Problems & Puzzles: Puzzles

 Puzzle 742. Prime-Golygons Lee Sallows invented and named the interesting and curious geometrical polygons named "golygons". In short "golygons" are defined as polygons with all of its vertexes being right angles and whose sides are the successive members of a predefined integer sequence. For example if the predefined sequence are the natural numbers: 1, 2, 3,... the minimal 8-sides golygon for that sequence is this one.   More info about these geometrical objects here: 1, 2. One important rule of construction of the golygons is that no one single side must cross over another side. Recently, in one site in the web -that I will omit just for this week- it was shown the following 16-sides prime-golygon, that I immediately discarded because of the use of the non-prime number "1" for the smallest side. Last week I started looking for minimal sum and proper prime golygons. At this very moment my claim is that I succeeded obtaining two "correct" prime-golygon solutions {p1, p2, p3, ...; p1=>3, p(i)=nextupperprimeto(p(i-1)) } already published in that site, solutions that I think they are the "minimal sum" for the quantity of sides used in such solutions (8 & 16 sides). I will only say here that the minimal prime solution for 8 sides goes from the prime 359 to the prime 401, for a sum of 3048, while the minimal prime solution for 16 sides goes from the prime 173 to the prime 257 for a sum of 3424. It's easy to show that the quantity of sides permitted by the rules of construction of the golygons -combined by use of the integer sequence of consecutive primes for the sides dimensions- are multiples of eight (8, 16, 24, 32, ...) Q. Can you get the "minimal sum" prime-golygons  for the quantity of sides 8*k, for k=1 to 4? ___________ Note: Each sequence of 8*k consecutive primes used need not to start in any particular prime number.

As I told last week, the subject of this puzzle came from, and my own results (n=8 & n=16) were already published in, this page from Claudio Meller:

http://simplementenumeros.blogspot.mx/2014/05/1315-goligonos-goliedros-y-demas.html

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Contributions came from Giovanni Resta, Jan van Delden, Emmanuel Vantieghem, Fred Schneider, Hakan Summakoglu and Carlos Rivera

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All of them pointed out my mistake when I stated "the quantity of sides permitted by the rules of construction of the golygons -combined by use of the integer sequence of consecutive primes for the sides dimensions- are multiples of eight". The truth is that they are multiple of four starting in eight.

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All the results and the drawings sent by Resta, van Delden & Summakoglu may be seen here.

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Resta wrote:

I computed the minimal starting prime p for N from 8 to 64. They are

N      p
8   359
12  59273
16   173
20    5
24,28,...,64  3

The minimal solution is unique for N=8,12,16,20. For N=24 there are 4 minimal solutions, 6 for N=28, 22 for N=32 and 348 for N=36.

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Jan wrote:

If n>20 all minimal solutions start with 3. [I tested untill 72 and computed n=100 as well].
For each admissible n I computed the solution with minimal startprime.
For this prime I computed the solution where the bounding box has minimal area.

If n<=20 we have: (n, minimal area bounding box, smallest prime, directions):

8 580608 359 [1, 2, 3, 4, 3, 4, 1, 2]
12 31703582881 59273 [1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 1, 2]
16 732592 173 [1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 3, 4, 1, 2, 1, 2]
20 21980 5 [1, 2, 1, 2, 3, 4, 3, 2, 3, 4, 1, 4, 3, 4, 1, 4, 3, 4, 3, 4]

