Petals Around the Rose

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The answer to this roll is six

Petals Around the Rose is a lateral thinking mind game traditionally played with five dice.

It is played by computer program or with physical dice that are rolled by a Potentate of the Rose, a person who knows the secret of the game. For each roll of all dice, there is a single numerical solution. The players then attempt to arrive at this solution. If they do not, the Potentate of the Rose will tell it to them, and it is their task to figure out the solution for the next roll.

There are three, and only three, rules:

The name of the game is "Petals Around the Rose", and the name is significant.
The answer is always zero or an even number.
Anyone who knows the game may give the answer to any roll, but they must not disclose the reasoning.

When a person discovers how the answer is computed, they become a Potentate of the Rose themselves. Much akin to magic tricks, an integral part of the puzzle is that those who have solved it are urged to keep the solution a secret.

[edit] Solutions

Spoiler warning: Those unfamiliar with the game may prefer not to read the following explanations.

The "rose" is the center dot on any face that has one (i.e. 1, 3, and 5) and the "petals" are all dots around it -- the 1 face has no petals, the 3 face has two petals and the 5 face has four petals. The die faces without center dots (i.e. 2, 4, and 6) do not count. Counting the total petal dots yields that round's answer.

Alternately, one may sum the faces of odd dice, and then subtract the number of them. Thus, in a roll of 1, 3, 4, 5, and 3, the odd faces sum to 12. There are four odd dice, so the solution is 12-4, or 8.

A PHP solution:

   function solve_petals_around_the_rose($dice) {
     $answer = 0;
     foreach($dice as $die) {
       $answer += $die % 2 == 0 ? 0 : $die - 1;
     }
     return $answer;
   }

Given the output from a program that will generate random rolls and score them, statistical analysis makes the underlying rule fairly obvious. For example in rolls with a score of zero, 3 and 5 never occur. And in rolls with a score of two, 5 never occurs. The other four numbers occur equally frequently and can occur in a roll with a score of zero.

1, 2, 4 and 6 are similarly distributed
3 and 5 do not occur when score is 0
5 does not occur when score is 2
maximum score is 20 and has all 5
next highest (18) has only 3 and 5 in ratio of 1:4
next highest (16) has ratio of (1,2,4,6),3,5 in a ratio of 1:0:4 or 0:2:3

[edit] Petals and Roots

One of the more obsure extensions to the game Petals Around the Rose is a similar game called "Petals and Roots", also known as "Fish under the Icehole" (in regards to the alternate title of Petals Around the Rose "Polar Bears Around an Icehole")

In this extension, the player is given three similar rules to those given in Petals around the Rose.