juggling sequences with number theory

Juggling Sequences with Number Theory

&

“A Tale of Two Kingdoms”

Juggling Sequences with Number Theory

Stephen Harnish

Professor of Mathematics

Bluffton University

harnishs@bluffton.edu

Miami University 35th Annual

Mathematics & Statistics

Conference: Number Theory

September 28-29, 2007

Theorem 1: (Euler) The sequence

has no equal initial and middle sums.

Theorem 2: (Dirichlet) The sequence

has no equal initial and

middle sums.

0(3k)(k+1) +1

k

Classical Results

3

05 ( 2) 5 (2 1) 1

kk k k k

Initial and Middle Sums of Sequences

• Note that sequence {1, 2, 3, 4, …} has numerous initial sums that equal middle sums:

(1 + 2) = 3 = (3)

(1 + 2 + 3 + 4 + 5) = 15 = (7 + 8)

(1 + 2 + 3 + 4 + 5 + 6) = 21 = (10 + 11)

(1 + 2 + 3 + 4 + 5 + 6) = 21 = (6 + 7 + 8)

Sequence Sums

Definition: For the sequence

an initial sum is any value of the form

for some integer k and

a middle sum is any value of the form

for some integers j and k, where

the length of a middle sum is .

1 2 3, , , , ,kx x x x

1 2 3k kI x x x x

1,

, 1 2j k j j j kM x x x x

,j kM 1k j 1;k j

Initial and Middle Sums of Sequences

• Note that sequence {1, 2, 3, 4, …} has numerous initial sums that equal middle sums:

(1 + 2) = 3 = (3)

(1 + 2 + 3 + 4 + 5) = 15 = (7 + 8)

(1 + 2 + 3 + 4 + 5 + 6) = 21 = (10 + 11)

(1 + 2 + 3 + 4 + 5 + 6) = 21 = (6 + 7 + 8)

Results

Theorem 3: Every initial sum of the sequence

is equal to some middle sum

Conjecture 1: Given the sequence

then for each integer k , some initial sum is

equal to some k-length middle sum.

1 2 3 41, 2, 3, 4, , ,kx x x x x k

1 2 3 41, 2, 3, 4, , ,kx x x x x k 1

Initial and Middle Sums of Sequences--Fibonacci

• Note that sequence {1, 1, 2, 3, 5, 8, 13…} has the following initial sums:(1) = 1 = (1)(1 + 1) = 2 = (2)(1 + 1 + 2) = 4(1 + 1 + 2 + 3) = 7 (1 + 1 + 2 + 3 + 5) = 12 (1 + 1 + 2 + 3 + 5 + 8) = 20

ResultsConjecture 2: Some initial sum of the sequence

is equal to a k-length middle sum for each k.

Theorem 4: The Fibonacci sequence

has only two instances of equal initial and

middle sums. Namely, middle sums (1) and (2).

(Hint: use the fact that

and compare with the magnitude of each middle sum

of length 1, 2, 3, etc.)

0 1 2 3 41, 1, 2, 3, 5,x x x x x

1 2 3 42, 4, 6, 8, , 2 ,kx x x x x k

0 1 2 2 1k kx x x x x

JugglingHistory• 1994 to 1781 (BCE)—first depiction on the 15th Beni Hassan tomb of an

unknown prince from Middle Kingdom Egypt.

The Science of Juggling• 1903—psychology and learning rates• 1940’s—computers predict trajectories• 1970’s—Claude Shannon’s juggling machines at MIT

The Math of Juggling• 1985—Increased mathematical analysis via site-swap notation

(independently developed by Klimek, Tiemann, and Day)

For Further Reference: • Buhler, Eisenbud, Graham & Wright’s “Juggling Drops and

Descents” in The Am. Math. Monthly, June-July 1994.• Beek and Lewbel’s “The Science of Juggling” Scientific American,

Nov. 95.• Burkard Polster’s The Mathematics of Juggling, Springer, 2003.• Juggling Lab at http://jugglinglab.sourceforge.net/

Juggling Patterns (via Juggling Lab)

Thirteen-ball Cascade

A 30-ball pattern of period-15

named:

“uuuuuuuuuzwwsqr”

using standard

site-swap notation

531

Several period-5, 2-ball patterns

90001 12223 30520 14113

A story relating juggling with number theory…

In the first year of the new century when the Kings of Onom and Laud each decreed the annual juggling period to be 1, a peace treaty was signed…

The Pact (1400 C.E.)

A Tale of Two Kingdoms(first studied by E. Tamref)

Values of Culture 1 (Onom)

1. Annual Juggling Ceremony

Values of Culture 2 (Laud)

1. Annual Juggling Ceremony

A Tale of Two Kingdoms(first studied by E. Tamref)

Values of Culture 1 (Onom)

1. Annual Juggling Ceremony

2. Orderly—1 period per year, starting with 1, then 2, 3, etc.

Values of Culture 2 (Laud)

1. Annual Juggling Ceremony

2. Orderly—1 period per year, starting with 1, then 2, 3, etc.

A Tale of Two Kingdoms(first studied by E. Tamref)

Values of Culture 1 (Onom)

1. Annual Juggling Ceremony

2. Orderly—1 period per year, starting with 1, then 2, 3, etc.

3. Sequential & Complete—Juggling performances start with all patterns for 0 balls, then 1, 2, 3, etc.

Values of Culture 2 (Laud)

1. Annual Juggling Ceremony

2. Orderly—1 period per year, starting with 1, then 2, 3, etc.

3. Sequential & Complete—Juggling performances start with all patterns for 0 balls, then 1, 2, 3, etc.

A Tale of Two Kingdoms(first studied by E. Tamref)

Values of Culture 1 (Onom)

1. Annual Juggling Ceremony

2. Orderly—1 period per year, starting with 1, then 2, 3, etc.

3. Sequential & Complete—Juggling performances start with all patterns for 0 balls, then 1, 2, 3, etc.

4. Individuality— Monistic presentation: 1 performer per ceremony

Values of Culture 2 (Laud)

1. Annual Juggling Ceremony

2. Orderly—1 period per year, starting with 1, then 2, 3, etc.

3. Sequential & Complete—Juggling performances start with all patterns for 0 balls, then 1, 2, 3, etc.

4. Complementarity— Dualistic presentation: 2 performers per ceremony

The Pact (1400 C.E.)

In the first year of the new century when the Kings of Onom and Laud each decreed the annual juggling period to be 1, a peace treaty was signed.

To strengthen this new union, the pact was to be celebrated each year at a banquet where each kingdom would contribute a juggling performance obeying its own principles. However, to symbolize their equal status and mutual regard, each performance must consist of an equal number of juggling patterns.

Year OneFor example, at the

end of the first year, the solo juggler of Onom performed all period-1 juggling patterns with 0, 1, 2, 3, and 4 balls, while the juggler duet from Laud first performed all period-1 patterns with 0 and 1 ball and then 0, 1, and 2 balls. (Total number of patterns for each: 5)

Year Two

00 11 20 02 22 31 13 40 04

33 42 24 51 15 60 06 44 53

35 62 26 71 17 80 08

Year TwoAlso, at the end of the second year, the following were performed at the banquet—the solo juggler of Onom performed all period-2 juggling patterns with 0 to 4 balls, while the juggler duet from Laud first performed all patterns with 0 to 2 & then 0 to 3 balls. (Total number of patterns for each: 25)

0 balls 1 ball

2 balls

3 balls 4 balls

A Tale of Two Kingdoms(first studied by E. Tamref)

Values of Culture 1 (Onom)

1. Annual Juggling Ceremony

2. Orderly—1 period per year, starting with 1, then 2, 3, etc.

3. Sequential & Complete—Juggling performances start with all patterns for 0 balls, then 1, 2, 3, etc.

4. Individuality— Monistic presentation: 1 performer per ceremony

Values of Culture 2 (Laud)

1. Annual Juggling Ceremony

2. Orderly—1 period per year, starting with 1, then 2, 3, etc.

3. Sequential & Complete—Juggling performances start with all patterns for 0 balls, then 1, 2, 3, etc.

4. Complementarity— Dualistic presentation: 2 performers per ceremony

Question

Will this harmonious arrangement continue indefinitely for the Kingdoms of Laud and Onom?

For years 3 and beyond, as the sanctioned periods continually increase by one, can joint ceremonies be planned so that each abides by their own rules and each presents the same number of juggling patterns?

Period-3 Juggling Patterns

0 balls 1 ball 2 balls…

1 7 19

Period-1

# of Balls: 0 1 2 3 4

# of Patterns: 1 1 1 1 1

Year TwoAlso, at the end of the second year, the following were performed at the banquet--the solo juggler of Onom performed all period-2 juggling patterns with 0 to 4 balls, while the juggler duet from Laud first performed all patterns with 0 to 2 & then 0 to 3 balls. (Total number of patterns for each: 25)

0 balls 1 ball 2 balls 1 pattern 3 patterns 5 patterns

3 balls 4 balls7 patterns 9 patterns

Period-2

# of Balls: 0 1 2 3 4

# of Patterns: 1 3 5 7 9

Period-3 Juggling PatternsWhere have we seen these numbers before?

0 balls 1 ball 2 balls…

1 7 19

Period-3

# of Balls: 0 1 2 3 4

# of Patterns: 1 7 19 37 61

Again, Period-2

• Patterns per ball are odd numbers

• A balanced juggling performance:

(1+3+5+7+9) = 25 = (1+3+5) + (1+3+5+7)

• Recall: (the sum of the first k odds) =

So:

=

• Initial sum = Middle sum

(1+3+5+7) = 16 = (7+9)

2k

25 2 23 4

02k +1 1,3,5,

k

Pythagorean Triples

Initial & Middle sums for

• Sequence: 1 7 19 37 61 91 …• Examples:

Initial Sums: 1, 8, 27, 64, 125,…Middle Sums: 7, 26, 63, …19, 56,117,…37,

…

• Euler: No initial and middle sums are equal.

(proven in the equivalent form of has no solutions in non-zero integers a, b, and c)

0(3k)(k+1) +1

k

3 3 3a b c

The future of the “Two Kingdoms” is resolved through number theory

T.F.A.E.:

1.

2.

3. For the specific sequences of the form

(initial sum) = (initial sum) – (initial sum)

(initial sum) = (middle sum)

n n na b c n n na c b

0

( 1)n n

kk k

ConclusionTheorem 5: (Graham, et. al., 1994)The number of period-n juggling patterns

with fewer than b balls is .

Theorem 6:

T.F.A.E.:

1. The monistic and dualistic sequential periodic juggling pact can not be satisfied for years 3, 4, 5, …

2. F.L.T.

nb

F.L.T.(It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.)

Fermat/TamrefConclusion: “Add one more to your list of applications of F.L.T.”

Thus ends our exercise in:Juggling Sequences with Number

Theory

&“A Tale of Two Kingdoms”

Stephen Harnishharnishs@bluffton.edu

Website sources

• Images came from the following sites:

http://www.sciamdigital.com/index.cfm?fa=Products.ViewBrowseList http://www2.bc.edu/~lewbel/jugweb/history-1.html http://en.wikipedia.org/wiki/Fermat%27s_last_theorem

http://en.wikipedia.org/wiki/Pythagorean_triple

http://en.wikipedia.org/wiki/Juggling

Another story-line from the 14th C

• Earlier in 14th C. Onom, there had emerged a heretical sect called the neo-foundationalists. They valued orderliness and sequentiality, but they also had more progressive aspirations—the solo performer’s juggling routine would be orderly and sequential but perhaps NOT based on the foundation of first 0 balls, then 1, 2, etc. These neo-foundationalists might start at some non-zero number of balls and then increase from there.

• However, they were neo-foundationalists in that they would only perform such a routine with m to n number of balls (where 1 < m < n) if the number of such juggling patterns equaled the number of patterns from the traditional, more foundational display of 0 to N balls (for some whole number N).

• For how many years (i.e., period choices) were these neo-foundationalists successful in finding such equal middle and initial sums of juggling patterns?

• (Answer: Only for years 1 and 2).