {3, 3, 3, 4, 4, 4};

Green {2, 2, 2, 3, 6, 6};

Blue {1, 1, 1, 6, 6, 6}

**Statistics:**

Red beats black (wins 21, draws 3, loses 12, winning factor 1.75)

Black beats green (wins 21, draws 3, loses 12, winning factor 1.75)

Green beats blue (wins 18, draws 6, loses 12, winning factor 1.5)

Blue beats red (wins 18, draws 6, loses 12, winning factor 1.5)

**Demonstration:**

You ask someone to examine the dice and you point out that the numbering is non-standard but that every die has the same number of spots, 21, as a standard die. You ask them to choose any one of the four dice then you select one of the remaining three according to the statistical order above. You both throw your dice at the same time and the highest number wins. If you do this several times, you will almost certainly win more often than your opponent. The more times you throw the dice, the more likely it is that your total number of wins will be more than that of your opponent. Swindle your friends and family with the power of mathematics !!

**Size :** 16mm (5/8inch)

**Shape :** rounded edges and corners

**Colour :** red, black, green and blue with black spots.

**Materials :** injection-moulded plastic.

**Other features :** The blank dice are made in England from injection-moulded plastic. I use a precision jig system to machine the pip indents into exact position then colour them with a mica infused acrylic paint which is extremely hard-wearing.

**Shipping Cost :** 3.50 within UK, Royal Mail (untracked, 1st class). Flat rate of 10.00 GBP to the rest of the world (tracked airmail).

**Shipping Times :** 95% of orders are shipped within 2 working days. If a delay is expected, you will be informed.

**Custom-made options :** The entire range is also available made-to-order in 36mm (1 and 1/2inch), red with white spots, rounded edges and corners, shipping time 1 – 2 weeks. Some other sizes and colour combinations are possible. Please **contact me** for availability and prices.