Shortcuts
 
corner

corner

•••

Related links

symbol puzzlePuzzles workshops for schools & museums.

symbol syndicationEditorial content and syndication puzzles for the media & publishers.

symbol numbersNumbers, just numbers facts & curiosities...

symbol questionsHave a Math question?
Ask Dr. Math!

•••
corner

corner

•••

•••
corner

corner

•••

SMILE!Smile!
"The best way to catch a train in time is to manage to miss the previous one"
"Le meilleur moyen de prendre un train à l'heure, c'est de s'arranger pour rater le précédent"
-- Marcel Achard

Math Gems
cosα ≈ 1 - α2/2
(when α is small)

•••
corner

corner

•••

•••
corner

 

corner top left

Previous Puzzles of the Month + Solutions

 
 
arrow back Back to Puzzle-of-the-Month page | Home arrow home
Send to a Friend Puzzle # 126
puzzle no. 126

Send to a Friend Suggest to a friend
Comment Contact us Stumbleupon Thumb up
this page
facebook Share it on
FaceBook
Twitter Follow us
on Twitter
RSS feeds View our
RSS feeds
Digg it Digg this
story
A troublesome sequence
  A number sequence is a set of numbers arranged in an orderly fashion, such that the preceding and following numbers are completely specified. Sometimes it is very easy to find in a series what number comes next, but usually it is not! Here is a tough example: try to replace the ‘X’ in the following sequence with the most appropriate number:
1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, X, ...
Can you guess the secret rule and the magic of the sequence above?

Difficulty level: bulbbulbbulb, general math knowledge.
Category: Number series.
Keywords: number sequence, progression, series.
Related puzzles:
- The Parrot sequence,
- Pacioli puzzle.


Source of the puzzle:
© G. Sarcone.
You cannot reproduce any part of this page without prior written permission.
cube separator
Solution

The answer is X = 6.

Remove every alternating (second) number of this special sequence, what do you have left? 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8... Exactly the same sequence! Which means if you remove every second term again, you get the same sequence. Over and over.
This sequence has in fact the property to contain itself as a proper subsequence, infinitely. This is why it is called ‘fractal sequence’ or ‘sandwich sequence’.

Vivisection of the sequence
Curiously enough, there are actually infinite positive integer subsequences embedded in this fractal sequence (see table further below):
Subsequence a: Starts from position n = 1 and increments by 1 while moving 2 numbers ahead in the sequence.
Subsequence b: Starts from position n = 2 and increments by 1 while moving 22=4 numbers ahead in the sequence.
Subsequence c: Starts from position n = 4 and increments by 1 while moving 23=8 numbers ahead in the sequence.
Subsequence d: Starts from position n = 8 and increments by 1 while moving 24=16 numbers ahead in the sequence.
Etc...
As you can see in the table below, each integer subsequence starts on the 2m-1 position and jumps ahead in the fractal sequence by 2m (with m≥0) positions and increments by 1.

n k a b c d e f
1 1 1
2 1 1
3 2 2
4 1 1
5 3 3
6 2 2
7 4 4
8 1 1
9 5 5
10 3 3
11 6 6
12 2 2
13 7 7
14 4 4
15 8 8
16 1 1
17 9 9
18 5 5
19 10 10
20 3 3
21 11 11
22 6 6
23 12 12
24 2 2
25 13 13
26 7 7
27 14 14
28 4 4
29 15 15
30 8 8
31 16 16
32 1 1
33 17 17
34 9 9
35 18 18
36 5 5
37 19 19
38 10 10
39 20 20
40 3 3
41 21 21
42 11 11
43 22 22
44 6 6
45 23 23
46 12 12
47 24 24
48 2 2
49 25 25
50 13 13
51 26 26
52 7 7
53 27 27
54 14 14
55 28 28
56 4 4
57 29 29
58 15 15
59 30 30
60 8 8

cup winnerThe 5 Winners of the Puzzle of the Month are:
Larry Bickford, USA USA flag - Emeline Luirard, France French flag - Denzil Gumbo, Zimbabwe zimbabwe flag - Jakub Nogly, Poland poland flag - Sarah Farooq, Pakistan Pakistan flag

Congratulations!

cube separator
Math fact behind the puzzle

Properties of the sequence
This particular fractal sequence is obtained from powers of 2. In fact, every number of the sequence occurs at 2m(2k - 1) position, with m≥0.
For instance, the number 6 occurs at the following positions (n):
n1 = 20(2x6 - 1) = 11
n2 = 21(2x6 - 1) = 22
n3 = 22(2x6 - 1) = 44
Etc...

Here is a simple program in "bc" (available on Unix, Linux, and Cygwin) sent by Larry Bickford that generates the sequence:

With i < 50, we obtain the following output:
1 1 2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5 10 3 11 6 12 2 13 7 14 4\
15 8 16 1 17 9 18 5 19 10 20 3 21 11 22 6 23 12 24 2 25 13

 

© 2011 G. Sarcone, www.archimedes-lab.org
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.

You're encouraged to expand and/or improve this article. Send your comments, feedback or suggestions to Gianni A. Sarcone. Thanks!
cube separator
Previous puzzles of the month...
contents+solutions
puzzle solver
Solved Puzzles

Puzzle 125: A mathematical shield (2011)
Apr-May 2010
:
A chopping problem
Oct-Nov 09:
The Mark of Zorro
July-Sept 09:
radiolarian's shell
May-June 09:
circle vs square

Jan-Feb 09
:
geometric mouse

Sept-Oct 08
:
perpendicular or not...

July-Aug 08:
ratio of triangles

May-June 08:
geometry of the bees

Febr-March 08
:
parrot sequence...

Dec 07-Jan 08
:
probable birthdates?

Oct-Nov 2007:
infinite beetle path
Aug-Sept 07:
indecisive triangle

June-July 07:
Achtung Minen!

April-May 07:
soccer balls

Febr-March 07:
prof Gibbus' angle

Jan 07:
triangles to square

Aug-Sept 2006
:
balance problem

June-July 06
:
squared strip

Apr-May 06:
intriguing probabilities

Febr-March 06:
cows & chickens
Dec 05-Jan 06:
red monad

Puzzle Archive
arrow back Back to Puzzle-of-the-Month page | Home arrow home
giflet Recommend this page 
 
Twitter Information Twitter Services & Products Twitter Follow us via... Twitter Support us...

About Us
Privacy & Terms
Copyrights

Contact us
Sitemap
Press Review
Products
Features
Workshops
For Publishers
Facebook
Newsletter
RSS feeds
Twitter
Blogs
Tell a Friend
Merchandising
Link to us
Sponsorship
line
© Archimedes' Laboratory™ | The web's best resource for puzzling and mental activitie
| italian flag Introduzione | francais flag Introduction | francais flag Einführung 
spacer spacer corner right bottom