The need to compare the relative difficulty of different juggling patterns arises in several contexts. For example, the Library of Juggling has arranged tricks by difficulty so it’s easy for learners to progress through increasingly complex tricks as their skill levels improve. The World Juggling Federation (WJF) and the International Jugglers’ Association (IJA) conduct juggling competitions which require a mechanism to grade the difficulty of various juggling routines. The WJF’s mechanism is a small move database that assigns points to the moves, and to transitions between moves, presumably based on their difficulty levels. At IJA, Juggling Difficulty is, “determined by the type and number of objects juggled; the speed of the juggling; the types of throws, catches, balances, or other object manipulations; the complexity of combinations of juggling tricks; and the transitions between juggling tricks.”
In all these examples, the difficulty rating seems to be guided by intuition and experience which is subjective and endlessly disputable. In this blog, we review some attempts at constructing mathematical formulae that could serve as the beginnings of an objective measure of juggling difficulty.
Dancey’s Doctrine
In the Encyclopaedia of Ball Juggling (EBJ), Charlie Dancey proposes a couple of formulae to calculate the difficulty of juggling patterns involving b balls and h hands. The formulae are meant only for “simple patterns, like the cascades and fountains[1]“. Dancey starts with an obvious formulation encapsulating the principle that the greater the objects-to-hands ratio, the greater the difficulty d(b,h):
d(b,h) = b/h
He then points out that this formula would suggest that juggling 4-balls in two hands and juggling 2-balls in one hand are equally difficult since d(4,2) = d(2,1) = 2, which does not feel right. He therefore proposes another definition, d'(b,h) for the difficulty:
d'(b,h) = b/(h + h/b)
As per this formula, d'(4,2) = 1.6 > d'(2,1) = 1.33 which seems more reasonable. The mathematically inclined reader may prove that in general, if b2 > b1 and d(b2,h2) = d(b1,h1), then d'(b2,h2) > d'(b1,h1). Thus, the d’ formula seems to have a “finer resolution” than the d formula.
Remarks
It turns out that d’ and d could lead to contradictory results. For example, consider the case (b1,h1) = (2,1) and compare it against the case (b2,h2) = (9,6). We get:
d(2,1) = 2.00 > d(9,6) = 1.50; whereas
d'(2,1) = 1.33 < d'(9,6) = 1.35
The d’ formula says that the pattern with the greater balls-to-hands ratio is the easier pattern, contradicting the principle we started out with. It may be argued that though the balls to hands ratio is lower in the 9-ball 6-hand case, it requires coordination between multiple jugglers, thus making the pattern harder, so d’ is still the correct formula. Again, the mathematically inclined reader may prove that whenever the two formulae contradict each other, the pattern deemed harder by the d’ formula will involve the added complication of at least one additional juggler (i.e., at least two more hands) than the pattern which it deems as the easier one.
The self-confessed limitation of Dancey’s formulae is that they work only for “simple” patterns. The relative difficulty of different siteswap sequences involving the same number of objects and hands cannot be determined using these formulae.
Beever’s Bids
In Siteswap Ben’s Guide to Juggling Patterns, Ben Beever first presents what he calls, “the simplest, credible [siteswap] difficulty formula I have come across”:
D1 = 2 + √[Σ(Vi-2)2/P]
where D1 is the difficulty of the pattern and Vi are the siteswap numbers representing the period P pattern. The siteswap number “0” should be ignored for the summation but should be counted towards the period. It is to be assumed that this formula applies to an individual juggler throwing alternately with right and left hands, i.e., with the Hand Siteswap Sequence (HSS) “2” (see Siteswap Notation II). Indeed, the (Vi-2) term seems to be designed specifically to eliminate the difficulty contribution of the siteswap number “2” which can be held when juggled in this fashion.
Table 1 shows the results of this formula when applied to some example patterns.
| Siteswap | Difficulty D1 |
|---|---|
| n ≥ 3 | n |
| 6 3 3 | 4.45 |
| 5 5 5 5 0 | 4.68 |
| 3 1 | 3 |
In general, Table 1 (as far as it goes) seems to match our intuition and experience: The difficulty of cascades and fountains increases as the number of objects n increases; 4-ball siteswaps, like “6 3 3” and “5 5 5 5 0”, are harder than the 4-ball fountain. An interesting and debatable result that Table 1 throws up is that “5 5 5 5 0” is harder than “6 3 3”. A decidedly absurd result from Table 1 is that the 2-ball shower “3 1” is as hard as the 3-ball cascade as both have a difficulty score of 3. A potential correction I propose to fix this discrepancy is to define D1‘ which ignores both the siteswap numbers “0” and “1”. Then the difficulty score of “3 1” goes down to 2.7. For the rest of this post, we will use D1‘ instead of D1.