If n=8k we have:
24 36795   3 [1, 2, 3, 4, 3, 2, 3, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 1, 4, 1, 2, 1, 4]
32 114276 3 [1, 2, 3, 4, 3, 4, 1, 4, 1, 2, 1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 1, 2]
40 189357 3 [1, 2, 3, 2, 1, 2, 3, 4, 3, 4, 1, 4, 3, 2, 3, 4, 1, 4, 1, 2, 1, 2, 1, 4, 3, 4, 1, 4, 3, 2, 3, 4, 3, 4, 3, 4, 1, 2, 3, 2]
48 358776 3 [1, 2, 3, 2, 1, 4, 1, 2, 1, 4, 3, 4, 1, 4, 3, 2, 3, 4, 1, 4, 3, 2, 3, 4, 3, 2, 1, 2, 3, 4, 3, 4, 3, 2, 1, 2, 1, 2, 3, 4, 3, 2, 3, 2, 3, 2, 1, 4]
56 590133 3 [1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 3, 4, 1, 4, 3, 4, 1, 2, 1, 4, 3, 4, 3, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 4, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2]
64 907412 3 [1, 2, 1, 4, 3, 4, 1, 4, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 3, 2, 1, 4, 1, 2, 1, 4, 3, 4, 3, 4, 3, 4, 1, 2, 1, 4, 3, 4, 1, 2, 1, 2, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 1, 4, 1, 2]
72 1372500 3 [1, 2, 1, 2, 1, 4, 3, 4, 1, 4, 3, 2, 3, 4, 1, 4, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 1, 4, 1, 2, 1, 4, 3, 4, 3, 4, 3, 4, 3, 4, 1, 4, 3, 4, 3, 2, 3, 2, 3, 4, 1, 4, 3, 4, 3, 4, 3, 4, 3, 4, 1, 2]

For 24<=n<=72 we have minimal area = 1,597.n^3,188 with a rather tight fit.

Reconstruct the primelist in descending order. Combine these primes with the given directions.
The directions I used were 1=North, 2=East, 3=South, 4=West.

If n=100 we have:
100 4128225 3 [1, 2, 1, 4, 3, 4, 1, 4, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 4, 1, 2, 1, 4, 3, 4, 1, 2, 1, 4, 3, 4, 1, 4, 3, 2, 3, 4, 1, 4, 3, 2, 3, 4, 1, 4, 3, 2, 3, 4, 1, 4, 1, 4, 3, 2, 3, 2, 3, 4, 1, 4, 3, 2, 3, 2, 3, 4, 1, 4, 3, 4, 1, 2, 1, 4, 1, 4, 3, 4, 1, 4, 3, 4, 3, 4, 3, 4]

Finding a solution for larger n is quite possible. Finding the minimal solution takes “some” time.

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Emmanuel wrote:

Here are some solutions.  They all are minimal (i.e. that the circumference of the polygons cannot be smaller) and non self intersecting.
I give only the value of  n, the first prime p, the circumference  c  and a sequence M of +/- ones from which one can draw the line.
(+1/-1  at an odd place means : move to the right/left;
+1/-1  at an even place means move up/down )

n = 8 : p = 359, c = 3048, M = {1,1,-1,-1,-1,-1,1,1}
n = 12 ; p = 59273, c = 712220, M = {1,1,-1,-1,-1,-1,-1,-1,1,1,1,1}  (it is allmost impossible to detect with the naked eye the non self intersecting property of this polygon)
n = 16 ; p = 173, c = 3424, M = {1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1}
n = 20 ; p = 5, c = 786, M = {1,1,1,1,1,-1,1,1,1,-1,1,1,-1,1,1,1,-1,-1,-1,-1}
n = 24 ; p = 3, c = 1058 , M = {1,1,1,1,-1,1,1,-1,1,-1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,1}
n = 28 ; p = 3, c = 1478, M = {1,1,-1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,-1,1,-1,-1,-1,-1,1}
n = 32 ; p = 3, c = 1986, M = {1,1,1,1,1,-1,1,1,1,-1,1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,-1,-1,-1,1,1,1,-1,1,1,1}
n = 36 ; p = 3, c = 2582, M = {1,1,1,1,1,-1,1,1,1,-1,1,-1,1,-1,1,1,1,-1,1,-1,1,-1,-1,-1,1,-1,1,-1,-1,1,-1,1,-1,1,-1,1}
n = 40 ; p = 3, c = 3264, M = {1,1,-1,1,-1,1,-1,1,1,1,-1,1,1,1,1,1,1,1,1,-1,1,-1,-1,-1,1,1,1,1,1,-1,1,-1,-1,1,-1,-1,-1,-1,-1,1}
n = 44 ; p = 3, c = 4026, M = {1,1,1,-1,-1,-1,1,-1,1,1,1,-1,-1,-1,1,-1,-1,-1,1,1,1,-1,1,-1,1,1,1,-1,1,-1,1,1,1,1,-1,-1,-1,1,-1,1,-1,1,-1,-1}
; p = 3,  c = 4026, M = {1,1,1,-1,-1,-1,1,-1,1,-1,1,-1,-1,-1,1,-1,-1,-1,1,1,1,1,1,-1,1,1,1,-1,1,1,1,1,1,1,-1,-1,-1,1,-1,-1,-1,1,-1,-1}

