25 Lotus Leaves, in the starting Layout

How to Play 25 Lotus Flowers

25 Lotus Flowers is a  rotating-piece puzzle I recently invented (at least I’m ignorant of any earlier versions). The object is to turn the 25 lotus Flowers to form a given pattern. The challenge is that each Flower prevents one of its neighbors from turning, so you usually have to turn several Flowers before you can turn the one you want to turn.

I called the puzzle 25 Lotus Flowers because the 25 puzzle pieces look to me like ancient Egyptian images of lotus flowers, and the addictive nature of the puzzle reminds me of the Lotus Eaters in The Odyssey.

Mesu Smelling a Lotus (detail), ca. 1525-1504 BCE. From a public domain image courtesy of The Metropolitan Museum of Art
Mesu Smelling a Lotus (detail), ca. 1525-1504 BCE. From a public domain image courtesy of The Metropolitan Museum of Art

I originally called the puzzle 25 Lotus Leaves, then found that I was actually thinking of Egyptian lotus flowers rather than leaves.

How to Participate

You can get started playing this puzzle by 3D printing your own Board and Flowers from my 25 Lotus Flowers design on Thingiverse, Join the discussion on that page!

Glossary

A few terms will help you make sense of the puzzle.

Flower One of the 25 puzzle pieces.  The center hole in the Flower fits one of the pegs of the Board. Any given Flower can be rotated to one of three Positions.

A Lotus Flower
One Lotus Flower

Position The orientation of a given FLower. That is, the direction that the top of the Flower points. Because the Board is hexagonal, there are 6 possible Flower Positions, but any given Flower can take only 3 Positions.

Positions for a 0°, 120°, 240° Flower
Positions for a 0°, 120°, 240° Flower
Poisitions for a 60°, 180°, 300° Flower
Positions for a 60°, 180°, 300° Flower

Board The surface that the Flowers sit on. The Board has 25 pegs, arranged in a hexagonal pattern. The Board holds 25 Flowers at all times.

The empty Board, with 25 pegs
The empty Board, with its 25 pegs
The underlying hexagonal pattern of the pegs
The underlying hexagonal pattern of the pegs

Layout  One state of the Board: The Positions of each of the 25 Flowers.

Independent Flower A Flower that can be turned without first turning some other Flower. In the figure below, the Flower marked “A” is an Independent Flower.

Independent (A) and Dependent (B) Flowers
Independent (A) and Dependent (B) Flowers

Dependent Flower A Flower that cannot be turned without first turning at least one other Flower. In the figure above, the Flower marked “B” is a Dependent Flower, because Flower “A” must be turned before Flower “B” can be turned.

Number of Dependents Given a Layout and a particular Flower in that Layout, the number of Flowers that must be turned before that particular Flower can be turned. In the figure below, Flower A has two Dependents, because A depends on B, which depends on C. That is, to turn Flower A, you have to turn two other Flowers (B and C) first.

A Flower (A) with 2 Dependents (B and C)
A Flower (A) with 2 Dependents (B and C)

Interestingly, the Flower to the right of A in the figure above has 5 dependents: you must turn the other 5 Flowers in the figure before you can turn that Flower. This is the sort of dependency that can make the puzzle difficult.

Loop A set of Flowers that have a circular dependency. That is, a set of Flowers where each Flower depends on the next, and the last Flower in the Loop depends on the first Flower in the Loop. As you can see from the figure below, none of the Flowers in a Loop can be turned.

A Loop of 6 Flowers
A Loop of 6 Flowers

A Layout that contains a Loop isn’t a valid Layout, because Flowers that can’t be turned are no fun.

The Starting Layout

The Starting Layout: diagonal lines
The Starting Layout: diagonal lines

I like this starting Layout because it’s easy to remember: it’s five diagonal lines, independent of each other, in which each Flower either points up (0°) or to the upper-right (300°).

I also like this starting Layout because it’s easy to work with: the Flowers in each line are independent of the Flowers in the other lines, and each line has only one Independent Flower.

The Goal

The object of the puzzle is to transform the board, by turning one Flower at a time, from the starting Layout to a another, given Layout, or a Layout with the given properties.

Here are some goals I’ve achieved or attempted:

Reverse This is a good starting goal. Reverse each line in the starting Layout, so that every Flower points either down (180°) or lower-left (120°). It takes 25 moves.

The reverse of the starting Layout
The reverse of the starting Layout

Mirror Transform a Layout into its left-right mirror image.

Left-right mirror of the starting Layout
Left-right mirror of the starting Layout

Horizontal Lines Transform the starting Layout to a set of horizontal lines, so that every Flower points upper-left (60°) or lower-left (120°).

Horizontal lines facing left
Horizontal lines facing left

Minimum Independent Flowers Create a Layout that contains the minimum possible number of Independent Flowers. See Solutions, below.

Maximum Independent Flowers Create a Layout that contains the maximum possible number of Independent Flowers. See Solutions, below.

Maximum Number of Dependents Create a Layout where the largest Number of Dependents of the Flowers in that Layout is the maximum possible. For example, in the starting Layout, the number of dependents in each line is 0, 6, 6, 6, 2. The largest number of Dependents in that Layout is 6. See Solutions, below.

Minimum Number of Dependents Create a Layout where  the largest Number of Dependents of the Flowers in that Layout is the minimum possible. See Solutions, below.

Most Work Create a pair of Layouts where is number of moves to transform the first Layout into the second Layout is the largest possible.

Questions

Since I only recently created the puzzle, I have a number of questions about it.

  1. How many different, valid Layouts are there? That is, of all the possible Layouts, how many don’t involve overlapping Flowers or Loops?
  2. How many disjoint sets of Layouts are there, where no Layout from one set can be transformed into a Layout from another set by turning Flowers?
  3. How many types of Loops are possible? (Of course, Loops are not allowed in playing the puzzle, but they’re interesting, mathematically).
  4. What is the minimum number of Independent Flowers in any Layout?
  5. What is the maximum number of Independent Flowers in any Layout?
  6. Is a Layout with the minimum number of Independent Flowers also a Layout with the maximum Number of Dependents?
  7. Is a Layout with the maximum number of Independent Flowers also a Layout with the minimum Number of Dependents?
  8. What is an example pair of Layouts where it takes the maximum number of moves to transform the first Layout into the second?
  9. What is that maximum number of moves?
  10. Can all Layouts be left-right mirrored?

Solutions, AKA Spoilers!

Maximum Independent Flowers. The best I’ve found so far is 16:

My maximum number of Independent Flowers solution
My maximum number of Independent Flowers solution

Minimum Independent Flowers. The best I’ve found so far is 3:

Fewest Independent Flowers (3)
Fewest Independent Flowers (3)

Maximum Number of Dependents. Interestingly, the Minimum Independent Flowers solution above also has a Flower (striped, above) with 24 dependents – the maximum possible in a board of 25 Flowers.

Minimum Number of Dependents. The best I’ve found so far is 4, which I suspect is the minimum possible, because the distance from the vertical center line to the outside of the pattern is 4 Flowers (including the center Flower).

Smallest maximum number of dependents (4)
Smallest maximum number of dependents (4)