Problems & Puzzles: Puzzles

Puzzle 747. A second follow up to Puzzle 742 (about golygons) Here we ask you to extend the golygons to the 3D space, as suggested independently by Fred Schneider and Jan van Delden. In this case, the figure produced is a "curve in the space 3D". Here is an example:   A golygon in the 3D space is valid if: a) Each side is orthogonal to the previous side. b) The complete curve is a closed one c) No one single side crosses or touch any other (except in the "start-end" point). d) In order to resemble better the original 2D golygons, we will add the following condition to the first one: the complete route of sides from the beginning to the end must follow a cyclic change of directions: X->Y->Z->X->Y->Z->... and so on.   Q. Send your minimal 3D-golygon solutions (data and drawing if possible) for the first five valid quantity of sides. a) using consecutive natural numbers as sides. b) using consecutive prime numbers as sides.

Conntributions came from Jan Delden and Fred Schneider

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Jan wrote:

The number of vertices:

Since we must have that the directions cycle we have n=0 mod 6.
Furthermore the lengths must differ, so in each direction, +/- (x,y,z) we must have at least 2 vertices.

So we have n=0 mod 6, n>=12.

 

The primes>2 all have the same sign, no further restrictions.
It is easy to see that prime 2 is not allowed.

 

If the natural numbers are considered, the lengths alternate in parity.
If one direction, say +x, starts with an even parity
the corresponding direction, –x, starts with an odd parity.
Therefore the number of lengths in the +x/-x direction should both be even.
So we have n=0 mod 12, n>=12.

 

Question a.
All solutions start with 1.

Question b.

n    smallest prime
12  137
18  60589 [No intersection, hard to see!]

24  13

30  3

36  5

 

The attached zip-file Golygons3D.zip contains:
Gygols.jar, a java application to display solutions.

Inputfiles of the form xyz#natural.txt (Question a) xyz#.txt (Question b).

 

A screenshot:

 

Current display: n=60, natural numbers.

It will display a set of points in 3D, connecting them by lines.
Dragging the mouse will rotate the model.
On the bottom row on the left one can change between:

• Perspective view (parallel projection)
• Red/Green glasses
• Red/Blue glasses

If you use colored glasses the slider at the toprow let’s you finetune the 3D-effect.
To the right you can change the position of the model with respect to the screen on which it is projected.

 

Open:    let’s you import a textfile describing the points in 3D.
Check:   check if your set of points are a proper solution to the puzzle (Question a/b).

Save:    save current image (*.png/*.gif), defaults to *.png.

The inputfile must be a text file of the following form:

 

#Part of cube
0 0 0
0 0 1
0 1 1
1 1 1
1 0 1
1 0 0

 

So # defines a comment-line. One point in 3D per line.

The program will close the polygon itself, i.e. there is no need to end with the starting point.
In fact if you do the [Check] will fail.

 

Installation:

If the java-control panel is installed on your computer, you’re done.
If not install the latest version of the JRE (Java Runtime Environment).

Note:

You won’t need SE unless you want to program in java yourself.

 

Copy gygols.jar anywhere on your harddrive and dubbelclick on the file.
Alternative 1: open a command-window on the directory where you put the file
Type: java –jar Gygols.jar [Enter]
Alternative 2: Create a desktop shortcut to the jar-file.

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Fred wrote:

Notation explanation:

3xP means move 3 in the "P"ositive "x" direction

139zN means move 139 in the "N"egative "z" direction

etc:

 

The solutions below are all minimal:

 

SOLUTION for a 3-d prime golygon of 12 consecutive primes: 137xP 139yP 149zP 151xN 157yN 163zN 167xN 173yN 179zN 181xP 191yP 193zP

 

SOLUTION for a 3-d prime golygon of 18 consecutive primes: 60589xP 60601yP 60607zP 60611xN 60617yN 60623zN 60631xP 60637yP 60647zP 60649xN 60659yN 60661zN 60679xN 60689yN 60703zN 60719xP 60727yP 60733zP

 

SOLUTION for a 3-d prime golygon of 24 consecutive primes: 13xP 17yP 19zP 23xP 29yP 31zN 37xP 41yN 43zN 47xN 53yP 59zP 61xP 67yP 71zP 73xN 79yP 83zN 89xP 97yN 101zN 103xN 107yN 109zP