Notice that D1‘ effectively calculates some sort of root mean square deviation, except that the deviation is measured from the siteswap number “2” rather than from the mean of the siteswap numbers. Also, the deviation is counted only if it is positive (siteswap numbers less than 2 are ignored). Inspired by this realization, I felt that a true root mean square formula might do better, but some of the results remain equally, if not more, debatable.
Multiplex and Synchronous
The D1‘ formula easily extends to multiplex patterns: the siteswap numbers within square brackets (representing multiplex throws) will count as additional Vi‘s but won’t lead to an increase in the period P. For example, difficulty for the 5-ball sequence [3 3] [3 3] 3, which is essentially juggled as a 3-ball cascade with multiplex throws, can be calculated as:
2 + √{[(3-2)2 + (3-2)2] + [(3-2)2 + (3-2)2] + (3-2)2}/3 = 3.29
This feels reasonable as the calculated difficulty rating is a little higher than that of a 3-ball cascade but far less than that of a 5-ball cascade.
The formula works similarly for synchronous patterns though one must keep in mind that in standard synchronous notation, there is an implicit empty beat after every synchronous throw. This empty beat contributes to the period of the pattern. The difficulty for the 3-ball synchronous shower (4x, 2x) will thus be calculated as:
2 + √{(4-2)2 + (2-2)2]}/2 = 3.4
Similarly, the difficulty for the (6x, 4) (4, 6x) pattern, also written as the (6x, 4)* pattern will be calculated as:
2 + √{[(6-2)2 + (4-2)2] + [(4-2)2 + (6-2)2]}/4 = 5.16
Ben Beever’s own tabulation of difficulty for (6x, 4)* however shows a value of 5.3, slightly higher than our calculation above. I’m not sure how this was arrived at. His result for the difficulty of (4x, 2x) on the other hand matches our calculation.
The D1‘ formula also makes a case for not using the standard synchronous notation but use the one where each beat is explicitly specified. For example, the representation of the 3-ball cascade in standard synchronous notation is “(6x, 0) (0, 6x)” for which the difficulty will be calculated as:
2 + √{(6-2)2 + (6-2)2}/4 = 4.83
If we instead use the notation with each beat explicitly specified (see Siteswap Notation V), we can represent the 3-ball cascade as “(3x, 0) (0, 3x)” in which case the difficulty correctly works out as:
2 + √{(3-2)2 + (3-2)2}/2 = 3
This approach can then be extended for mixed sync-async patterns.
Passing
Though Beever does not comment on applying this formula to passing patterns, it possibly can be by identifying the effective siteswap sequence to be juggled by the individual jugglers involved in the passing pattern. For example, difficulty of the 7-ball passing pattern “4p 3” for each juggler would be:
2 + √{(4-2)2 + (3-2)2}/2 = 3.58
The formula thus suggests that “4p 3” is somewhere between the difficulty levels of the 3-ball cascade and the 4-ball fountain.
A debatable corollary of evaluating passing patterns this way is that, for example, 6-ball passing between 2-jugglers is equally hard whether they do 4-count (3p 3 3 3) or 1-count (3p) passing. Moreover, these passing patterns have the same difficulty as the individual 3-ball cascade.
State Based Formula
A second measure of difficulty suggested by Beever is based on the “average excitation level[2] of the siteswap”. He defines the difficulty D2 of the siteswap sequence as:
D2 = L/2 + B
where L is the average excitation level of the siteswap sequence and B is the number of balls involved in the corresponding pattern. The factor of 2 dividing L is chosen somewhat arbitrarily, the aim being to get D2 values to match D1 values as closely as possible for the list of siteswap sequences chosen for evaluation by Beever. D2 however, returns absurd values for multiplex patterns. It does a somewhat better job for synchronous patterns but Beever himself ignores these results when he compares the D1 and D2 formulae.
Body Moves
Beever also considered the problem of body moves and assigned a difficulty score for various body throws and catches[3] based on his judgement. He then suggests that this difficulty rating can be incorporated into the difficulty of a juggling pattern by using the formula:
X = D*[1+{ΣR/(10*P)}]
where X is the overall difficulty of the pattern, D is the siteswap difficulty (either D1 or D2 could be used), ΣR is the sum of the difficulty scores of the body throws/catches involved and P is the period.
Remarks
Let us analyze a couple of scenarios to check how well the D1‘ formula holds up against the real life experience of jugglers.
Mike Day while attempting to juggle the “7 2 6” found himself naturally falling into a synchronous rhythm and ended up discovering the (6x,4)*. This would seem to indicate that (6x,4)* is an easier pattern to juggle than “7 2 6”. Indeed, the D1‘ formula for (6x,4)* yields 5.16 while for “7 2 6” it yields a higher difficulty rating of 5.69.
