Petals Around the Rose

The answer to this roll is six

Petals Around the Rose is a mathematical challenging puzzle in which the object is to work out the formula by which a number is derived from the roll of a set of five or six dice. It is often used as an exercise in inductive reasoning. The puzzle became popular in computer circles in the mid 1970s, particularly through an anecdote recounted in Personal Computing which depicts Bill Gates working out the solution in an airport.

Puzzle play

The puzzle is commonly presented as a game in which one person rolls the dice and announces a numerical result for each roll. Players are instructed to work out the pattern and announce the result of each roll themselves, but they are supposed to keep the formula for that result to themselves.

The formula

The key to the formula is given by the name of the puzzle. The solution for a throw is found by counting the "petals around the rose", where a "rose" is any die face with a center dot. On a standard 6-sided die, this corresponds to the three odd faces—1, 3, and 5. The rose's "petals" are the dots which surround the center dot. There is no rose on the 2, 4, or 6 faces, so these count as zero. There are no petals on the 1 face, so it also counts as zero. There are two petals and four petals on the 3 and 5 faces, respectively. Thus, the solution to a given throw can be found by:

• Adding the total petals, or;
• Starting with 2 times the number of dice that land on the 3 face and adding to it 4 times the number of dice that lands on the 5 face.

In the example above, there is one 3 and one 5 face, so the result is two plus four, or six.