Shortcuts

 Sitemap Contact Newsletter Store Books Features Gallery E-cards Games

•••

 Related links Puzzles workshops for schools & museums. Editorial content and syndication puzzles for the media & publishers. Numbers, just numbers facts & curiosities... Have a Math question? Ask Dr. Math!
•••

 ••• •••

•••

 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)
•••

 ••• •••

# Previous Puzzles of the Month + Solutions

Back to Puzzle-of-the-Month page | Home
Puzzle # 126

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

 Français

Source of the puzzle:
You cannot reproduce any part of this page without prior written permission.

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

The 5 Winners of the Puzzle of the Month are:
Larry Bickford, USA - Emeline Luirard, France - Denzil Gumbo, Zimbabwe - Jakub Nogly, Poland - Sarah Farooq, Pakistan

Congratulations!

 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: for( i = 0; i < 50; ++i){ j = i; while( (j % 2) == 1){ j = (j-1)/2; } j /= 2; print " ", j+1; } 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!