Problems & Puzzles: Puzzles

Puzzle 991. Peculiar n-gons JM Bergot sent the following nice puzzle:   Plot the points (2,3), (3,5), (5,7)...(17,19) to create an irregular non-convex n-gon.  Its computed area is just 19. (See also 1 and 2)   Q. Are there more n-gons like this one? If so, send your solutions.   "like this one" means here that at least all the vertices are of the form (p,q) such that p and q are consecutive primes. The second condition [two contiguous points (p,q) & (r,s) are such that r=q] can be avoided if needed.  Finally, the area of the polygon should be equal to the largest prime employed.

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Contributions came from Jan van Delden, Paul Cleary and Oscar Volpatti, during the week 23-29 Feb, 2020

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Jan wrote:

I found 3 different solutions to 19. The other two are:

[[2, 3], [3, 5], [5, 7], [7, 11], [17, 19]] and [[2, 3], [3, 5], [5, 7], [13, 17], [17, 19]].

Prime 23,29 have no solutions.

Prime 31 has 8 solutions with no intersections. The longest list is: [[2, 3], [3, 5], [5, 7], [7, 11], [11, 13], [17, 19], [19, 23], [29, 31]].
Prime 37 has 1 solution, no intersection.  It is: [[2, 3], [5, 7], [7, 11], [23, 29], [31, 37]].
Prime 41 has 17 solutions, but all have intersections. A short one is: [[2, 3], [7, 11], [17, 19], [31, 37], [37, 41]].
Prime 43 has 37 solutions with no intersections.  A short one is: [[2, 3], [3, 5], [29, 31], [31, 37], [41, 43]].
Prime 47 has 71 solutions, 8 have no intersections, 63 have intersections.
Prime 53 has 23 solutions, 8 have no intersections, 15 have intersections.
Prime 59 has 97 solutions, 11 have no intersections, 86 have intersections.
Prime 61 has 6 solutuions with no intersections.
Prime 67 has 453 solutions. 42 have no intersections, 411 have intersections. A short one is: [[2, 3], [11, 13], [31, 37], [61, 67]]

And here I stopped my research.

I also tested lists where all primes are consecutive, starting at 2 where we do have that primes are equal where the pairs touch, like in the given example. I found no other solutions of this type, I searched until p[1000]. There could be more.

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Paul wrote

All polygons were tested for no intersections, all were graphed in Mathematica as a test and non were found and a random sample were manually checked in GeoGebra, so to the best of my ability I recon they are all valid.  I have also extended the range of primes and found new solutions, I have collated them and created a new table, sorted by area (last element in a row). 

See here 441 solutions

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Oscar wrote:

I found some solutions with 13 <= pmax <= 37.

For 13 and 23, I needed to swap the coordinates of at least one point.

In some cases, three consecutive points are aligned, but there are no self-intersecting polygons.

 

pmax = 13:
(2,3) (3,5) (7,11) (5,7) (13,11)

 

pmax = 17:
 

(2,3) (5,7) (3,5) (7,11) (13,17) (11,13)

 

pmax = 19:
(2,3) (3,5) (5,7) (7,11) (11,13) (13,17) (17,19)
(2,3) (3,5) (5,7) (7,11) (13,17) (11,13) (17,19)
(2,3) (3,5) (5,7) (11,13) (7,11) (13,17) (17,19)

pmax = 23:
(3,2) (3,5) (7,11) (13,17) (19,23) (11,13) (17,19) (5,7)

 

pmax = 29:
(2,3) (3,5) (5,7) (7,11) (13,17) (11,13) (23,29) (19,23) (17,19)
(2,3) (3,5) (7,11) (5,7) (13,17) (23,29) (11,13) (19,23) (17,19)
(2,3) (3,5) (7,11) (13,17) (19,23) (23,29) (17,19) (11,13) (5,7)
(2,3) (3,5) (7,11) (13,17) (23,29) (17,19) (11,13) (19,23) (5,7)
(2,3) (3,5) (7,11) (13,17) (23,29) (17,19) (19,23) (11,13) (5,7)
(2,3) (3,5) (7,11) (23,29) (17,19) (11,13) (19,23) (13,17) (5,7)
(2,3) (3,5) (7,11) (23,29) (17,19) (19,23) (11,13) (13,17) (5,7)
(2,3) (3,5) (7,11) (23,29) (17,19) (19,23) (13,17) (11,13) (5,7)
(2,3) (3,5) (13,17) (7,11) (23,29) (17,19) (11,13) (19,23) (5,7)
(2,3) (3,5) (13,17) (7,11) (23,29) (17,19) (19,23) (11,13) (5,7)
(2,3) (3,5) (19,23) (13,17) (7,11) (23,29) (17,19) (11,13) (5,7)
(2,3) (5,7) (3,5) (7,11) (23,29) (17,19) (11,13) (19,23) (13,17)
(2,3) (5,7) (3,5) (7,11) (23,29) (17,19) (19,23) (11,13) (13,17)

 

pmax = 31:
(2,3) (3,5) (7,11) (5,7) (13,17) (11,13) (23,29) (19,23) (17,19) (29,31)
(2,3) (3,5) (7,11) (13,17) (5,7) (11,13) (23,29) (19,23) (17,19) (29,31)

 

pmax = 37:
(2,3) (3,5) (5,7) (13,17) (7,11) (19,23) (23,29) (31,37) (11,13) (17,19) (29,31)
(2,3) (3,5) (7,11) (5,7) (13,17) (23,29) (19,23) (31,37) (11,13) (17,19) (29,31)
(2,3) (3,5) (7,11) (23,29) (13,17) (5,7) (19,23) (31,37) (11,13) (17,19) (29,31)

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