V. Reinhart wrote on Sat, Apr 8, 2017 05:40 PM UTC:
A huygens is chess piece that jumps in the directions of a rook any prime number of squares. In this discussion, I also impose the limit that it has a minimum jump distance of 5 or more squares (as it is used in Trappist-1 ).
So this huygens jumps distances of 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97... and so on. Its icons are shown here:
Icon 1 - by Fergus Duniho.
Icon 2 - Scientific Version
Just like a knight sometimes has trouble moving to a certain square (like requiring 4 jumps to move to a square 2 squares up and 2 left), moving a huygens can also take a few jumps to move to certain squares. Moving an odd number of squares can be tricky if the number isn't prime, because the sum of two primes is always even (unless one of the numbers is 2, but the huygens here can't jump 2 squares). So in these cases, a huygens needs to make 3 jumps to get to a particular square.
When moving an even number of squares, I think it would usually take 2 jumps. But I don't know if there is a way to prove this for every even-numbered move. It is currently unknown if every even integer can be expressed as the sum of two primes. In the 1700's Christian Goldbach believed it was true but couldn't prove it. Today it is still an unsolved problem and is known as the Goldbach Conjecture.
So if you are playing a game of chess with the huygens, don't always assume that you can move an even number of squares in two jumps. There may be some rare cases where three jumps are required. But shorter moves are usually not a problem to figure out. Here's a summary I believe is usually true:
If the distance is prime (5 or more) the huygens can move there in one jump.
If the distance is even, the huygens can get there in two jumps (always or almost always true)
If the distance is odd and not prime, it will require three jumps
The list below shows how to do it for distances up to 40. This may not include every possible method for each distance. For some short moves, it is necessary to overjump the destination, and them move back.
(Move/Leap distances to make the move):
1 (5,7,-11)
2 (7,-5)
3 (5,11,-13)
4 (11,-7)
5 (5)
6 (11,-5)
7 (7)
8 (13,-5)
9 (5,11,-7)
10 (5,5) or (17,-7)
11 (11)
12 (5,7)
13 (13)
14 (7,7)
15 (5,5,5)
16 (5,11)
17 (17)
18 (5,13) or (7,11)
19 (19)
20 (7,13)
21 (7,7,7)
22 (5,17) or (11,11)
23 (23)
24 (5,19) or (7,17) or (11,13)
25 (5,7,13)
26 (7,19) or (13,13)
27 (5,11,11) or (5,5,17) or (7,7,13)
28 (5,23) or (11,17)
29 (29)
30 (7,23) or (11,19) or (13,17)
31 (31)
32 (13,19)
33 (11,11,11)
34 (17,17) or (11,23)
35 (11,11,13)
36 (17,19)
37 (37)
38 (11,11,11,5)
39 (13,19,7)
40 (11,29) or (17,23)
If anyone finds an error or a faster way for any of these moves please leave a reply.
A huygens is chess piece that jumps in the directions of a rook any prime number of squares. In this discussion, I also impose the limit that it has a minimum jump distance of 5 or more squares (as it is used in Trappist-1 ).
So this huygens jumps distances of 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97... and so on. Its icons are shown here:
Icon 1 - by Fergus Duniho.
Icon 2 - Scientific Version
Just like a knight sometimes has trouble moving to a certain square (like requiring 4 jumps to move to a square 2 squares up and 2 left), moving a huygens can also take a few jumps to move to certain squares. Moving an odd number of squares can be tricky if the number isn't prime, because the sum of two primes is always even (unless one of the numbers is 2, but the huygens here can't jump 2 squares). So in these cases, a huygens needs to make 3 jumps to get to a particular square.
When moving an even number of squares, I think it would usually take 2 jumps. But I don't know if there is a way to prove this for every even-numbered move. It is currently unknown if every even integer can be expressed as the sum of two primes. In the 1700's Christian Goldbach believed it was true but couldn't prove it. Today it is still an unsolved problem and is known as the Goldbach Conjecture.
So if you are playing a game of chess with the huygens, don't always assume that you can move an even number of squares in two jumps. There may be some rare cases where three jumps are required. But shorter moves are usually not a problem to figure out. Here's a summary I believe is usually true:
If the distance is prime (5 or more) the huygens can move there in one jump.
If the distance is even, the huygens can get there in two jumps (always or almost always true)
If the distance is odd and not prime, it will require three jumps
The list below shows how to do it for distances up to 40. This may not include every possible method for each distance. For some short moves, it is necessary to overjump the destination, and them move back.
(Move/Leap distances to make the move):
1 (5,7,-11)
2 (7,-5)
3 (5,11,-13)
4 (11,-7)
5 (5)
6 (11,-5)
7 (7)
8 (13,-5)
9 (5,11,-7)
10 (5,5) or (17,-7)
11 (11)
12 (5,7)
13 (13)
14 (7,7)
15 (5,5,5)
16 (5,11)
17 (17)
18 (5,13) or (7,11)
19 (19)
20 (7,13)
21 (7,7,7)
22 (5,17) or (11,11)
23 (23)
24 (5,19) or (7,17) or (11,13)
25 (5,7,13)
26 (7,19) or (13,13)
27 (5,11,11) or (5,5,17) or (7,7,13)
28 (5,23) or (11,17)
29 (29)
30 (7,23) or (11,19) or (13,17)
31 (31)
32 (13,19)
33 (11,11,11)
34 (17,17) or (11,23)
35 (11,11,13)
36 (17,19)
37 (37)
38 (11,11,11,5)
39 (13,19,7)
40 (11,29) or (17,23)
If anyone finds an error or a faster way for any of these moves please leave a reply.