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Fred wrote:

I went a little crazy and found solutions for up to 704(!).  After 16 (through 704), the minimum prime is 3 (This is equivalent to the minimum sum as we are dealing with a sequence of consecutive primes).  Here are example solutions through 96; primes are followed by their direction: R(ight), L(eft), U(p), D(own).

8: 359R 367U 373L 379D 383L 389D 397R 401U

16: 173R 179U 181R 191U 193L 197D 199L 211D 223L 227D 229L 233D 239R 241U 251R 257U

24: 3R 5U 7L 11U 13R 17U 19L 23D 29L 31D 37L 41U 43R 47U 53R 59U 61R 67D 71L 73D 79R 83D 89L 97U

32: 3R 5U 7L 11D 13L 17D 19L 23D 29L 31D 37L 41D 43L 47U 53R 59U 61L 67U 71R 73U 79L 83D 89L 97D 101L 103D 107R 109D 113R 127U 131R 137U

40: 3R 5U 7L 11D 13L 17D 19L 23D 29L 31D 37L 41D 43L 47U 53L 59D 61L 67D 71L 73D 79L 83D 89L 97D 101R 103D 107R 109D 113R 127D 131R 137U 139L 149U 151L 157U 163R 167U 173R 179U

48: 3R 5U 7L 11D 13L 17D 19L 23D 29L 31D 37L 41D 43L 47D 53L 59D 61L 67D 71L 73U 79R 83U 89L 97D 101L 103U 107R 109U 113L 127D 131L 137U 139R 149U 151R 157U 163R 167D 173R 179D 181R 191D 193R 197D 199L 211U 223L 227U

56: 3R 5U 7L 11D 13L 17D 19L 23D 29L 31D 37L 41D 43L 47D 53L 59D 61L 67D 71L 73D 79L 83D 89L 97D 101L 103U 107L 109U 113L 127D 131R 137D 139R 149D 151L 157D 163R 167D 173R 179U 181R 191U 193L 197U 199R 211D 223R 227U 229R 233U 239R 241U 251L 257D 263L 269U

64: 3R 5U 7L 11D 13L 17D 19L 23D 29L 31D 37L 41D 43L 47D 53L 59D 61L 67D 71L 73D 79L 83D 89L 97D 101L 103D 107R 109D 113L 127D 131R 137D 139L 149U 151L 157U 163L 167U 173R 179U 181R 191U 193R 197U 199R 211U 223L 227U 229R 233D 239R 241U 251R 257D 263L 269D 271R 277D 281R 283D 293L 307U 311L 313U

72: 3R 5U 7L 11D 13L 17D 19L 23D 29L 31D 37L 41D 43L 47D 53L 59D 61L 67D 71L 73D 79L 83D 89L 97D 101L 103D 107L 109D 113L 127D 131L 137D 139L 149D 151L 157D 163L 167D 173R 179D 181R 191D 193R 197D 199R 211D 223R 227D 229R 233D 239R 241U 251R 257U 263L 269U 271R 277U 281R 283U 293L 307D 311L 313U 317L 331U 337R 347U 349L 353U 359R 367U