Table 2 compares the score assigned to some 5-ball siteswaps in the WJF move database against the difficulty of these siteswaps as calculated by the D1‘ formula.
| Siteswap | WJF Score | D1‘ |
|---|---|---|
| (6x, 4)* | 0.10 | 5.16 |
| (6, 4x)* | 0.12 | 5.16 |
| 7 4 4 | 0.12 | 5.32 |
| 6 4 7 4 4 | 0.15 | 5.26 |
| 11 9 7 5 3 0 0 | 0.17 | 6.86 |
| 9 7 5 3 1 | 0.2 | 6.10 |
| 6 7 8 9 1 2 3 4 5 | 0.3 | 5.94 |
In this case, there are several disagreements between what the WJF move database considers harder vs. what the formula evaluates as harder. Indeed, the formula may well be wrong, but I feel it at least lends some objectivity to the debate.
Interpreting Difficulty
Notice that according to the D1‘ formula, the difference in difficulty score going from 3-ball cascade to 4-ball fountain = 4-3 = 1, is the same as the difference when going from the 4-ball fountain to the 5-ball cascade (5-4 = 1). However, most jugglers will testify that going from 4 to 5 is much harder than going from 3 to 4. Beever explains that, “the practice-time it takes, to be able to do patterns of a certain difficulty score, increases (highly) exponentially.” For example, if the practice-time required to master the b-ball cascade/fountain varies as the cube of its difficulty value, then the 3-ball cascade will need to be practiced for 33 = 27 days (~1 month), the 4-ball fountain for 43 = 64 days (~2 months) and the 5-ball cascade for 53 = 125 days (~4 months). The cubic relation has been chosen simply to illustrate the concept and is not necessarily true.
Interestingly, Dancey’s d’ formula does return a higher difference in difficulty when going from b balls to b+1 balls as compared to the difference when going from b-1 to b balls juggled with two hands. Again, the mathematically inclined may derive some pleasure from proving that statement.
Conclusion
As Ben Beever remarks, “the difficulty of a pattern is quite subjective.” The results generated by the mathematical formulae presented here remain highly debatable, even when we leave body moves out of the discussion. A probable reason for this is that the formulae ignore some key physiological factors. For example, it may be argued that as number of objects increase, the difficulty of fountain throws increases faster than the difficulty of cascade throws[4]. Another possible consideration is a measure of the asymmetry of the pattern – how different are the successive throws from the same/other hand? Both these factors could, for example, play a part in determining the relative difficulty of “5 5 5 5 0” and “6 3 3”, which we considered in Table 1.
For multiplex and synchronous patterns too, similar physiological considerations may apply. For example, a multiplexed [3 6] throw, with a low crossing “3” and a high non-crossing “6”, may be more difficult than a multiplexed [3 7] throw where both the “3” and the “7” are crossing throws. A formula based purely on the magnitude of the siteswap numbers though, would rate the [3 7] throw as more difficult than the [3 6] throw. Similarly, in the case of synchronous throws, the contribution of a (6x,4) throw to the difficulty of a pattern is calculated to be exactly the same as that of a (6,4x) throw by the D1‘ formula. Yet, the WJF move database seems to rate the (6,4x)* pattern to be slightly harder than the (6x,4)* pattern. This could tie in well with a formula which appropriately weighs the difficulty of fountain throws as discussed above.
Patterns that involve holds or empty hands (i.e., siteswap numbers “2” or “0”) may also need special consideration as they tend to disrupt the rhythmic nature of the throws, thus making the overall pattern more difficult to juggle even though these throws contribute nothing to the difficulty of the pattern in the D1‘ formula. This may be the real reason why Mike Day found it more natural to juggle the (6x,4)* than the “7 2 6”.
The formulae presented here, by the very act of generating debatable results, prompt a deeper enquiry into the factors that influence the difficulty of juggling a given siteswap sequence apart from just the magnitude of the siteswap numbers. As we identify and isolate these factors, we may one day be able to objectively rate the relative difficulties of various siteswap patterns. If nothing else, such formulae provide an intriguing diversion for the mathematically inclined.
Footnotes
- In EBJ, Dancey defines cascades and fountains in the context of two-handed juggling, but his difficulty formulae seem to be intended for the generalized case where cascades occur when b and h are mutually prime while fountains occur otherwise.
- See “Excitation Levels” on pages 26-27 of Guide to Juggling Patterns.
- See page 12 of Guide to Juggling Patterns.
- See Jack Kalvan: Optimal Juggling and Jack Boyce: The Physical Demands of Numbers Juggling.