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.

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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