 

SOLUTION for a 3-d prime golygon of 30 consecutive primes: 3xP 5yP 7zP 11xN 13yN 17zP 19xN 23yN 29zN 31xP 37yN 41zP 43xN 47yN 53zP 59xP 61yP 67zP 71xP 73yN 79zN 83xN 89yN 97zN 101xP 103yP 107zN 109xN 113yP 127zP

 

SOLUTION for a 3-d prime golygon of 36 consecutive primes: 3xP 5yP 7zP 11xN 13yN 17zN 19xN 23yN 29zN 31xN 37yN 41zN 43xP 47yP 53zP 59xN 61yP 67zP 71xN 73yP 79zP 83xN 89yN 97zP 101xP 103yP 107zN 109xP 113yN 127zN 131xN 137yP 139zN 149xP 151yN 157zP

 

Minimal solutions begin with 3 after this:

 

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Balanced 3-d golygons:

 

In each dimension, you must move increase and decrease the same number of times.  This is more challenging and seems to forestall the 3 convergence:

SOLUTION for a 3-d prime golygon of 12 consecutive primes: 137xP 139yP 149zP 151xN 157yN 163zN 167xN 173yN 179zN 181xP 191yP 193zP

 

SOLUTION for a 3-d prime golygon of 18 consecutive primes: 60589xP 60601yP 60607zP 60611xN 60617yN 60623zN 60631xP 60637yP 60647zP 60649xN 60659yN 60661zN 60679xN 60689yN 60703zN 60719xP 60727yP 60733zP

 

SOLUTION for a 3-d prime golygon of 24 consecutive primes: 2213xP 2221yP 2237zP 2239xN 2243yN 2251zN 2267xN 2269yN 2273zN 2281xP 2287yP 2293zP 2297xN 2309yN 2311zP 2333xP 2339yP 2341zN 2347xP 2351yP 2357zN 2371xN 2377yN 2381zP

 

SOLUTION for a 3-d prime golygon of 30 consecutive primes: 463xP 467yP 479zP 487xN 491yN 499zN 503xN 509yN 521zP 523xP 541yN 547zN 557xN 563yP 569zN 571xP 577yP 587zP 593xN 599yP 601zN 607xP 613yP 617zP 619xP 631yN 641zN 643xN 647yN 653zP

 

SOLUTION for a 3-d prime golygon of 36 consecutive primes: 29xP 31yP 37zP 41xN 43yN 47zN 53xN 59yP 61zN 67xN 71yN 73zP 79xP 83yN 89zN 97xP 101yP 103zN 107xP 109yP 113zP 127xP 131yN 137zP 139xN 149yN 151zP 157xN 163yP 167zP 173xN 179yN 181zN 191xP 193yP 197zN

 

SOLUTION for a 3-d prime golygon of 42 consecutive primes: 19xP 23yP 29zP 31xN 37yN 41zN 43xN 47yN 53zN 59xN 61yP 67zN 71xN 73yN 79zP 83xP 89yN 97zP 101xP 103yP 107zN 109xP 113yP 127zP 131xP 137yP 139zP 149xP 151yN 157zN 163xN 167yP 173zP 179xP 181yN 191zN 193xN 197yP 199zP 211xN 223yN 227zN

 

SOLUTION for a 3-d prime golygon of 48 consecutive primes: 31xP 37yP 41zP 43xN 47yN 53zN 59xN 61yN 67zN 71xN 73yN 79zN 83xP 89yP 97zP 101xP 103yP 107zN 109xN 113yP 127zP 131xN 137yN 139zN 149xP 151yP 157zP 163xP 167yN 173zP 179xP 181yP 191zP 193xP 197yN 199zP 211xN 223yP 227zP 229xP 233yN 239zN 241xN 251yN 257zN 263xN 269yP 271zN

Graphs can be made using tools from www.geogebra.org. (As a matter of fact the one I used when I posted this puzzle was provided by Fred to me by email)

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Hakan Summakoglu wrote on 20 July 2014:

a) I found the minimal starting prime for

N       Minimal prime
12      137
18      60589
24      13
30,36,42,48      3

b) I found the minimal starting natural number 1 for all N.

Here are my graph-results.

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