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Maneuvering a Huygens on a Chessboard[Subject Thread] [Add Response]
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.


V. Reinhart wrote on Sun, Apr 9, 2017 01:49 PM UTC:

Is this maneuvering problem similar to the knight's tour problem (first discussed in the 9th century)?


George Duke wrote on Mon, Apr 10, 2017 07:40 PM UTC:

There are other integer number sequences.  You could add diagonal directions and make these sliders too.  Then the board could be fixed at 30x30.  Huygens is orthogonal leaper but a variant piece would be Queenlike to the prime number squares five and over or more inclusively three and over. 

Then there are more pieces to expand the idea to other sequences.  Fibonacci moves Queenlike to 3,5,8,13, and 21 distance.  Triangular number piece moves along radial lines exactly 3,6,10,15, 21, or 28.  Square number piece to 4,9,16, and 25, a weak piece.  Deficient number sequence piece (since perfect numbers are so rare) is the strongest going to 4,5,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29. Tetrahedral number 4,10,20, weak mover again.  Abundant number to 12,18,20,24 (keeping all these less than 30 to fit the board). Lucas to 3-distance, or 4, 7, 11,18 or 29.  Pawns should be Man to all eight directions one or two steps, squares which the mathematical pieces cannot any of them reach from the same starting square.

Lucky number unit can go radially 3 or 7 or 9  or 13 or 15 or 21 or 25 only.  Pancake number type moves 4, 7, 11, 16, 22, or 29.

However, never design a chess piece based on Weird Numbers. They are too few and too large.  '70' is the first weird number because it is abundant being less than (1+2+5+7+10+14+35, its factors), but no set of those divisors sum to 70 itself.  The only other less than 4-digit weird number  is 836, and the sequence 70, 836, 4030, 5830 is unsuitable -- except on that infinite board.


V. Reinhart wrote on Mon, Apr 10, 2017 09:44 PM UTC:

I've heard of some of those sequences, but not all of them. I had to look up the pancake numbers.

For example, for 4 pancakes, there's 3 ways it might be in an unorganized stack so that it requries 4 flips with a spatula to organize it (from large to small), 11 that require 3, 6 for 2, 3 for 1, and 1 for 0. So a 4 pancake stack gives a pancake sequence of 3, 11, 6, 3, and 1. (Or 1, 3, 6, 11, 3 in reverse order).

But I don't understand the pancake sequence that you showed. It's not a sequence for any stack of pancakes. Am I not on the right path to what a pancake sequence is? Were some pancakes burned and thrown away? Let me know!


George Duke wrote on Mon, Apr 10, 2017 10:12 PM UTC:

The maximum number of pieces into which a pancake can be cut with n slices: 1 slice 2, 2 slices 4, 3 slices 7, 4-11, 5-16, 6-22, 7-29, 8-37....

The Lucky numbers in texts come about by striking out every other number, then from remaining 13579... strike out every third number, because that's the next one left. Then since '5' is stricken and what remains is 1,3,7,9,13,15,19..., now strike out every seventh number starting with 19. '1,3,7,9,13,15,21,25,31,33,37,43,49,51....' never get stricken and are Lucky. It's related to the Sieve of Erastosthenes to generate the prime numbers. A Lucky Chess Piece allowing one-step should be Bishop value on 10x10 and up.

 


V. Reinhart wrote on Tue, Apr 11, 2017 02:55 PM UTC:

That's funny. Pancake numbers can come from two ways: stacking them and cutting them! It's making me hungry.

I'll eat and then enjoy web-surfing to learn more about some of the other number sequences you listed!


V. Reinhart wrote on Thu, Apr 13, 2017 04:22 PM UTC:

OK, today I'll study the lucky numbers, and the wierd numbers from George Duke. (Possible new chess pieces for large chessboards and infinite chess).

I like the Lucky numbers. Once a number is stricken from the list, it can never be added back. The lucky ones remain!


George Duke wrote on Thu, Apr 13, 2017 07:54 PM UTC:

There could be other sequences used for where a radial piece is allowed to stop. A compound piece of pentagonal, hexagonal, heptagonal and octagonal numbers can move: 5, 6, 7, 8, 12, 15, 18, 21, 22 or 28..., keeping to the lower lengths for 30x30 board. Aligning the primes and Fibonacci and Deficient and others pairwise, both orthogonally and diagonally, may discover hidden relationships just by tooling around, for applicability beyond the chessboards. For example, applying some Knights Tour like V. Reinhart mentions. Another piece can have numbers of distinct integral squares dividing a rectangle: 1, 4, 7, 8, 9, 10, 14, 15, 18. Call that one Squarec, and it is good piece to cross quickly the boundaries of boards either 12x12 or 16x16. With 15 steps specifically allowed it traverses 16x16 diagonally, or orthogonally, all the way without being just plain Queen


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