# Farmland Gear

Farmland Gears are fairly simple puzzles which consist of two overlapping circular disks, reminiscent of other circle puzzles such as Turnstile, Rashkey, but especially Circle Puzzle because of its 4-fold symmetry. In common with the latter each disk has four pieces around the outside, one of which is shared with the other disk. What makes these puzzles so special is that they have several smaller wheels inside the disks' centres. Turning one disk will also rotate any inner wheel of the other disk that it touches.

These puzzles were invented by Douglas A. Engel, who also invented various other puzzles such as:

## The number of positions:

The two centres each have 4 orientations. For a given centre orientation, the 7 main pieces can be rearranged by any even permutation, thus we have 42·7!/2 = 40,320 positions excluding the internal wheels.
There are 3 types of internal wheels these puzzles may have. A small wheel (8 teeth) only has 2 states, normal or reversed, and turning one of these wheels will also affect one of the other internal wheels. A large wheel (11 teeth) can have any of the 11 orientations, and can be twisted in isolation. The 'mulcher', which has two small overlapping disks with 5 pieces of 2 colours, can have 5!/(3!2!) = 10 positions. It can also be moved without affecting anything else.
The table below shows the four different variants.

Name Small wheels Large wheels Mulcher Positions calculation Result
Plow Deep 1 1 1 40320 · 2 · 11 · 10 8,870,400
Big Crop 2 2 0 40320 · 22 · 112 19,514,880
Hay Ride 3 1 0 40320 · 23 · 11 3,548,160
Feed Lots 4 0 0 40320 · 24/2 322,560

## Notation:

Let a clockwise quarter turn of the left disk be denoted by L1. A half turn is denoted by L2, and an anti-clockwise turn by L3. Turns of the right disk are denoted in the same way, but using the letter R.

## Solution:

Phase 1: Solve the main pieces.
This part is quite simple, especially if you first solve the top and bottom pieces, and the detailed steps below only make it seem more complicated than it is.

1. Rotate the disks so that their centres are the right side up.
2. Find the piece that belongs at the top of the left disk. Do the appropriate move sequence from the following list:
Top-left: -
Left: L2 R2 L3 R2 L3
Bottom-left: L3 R2 L2 R2 L3
Centre: R2 L1 R2 L3
Top-right: L1 R3 L3 R1
Right: L1 R2 L3 R2
Bottom-right: L1 R1 L3 R3
3. Solve the top-right, bottom-left, and bottom-right locations. These are done in exactly the same way as step b above - simply hold the puzzle such that the location you are solving is at the top-left.
4. If the pieces at the left, centre, and right are not correct, then do R2 L2 R2 L2. Repeat this again if they are still not correct.

Phase 2: Solve small internal wheels.

1. Choose the two little wheels that you want to reverse. If there is only one little wheel that you want to change, use any of the other types of internal wheel as the second of your two wheels.
2. If the two wheels lie in different disks, then
1. If the left wheel of the left disk is one of your chosen wheels, do L2.
2. If the right wheel of the right disk is one of your chosen wheels, do R2.
3. Do L1 R3 L3 R1 three times to affect only the two wheels near the centre.
4. If you did an R2 move in step 2, undo it now.
5. If you did an L2 move in step 1, undo it now.
3. If the two wheels you want to reverse lie in the left disk, then do R1 L2 R3 L2 three times. Similarly if they both lie in the right disk, do L1 R2 L3 R2.
4. Repeat the steps above until all the small wheels are correct.

Phase 3: Solve the 'Mulcher'.

1. If the mulcher's left wheel has a piece of the wrong colour, turn the left disk full circle once or twice to bring that incorrect piece to the centre of the mulcher.
2. If the mulcher's right wheel has a piece of the wrong colour, do R2, turn the left disk full circle once or twice to bring that incorrect piece to the centre of the mulcher, and then undo the R2 move.
3. Repeat steps a-b until the mulcher is solved.

Phase 4: Solve large internal wheels.

1. To adjust a big internal wheel in one disk, rotate the other disk a full circle.
2. Repeat step a if necessary, until the wheel is correct. After at most 10 full turns, the wheel will be correct.