Take, say, 3x3 board with digits arranged e.g. in order:
1 2 3
4 5 6
7 8 9
Starting from any cell,
King (Chess King!) moves for primes.
How many primes can he (or you) find by rules:
King moves for any neighbor cell,
(* e.g. from cell with 5,
he can move to any of 8 other cells,
from cell with 1, he can go to cell with 2, 5, or 4 etc*)
then again for any neighbor cell (even returning back).
Each time digits contacenate to form
(finally - maximum after say 10 moves) a prime.
Then you may start from any other cell.
Ok, let me show examples
a) for 1-move version (two-digit primes) we have :
starting from 5, two primes: 53 and 59,
starting from 4, two primes: 41 and 41,
starting from 2, one prime : 23,
total 5 two-digit primes.
b) for 2-move version (three-digit primes) we have:
starting from 5, six primes: 541, 547, 521, 523, 563, 569, 587, etc
With increasing number of moves, the problem should be
solved by program.
Several Qs:
What is the maximal number 2-,3-,4-,....10-digit primes,
using the given start position?
What is the best start position (giving more primes).
What about larger board?
In a message dated 13/07/03 15:35:51 GMT Daylight Time, seidovzf@... ... Interesting idea, Zak. We could replace your 3rd Q with another: What about...
[Zak Seidov] King's moves taking maximum three cells give the next primes: [Jon Perry] Why precisely is the question forcing a King? A Queen or Rook should...
I have programmed the 2x2 board version. The method used is exhaustive search, calling recursively (for increasing no. of moves) a simple "evaluate" procedure....
OK, Mike, you did a JOB, now i think that may be only distinct primes count, what do you think about this in your program - rejecting repeating primes, thanks,...
At 2x2 board, the number of n-digit numbers (after n-1 moves) is 4*3^(n-1), so according to Mike's calcs, the percentage of n-digit primes gradually decreases ...
In a message dated 14/07/03 14:51:36 GMT Daylight Time, seidovzf@... ... What I think is: let's stick with the problem you originally defined. It's a...
In a message dated 15/07/03 18:52:24 GMT Daylight Time, seidovzf@... ... Zak: I agree with your formula for the count of numbers. But where does 2187...
Here's the relationship 3 ways - as a Matrix recursion, as an explicit formuula, and as a 3-term recursion. MATRIX VERSION: There are three kinds of squares -...
Are you sure that there are no misprints: ... a(-2)^N + b*(2+2sqrt(2))^N + c*(2-2sqrt(2))^N The values turn out to be a=1/2 b=17/4 + 3sqrt(2) c=17/4 - 3sqrt(2)...
... There's a missing "*" between the "a" and the "(-2)". There's a fencepost problem about whether we count the number of moves the king makes or the number...
yes, you are right, misteriously my Mma program kept giving 5s not 3s?? now at last i've found the error, thanks, zak ... __________________________________ Do...
I get the same values. For the 4x4 board the eigenvalues are 0 (5+3sqrt(5))/2 (5-3sqrt(5))/2 The eigenvalue of zero results in a two-term recursion that is...
On 16/07/03 I wrote:- ... After playing around with these factorisations a bit more today, I have found the right way to express them. Sit back and I will show...
A final despatch from the programming front:- I have speeded up the program by a factor of about 30, by 3 independent tricks:- (a) If you distribute the...
Mike, it's a real pity that your ... I don't think that none else can meet the challenge... Still, if you keep your results, can your present number of...
... At this night i did it myself, and here are some results: i consider 3x3 board with figures 1,2,3,4,5,6,7,8,9 each in one cell, then (for a time being) not...
A novel 'call me Carlos' extension, which 3*3(*3) board produces the most primes where each square (cube) is itself a distinct prime? The obvious extension of...
... Hash: SHA1 ... Was the primality testing slowing down your program? Because there are only 900k 6-digit numbers (assuming no other restrictions placed on...
Décio Luiz Gazzoni...
decio@...
Jul 20, 2003 5:35 pm
In a message dated 20/07/03 18:35:59 GMT Daylight Time, ... The primality test was indeed simply a 1-bit test in a bit-array of all odd primes (as you...
In a message dated 21/07/03 06:52:46 GMT Daylight Time, seidovzf@... ... [snip] ... Nice work, Zak, and everything you posted seems to be correct. Your...
Mike and all, I have expansion in the direction of larger board but only for N=2 routes (that is concatenations of any two adjacent cells). Take MxM board with...
Jon, here my initial results: Take 3x3 board with 9 first primes {2,3,5,7,11,13,17,19,23} each in one cell, then 2-move routes give minimal 6 primes (in 224 ...
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