The easiest natural and unnatural Eulerian Circuit for all possible club exchanges between two people

I took photos of clubs to demonstrate the 16 possible club exchanges between two people. The knobs indicate the starting state and the tops indicate the next state. A natural pass throws from the hand that is receiving a club. An unnatural pass throws from the hand that is not receiving a club and requires an immediate hand-across. The terms natural and unnatural passes were coined by Mario Battaglia-Winfield at a juggling meetup in Morro Bay in 2024.

The center four exchanges are likely collisions, although it's possible to throw them with some fudging. It's also possible to avoid any collision by having one juggler throw an unnatural pass with the opposite hand (toggling its regularity so that a straight pass becomes a diagonal pass and a diagonal pass becomes a straight pass) and then immediately do a hand-across. The resulting club exchange remains the same.

A natural Eulerian Circuit chains the 12 outside passes (all but the collisions) together so that all are used, none are repeated, the final state matches the initial state, and the next state of the previous pass matches the initial state of the current pass. No hand-acrosses are necessary.

An unnatural Eulerian Circuit chains all 16 exchanges together so that all are used and none are repeated. These can go in any order and hand-acrosses are sometimes necessary.

Jim's Jam and synchronous variations use 1, 2, 3, 4 aka the usual suspects. I was curious how many were used by an asynchronous Jim's Jamboree pattern.

Doing Top: Chutney-Jam-Marmalade-Compote vs. Bottom: Jam-Marmalade-Compote-Chutney uses 6, 4, 8, 2, 1, 8, 3, 6, 10, 3, 12, 1, 2, 8, 4, 6 (misses 5, 7, 9, 11). The misses are because passes are always thrown to the same shoulder. 5, 7, 9 and 11 are thrown to opposite shoulders (mirrored).

Doing Top: Jam-Marmalade-Compote-Chutney vs. Bottom: Chutney-Jam-Marmalade-Compote uses 10, 4, 5, 2, 1, 12, 3, 10, 6, 3, 8, 1, 2, 8, 4, 6 (misses 5, 7, 9, 11 ).

I was curious if there was a natural Eulerian Circuit for the non-collision passes and discovered this one which is relatively easy to remember. I labeled my graphic with these numbers.

The Usual Suspects

1. BR-TR → BL-TL : BR→TL, TR→BL pass-pass

2. BL-TL → BL-TL : BL→TL, TL→BL scross-scross

3. BL-TL → BR-TR : BL→TR, TL→BR spass-spass

4. BR-TR → BR-TR : BR→TR, TR→BR cross-cross

Jim's 1-count Top is Mad:

5. BR-TR → BR-TL : BR→TL, TR→BR poss-crass

6. BR-TL → BL-TL : BR→TL, TL→BL pass-scross

7. BL-TL → BL-TR : BL→TR, TL→BL sposs-scrass

8. BL-TR → BR-TR : BL→TR, TR→BR spass-cross

Jim's 1-count Top is Mild:

9. BR-TR → BL-TR : BR→TR, TR→BL crass-poss

10. BL-TR → BL-TL : BL→TL, TR→BL scross-pass

11. BL-TL → BR-TL : BL→TL, TL→BR scrass-sposs

12. BR-TL → BR-TR : BR→TR, TL→BR cross-spass

The progression for practicing this is to do all the passes as a three count (36 throws). Once that's good to go, practice the passes in Chocolate Bar (pass pass self self) for a total of 24 throws. Once that's successful, practice ultimates (all passes) for a total of 12 throws.

The easiest unnatural Eulerian Circuit I came up with keeps the first 11 passes the same. Then:

The Unusual Suspects:

13. BR-TL → BR-TL : BR→TL, TL→BR poss-sposs (beauty: scrasso-sposs)

14. BR-TL → BL-TR : BR→TR, TL→BL crass-scrass (truth: crass-posso)

15. BL-TR → BL-TR : BL→TR, TR→BL sposs-poss (beauty: crasso-poss)

16. BL-TR → BR-TL : BL→TL, TR→BR scrass-crass (truth: scrass-sposso)

the last pass of Jim's 1-count Top is Mild:

12. BR-TL → BR-TR : BR→TR, TL→BR cross-spass

For The Unusual Suspects I arbitrarily chose the Top juggler to do the first hand-across and then alternated. See The Unusual Suspects.

"beauty" indicates an unnatural pass from the Bottom juggler. "truth" indicates an unnatural pass from the Top juggler. These can both be done in non-collision passes but if they are both done in collision passes the collision happens. If only one is done in a non-collision pass a collision may occur.

I've previously referred to Mild Madness as Jim's two out of three count. I now consider it to be it's own unique pattern which uses an unnatural pass to avoid a collision. I feel that all Jim's n-count patterns are natural and plan to examine this in more detail in a future post.

Easiest natural Eulerian Circuit of juggling passes: