# 1) Moessner's Magic (1)

```Welcome
Art of Numbers
CCC-801
Instructor : Neha Gupta
Email id : [email protected]
Office : B-122 (c)
Art of Numbers : Overview
• This course deals with two aspects of numbers. In the first part
of the course we take up some unexplored patterns that exist
in ‘nature’, study them and see some of their applications.
• The second part looks at numbers as carriers of information
about our lives. Here we learn how to analyze and present data
in ways that make sense from the raw data. We use the
spreadsheet program in Open Office to present and analyze
the data in depth.
Detailed Syllabus
Part A: Fun with Numbers (~ 12 lectures)
•
•
•
•
•
•
Moessner’s Magic
Golden Ratio
Pascal Triangle, Binomial Theorem
Fibonacci Sequence
Introductory Number Theory Concepts
Continued Fractions and some applications
Detailed Syllabus
Part B: Handling Data (≤ 3 lectures)
• Descriptive Statistics like mean, median, mode, range,
standard deviation, percentiles, quartiles
• Charts – Bar Charts, column charts, Line Charts, Pie
Charts, scatter plots
References
• The Book of Numbers by John Horton Conway, Richard K. Guy. 2nd edition,
Copernicus.
• The Heart of Mathematics: An Invitation to Effective Thinking by Edward B.
Burger and Michael Starbird. 3rd edition, Wiley.
• The Visual Display Of Quantitative Information by Edward Tufte. 2nd
edition, Graphics Press.
• Excel 2007 for Starters: The Missing Manual by Matthew MacDonald.
Shroff/O'Reilly.
• Analyzing Business Data with Excel by Gerald Knight. Shroff/O'Reilly.
Assessment Scheme
Components
Attendance
Assignments
Quizzes
Final Term Report
Total
%
10
10 x 2
15 x 3
25
100
Assignments : 2 ( 10 each )
Quizzes : 3 (15 each)
Total marks = 10 + 20 + 45 + 25 = 100
Plan for the Course
Part A: Fun with numbers : mostly lectures on Slides/Board
Part B: Handling Data (2/3 classes for data Handling using Excel)
Quizzes : during Lecture timings
Assignments : Submit hard copy on the due date.
Final Exam Report: (you can work in groups of ≤ 4 members)
• A pool of topics will be uploaded on BB near to the date of
submission.
• Submission of report(hard copy) by 17th September (Monday), 2018.
• Group Members name & topic to be reported by 10th September,
2018.
Lets get started
Yayyyyy!!!
Here, step size = 2
Here, step size = 3
Here, step size = 4
Moessner’s theorem says that the final sequence is 1n , 2n , 3 n, …
where n = step size.
• This construction is an interesting combinatorial curiosity
that has attracted much attention over the years.
Moessner’s theorem was never proved by its eponymous
discoverer.
• 1951 A.Moessner conjures it: (Works for all n ϵ N)
• 1952 O.Perron proves it
• 1952 I.Paasche and H.Salie generalize it
• 1966 Long presents an alternative proof ( and generalizes it)
• 2010 Hinze, Rutten & Niqui present new proofs of the thm
The initial step size n is constant. What happens if we increase it in each step?
Let us repeat the construction starting with a step size of one and increasing the step size by
one each time.
…, K+1C2 , …
First generalization : increasing the step size by one each time
Step
Size
1
2
4
3
Step Size increasing by 1
5
( Paasche’s Theorem )
• The proof of the Moessner’s and Paasche’s theorem can be
found on a recent paper titled
“On Moessner’s Theorem “ by Dexter Kozen and Alexandra
Silva
https://www.cs.cornell.edu/~kozen/papers/Moessner.pdf
Second generalization : Let us now increment the increment by one in each
step.
Thus incrementing the step size by 1, 2, 3, 4, . . . in successive steps
crossing out 1, 4, 10, 20, . . . , k+2C3 , …
3
Step
Size
1
4
1 +2 = 3
6
15
3 +3 = 6
Step size increasing by 1, 2, 3, …
...
20
6 + 4 = 10
• Let us now increment the increment by one in each step, thus incrementing the
step size by 1, 2, 3, 4, . . . in successive steps, crossing out 1, 4, 10, 20, . . . , k+2C3 ,
....
90 105
362
• The final sequence consists of the superfactorials
• 1, 2, 12, 288, . . . = 1!, 2!1!, 3!2!1!, 4!3!2!1!, . . . = 1!!, 2!!, 3!!, 4!!, . . . .
• The generalization of Moessner’s theorem that handles these cases is one of the
particular cases of Paasche’s theorem.
Summary :
• Moessner’s theorem describes a procedure for generating a sequence of n
integer sequences that lead unexpectedly to the sequence of nth powers 1n ,
2n , 3n , . . . .
• Paasche’s theorem is a generalization of Moessner’s; by varying the
parameters of the procedure, one can obtain the sequence of factorials 1!,
2!, 3!, . . . or the sequence of super factorials 1!!, 2!!, 3!!,
• Long’s theorem generalizes Moessner’s in another direction, providing a
procedure to generate the sequence a · 1 n-1 ,(a + d) · 2n-1 , (a + 2d) · 3n-1 , . . .
• Proofs of these results in the literature are typically based on combinatorics
of binomial coefficients or calculational scans.
Finish.
Yayyyyyy 
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