80: 3R 5U 7L 11D 13L 17D 19L 23D 29L 31D 37L 41D 43L 47D 53L 59D 61L 67D 71L 73D 79L 83D 89L 97D 101L 103D 107L 109D 113L 127D 131L 137D 139L 149D 151L 157D 163L 167D 173L 179D 181L 191D 193L 197D 199R 211D 223R 227D 229L 233D 239R 241U 251R 257U 263R 269U 271R 277U 281R 283U 293R 307U 311R 313U 317R 331U 337R 347D 349L 353D 359R 367U 373R 379U 383L 389U 397L 401D 409L 419U

88: 3R 5U 7L 11D 13L 17D 19L 23D 29L 31D 37L 41D 43L 47D 53L 59D 61L 67D 71L 73D 79L 83D 89L 97D 101L 103D 107L 109D 113L 127D 131L 137D 139L 149D 151L 157D 163L 167D 173L 179D 181L 191D 193L 197U 199L 211D 223L 227U 229L 233U 239L 241D 251R 257D 263R 269D 271R 277U 281R 283D 293R 307U 311R 313U 317R 331D 337R 347U 349R 353U 359R 367U 373R 379D 383R 389U 397R 401U 409R 419U 421L 431U 433L 439D 443L 449D 457L 461U

96: 3R 5U 7L 11D 13L 17D 19L 23D 29L 31D 37L 41D 43L 47D 53L 59D 61L 67D 71L 73D 79L 83D 89L 97D 101L 103D 107L 109D 113L 127D 131L 137D 139L 149D 151L 157D 163L 167D 173L 179D 181L 191D 193L 197D 199L 211D 223L 227D 229L 233D 239L 241D 251L 257D 263L 269D 271L 277U 281L 283U 293R 307U 311R 313D 317R 331U 337L 347U 349R 353U 359L 367U 373R 379U 383R 389U 397R 401U 409R 419U 421R 431U 433R 439D 443R 449U 457L 461U 463R 467D 479R 487D 491R 499D 503L 509U

Longer solutions are available at: https://sites.google.com/site/grandpascorpion/home/math

I began each golygon my moving right from the origin (0, 0) with the first prime and then up with the second prime. This movement is arbitrary and just serve to anchor the shape. (Otherwise, you would search space would be 4*4 times larger than it need to be).

I optimized my search by determining at each step what coordinates could lead to a solution by working my way backwards  from the last prime in the sequence ending at the origin.  Because we alternate between horizontal and vertical movement, those X and Y legal values can could be considered independently. Primes that did not have a road back to the origin were skipped.  (This ended up only being relevant to the 8 and 16 search.)  Then, it's just a matter of stepping through the search space, cutting short any path that crosses itself.  Initially, I quit searching as soon as I found a solution.

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Hakan wrote:

Minimal prime solution for 24 sides goes from the prime 3 to the prime 97, for a sum of 1058.

Minimal prime solution for 32 sides goes from the prime 3 to the prime 137, for a sum of 1986.

I sent my solution for k=1 to 4 in additional file.(p742.rar)

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Rivera wrote:

I tried to make golygons for certain kind of primes; for example y we restrict the sides to the series of consecutive twin primes...

The smallest solution for n=8 is this one:

29N, 31E, 41S, 43W, 59S, 61W, 71N, 73E

Here is one for n=12 (not sure if this is the minimal)

1019N 1021E 1031N 1033E 1049S 1051W 1061S 1063W 1091S 1093W 1151N 1153E

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Some days later, Hakan Summakoglu sent all the solutions n mod 4 from 8 to 40.

More or less at the same time, Jan van Delden sent a 100-twin primes solution.

All of these have been added to the pdf file linked above.

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