[math-fun] hexadoku grids
Hi, I'm looking for solved hexadoku grids (like sudoku, but 16x16 instead of 9x9). Ideally, I'd like to find symmetric grids or grids where each contiguous 4x4 block uses each character only once, not just the main 4x4 blocks. Any hints? Thanks, Kerry Mitchell -- lkmitch@gmail.com www.fractalus.com/kerry
The contiguous 4x4 restriction is impossible. If you have a valid 4x4 block, and shift it right one column, the four entries in the new column must match (as a group) the four entries in the departed column. This implies any 1x4 vertical box has the same entries as the box 4 cells to the right, although possibly rearranged. abcdw efghx --> {a,e,i,m} = {w,x,y,z} ijkly mnopz q...v Now consider shifting the 1x4 boxes down one row. They must still be equal as sets, so the operation of bringing in one new element into each set, and deleting the old ones, must preserve the set equality. {e,i,m,q} = {x,y,z,v} This is only possible if either the two removed elements are equal (a=w) and the two added elements are equal (q=v), or, in each set separately, the element added equals the element removed (a=q and w=v). But these possibilities are forbidden by the sudoku row and column no-repeat rule. This suggests a new kind of combinatorial object, keeping the contiguous box rule, but dropping the row & column restrictions. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of Kerry Mitchell Sent: Sat 4/14/2007 4:14 PM To: math-fun Subject: [math-fun] hexadoku grids Hi, I'm looking for solved hexadoku grids (like sudoku, but 16x16 instead of 9x9). Ideally, I'd like to find symmetric grids or grids where each contiguous 4x4 block uses each character only once, not just the main 4x4 blocks. Any hints? Thanks, Kerry Mitchell -- lkmitch@gmail.com www.fractalus.com/kerry _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Thanks for your reply, Rich. I realized shortly after I posted that I mispoke. I had use the sum of the row/column/block to double-check my entries (I know that that is not a rigorous check of correctness, but it will generally catch one or two typographical errors). I have found a 9x9 grid where the *sums* are preserved for each contiguous 3x3 block, although the numerals are not. Is there a relatively simple way to find a 16x16 grid using the numbers 0-15 that satisfies the standard sudoku/hexadoku requirements and has the sum of each contiguous 4x4 block the same? Kerry On 4/15/07, Schroeppel, Richard <rschroe@sandia.gov> wrote:
The contiguous 4x4 restriction is impossible.
---- Kerry asked: Is there a relatively simple way to find a 16x16 grid using the numbers 0-15 that satisfies the standard sudoku/hexadoku requirements and has the sum of each contiguous 4x4 block the same? Kerry, this example should satisfy your requirements. Sum of each contiguous 4x4 block = 120. And using a simple way, as you asked. Christian. www.multimagie.com/indexengl.htm 0 1 2 3 | 4 5 6 7 | 8 9 10 11 | 12 13 14 15 4 5 6 7 | 8 9 10 11 | 12 13 14 15 | 0 1 2 3 8 9 10 11 | 12 13 14 15 | 0 1 2 3 | 4 5 6 7 12 13 14 15 | 0 1 2 3 | 4 5 6 7 | 8 9 10 11 ----------------------------------------------------- 1 2 3 0 | 5 6 7 4 | 9 10 11 8 | 13 14 15 12 5 6 7 4 | 9 10 11 8 | 13 14 15 12 | 1 2 3 0 9 10 11 8 | 13 14 15 12 | 1 2 3 0 | 5 6 7 4 13 14 15 12 | 1 2 3 0 | 5 6 7 4 | 9 10 11 8 ----------------------------------------------------- 2 3 0 1 | 6 7 4 5 | 10 11 8 9 | 14 15 12 13 6 7 4 5 | 10 11 8 9 | 14 15 12 13 | 2 3 0 1 10 11 8 9 | 14 15 12 13 | 2 3 0 1 | 6 7 4 5 14 15 12 13 | 2 3 0 1 | 6 7 4 5 | 10 11 8 9 ----------------------------------------------------- 3 0 1 2 | 7 4 5 6 | 11 8 9 10 | 15 12 13 14 7 4 5 6 | 11 8 9 10 | 15 12 13 14 | 3 0 1 2 11 8 9 10 | 15 12 13 14 | 3 0 1 2 | 7 4 5 6 15 12 13 14 | 3 0 1 2 | 7 4 5 6 | 11 8 9 10
participants (3)
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Christian Boyer -
Kerry Mitchell -
Schroeppel, Richard