Pivot swap permutations in a 4 ball break

Basic 4 ball break

In a 4 ball break the first pioneer (P1) and forward pioneer (Pf) swap while the pivot (Pi) stays the same

4 ball break

Shot
1Roquet P1
2Hoop approach. Striker ball position critical. Reception ball less so
3Run hoop
4Roquet reception ball
5Croquet reception ball to new Pf. Striker ball within range of Pi
6Roquet pivot
7Take off to old Pf ≡ next P1

Pivot swap

Start with roquet of P1. P1 is rushed to near Pi after it becomes the reception ball.
It must have been accurately placed in a different position than for a normal 4 ball break hoop, so you need to know 2 hoops in advance that a pivot swap is required.

Pivot swap

The sequence of shots from n-back is a reflection in the WE centre line of that from n for n ∈ [1,4].
The last shot of a pivot swap is the same as the last shot of a 3 ball break.
Pivot swaps at even numbered hoops are much harder than those at odd numbered hoops.

Arbitrary P1, Pf and Pi permutations

The objective is to have the required balls at P1, Pf and Pi at, say, 2 back.
Start at hoop 1 with balls B1, B2 and B3 at P1, Pi and Pf respectively.

HoopMoveP1PiPf
1NormalB1B2B3
2NormalB3B2B1
…
2bNormalB3B2B1
HoopMoveP1PiPf
1P swapB1B2B3
2NormalB3B1B2
…
2bNormalB3B1B2
HoopMoveP1PiPf
1NormalB1B2B3
2P swapB3B2B1
3NormalB1B3B2
…
1bNormalB1B3B2
2bNormalB2B3B1
HoopMoveP1PiPf
1NormalB1B2B3
2P swapB3B2B1
3P swapB1B3B2
4NormalB2B1B3
…
2bNormalB2B1B3
HoopMoveP1PiPf
1P swapB1B2B3
2P swapB3B1B2
3NormalB2B3B1
…
1bNormalB2B3B1
2bNormalB1B3B2
HoopMoveP1PiPf
1P swapB1B2B3
2P swapB3B1B2
3P swapB2B3B1
4NormalB1B2B3
…
2bNormalB1B2B3

Example 1

Striking ball R arrives at hoop 5 with Y as P1, K as Pi and B as Pf.
We want to reach 2 back with B as P1, Y as Pi and K as Pf to make a 2 hoop leave after running 3 back.
This set of moves is required:
HoopMoveP1PiPf
5P swapYKB
6NormalBYK
1bNormalKYB
2bBYK

Example 1

Croquet home