The New Sudoku Players' Forum

by **hkociemba1** » Tue Aug 21, 2018 8:15 pm

The following formula is an upper bound for the minimum number min(N) of givens for N^2*N^2 sudoku:

min(N) <= (N^2+1)*N*(N-1)/2 , N>=2

For n= 2, 3, 4...15 this gives 5, 30, 102, 260, 555, 1050, 1820, 2952, 4545, 6710, 9570, 13260, 17927, 23730
I noticed that this sequence is also in the OEIS https://oeis.org/A071252

Code: Select all: +-------+-------+-------+ | . . . | . . . | . . . | | . . . | . . . | 1 2 3 | | . . . | 1 2 3 | 4 5 6 | +-------+-------+-------+ | . . . | . . . | . . . | | . . . | . . . | 2 3 4 | | . . 1 | 2 3 4 | 5 6 7 | +-------+-------+-------+ | . . . | . . . | . . . | | . . . | . . . | 3 4 5 | | . 1 2 | 3 4 5 | 6 7 8 | +-------+-------+-------+ +-------------+-------------+-------------+-------------+ | . . . . | . . . . | . . . . | . . . . | | . . . . | . . . . | . . . . | 1 2 3 4 | | . . . . | . . . . | 1 2 3 4 | 5 6 7 8 | | . . . . | 1 2 3 4 | 5 6 7 8 | 9 10 11 12 | +-------------+-------------+-------------+-------------+ | . . . . | . . . . | . . . . | . . . . | | . . . . | . . . . | . . . . | 2 3 4 5 | | . . . . | . . . . | 2 3 4 5 | 6 7 8 9 | | . . . 1 | 2 3 4 5 | 6 7 8 9 | 10 11 12 13 | +-------------+-------------+-------------+-------------+ | . . . . | . . . . | . . . . | . . . . | | . . . . | . . . . | . . . . | 3 4 5 6 | | . . . . | . . . . | 3 4 5 6 | 7 8 9 10 | | . . 1 2 | 3 4 5 6 | 7 8 9 10 | 11 12 13 14 | +-------------+-------------+-------------+-------------+ | . . . . | . . . . | . . . . | . . . . | | . . . . | . . . . | . . . . | 4 5 6 7 | | . . . . | . . . . | 4 5 6 7 | 8 9 10 11 | | . 1 2 3 | 4 5 6 7 | 8 9 10 11 | 12 13 14 15 | +-------------+-------------+-------------+-------------+ +----------------+----------------+----------------+----------------+----------------+ | . . . . . | . . . . . | . . . . . | . . . . . | . . . . . | | . . . . . | . . . . . | . . . . . | . . . . . | 1 2 3 4 5 | | . . . . . | . . . . . | . . . . . | 1 2 3 4 5 | 6 7 8 9 10 | | . . . . . | . . . . . | 1 2 3 4 5 | 6 7 8 9 10 | 11 12 13 14 15 | | . . . . . | 1 2 3 4 5 | 6 7 8 9 10 | 11 12 13 14 15 | 16 17 18 19 20 | +----------------+----------------+----------------+----------------+----------------+ | . . . . . | . . . . . | . . . . . | . . . . . | . . . . . | | . . . . . | . . . . . | . . . . . | . . . . . | 2 3 4 5 6 | | . . . . . | . . . . . | . . . . . | 2 3 4 5 6 | 7 8 9 10 11 | | . . . . . | . . . . . | 2 3 4 5 6 | 7 8 9 10 11 | 12 13 14 15 16 | | . . . . 1 | 2 3 4 5 6 | 7 8 9 10 11 | 12 13 14 15 16 | 17 18 19 20 21 | +----------------+----------------+----------------+----------------+----------------+ | . . . . . | . . . . . | . . . . . | . . . . . | . . . . . | | . . . . . | . . . . . | . . . . . | . . . . . | 3 4 5 6 7 | | . . . . . | . . . . . | . . . . . | 3 4 5 6 7 | 8 9 10 11 12 | | . . . . . | . . . . . | 3 4 5 6 7 | 8 9 10 11 12 | 13 14 15 16 17 | | . . . 1 2 | 3 4 5 6 7 | 8 9 10 11 12 | 13 14 15 16 17 | 18 19 20 21 22 | +----------------+----------------+----------------+----------------+----------------+ | . . . . . | . . . . . | . . . . . | . . . . . | . . . . . | | . . . . . | . . . . . | . . . . . | . . . . . | 4 5 6 7 8 | | . . . . . | . . . . . | . . . . . | 4 5 6 7 8 | 9 10 11 12 13 | | . . . . . | . . . . . | 4 5 6 7 8 | 9 10 11 12 13 | 14 15 16 17 18 | | . . 1 2 3 | 4 5 6 7 8 | 9 10 11 12 13 | 14 15 16 17 18 | 19 20 21 22 23 | +----------------+----------------+----------------+----------------+----------------+ | . . . . . | . . . . . | . . . . . | . . . . . | . . . . . | | . . . . . | . . . . . | . . . . . | . . . . . | 5 6 7 8 9 | | . . . . . | . . . . . | . . . . . | 5 6 7 8 9 | 10 11 12 13 14 | | . . . . . | . . . . . | 5 6 7 8 9 | 10 11 12 13 14 | 15 16 17 18 19 | | . 1 2 3 4 | 5 6 7 8 9 | 10 11 12 13 14 | 15 16 17 18 19 | 20 21 22 23 24 | +----------------+----------------+----------------+----------------+----------------+

These examples above are all solvable by singles only (and they are also minimal). The number of givens for the last case N=5 is
1+2+3+4 (first stack) +5*(1*5) (second stack) +...5*(4*5) (last stack), which is
1+2+3+4 +5^2(1+2+3+4) = (5^2+1)(1+2+3+4) = (5^2+1)*5*4/2

Generalization for arbitrary N gives the formula.

The upper bound is not very sharp for small N but I think that it is quite difficult to generate a random sudoku for N>6 which has less clues. The 144x144 sudokus in this thread have for example more than 12000 givens while min(12)<=9570.

by **m_b_metcalf** » Wed Aug 22, 2018 8:50 am

hkociemba1 wrote:The upper bound is not very sharp for small N but I think that it is quite difficult to generate a random sudoku for N>6 which has less clues. The 144x144 sudokus in this thread have for example more than 12000 givens while min(12)<=9570.

You'd appear to be right. My best shot for a 49x49 was 1104 clues, still quite a lot higher than 1050 (the puzzle would have an SE rating of about 5).

Regards,

Mike Metcalf

Code: Select all: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . . . . . . . . . . . . . . . . . . . 31 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 16 23 . 40 . 38 . 1 . . . 47 15 39 4 . . . . . . . . . . . . . . . . . . . . . . 13 36 . . . . . 46 . . . 40 . 15 4 21 . . 16 . 2 17 34 . 33 37 3 . . . . . . . . . . . . . . . 11 . . . . . . 42 40 49 14 15 16 . 39 6 20 . . 47 . . . 13 . 33 31 . 26 28 41 38 30 . 27 . . . . . . . . . . . . . . . . 34 4 15 . 47 . 30 39 2 25 9 46 . 21 . 36 . . 24 48 28 49 . 26 29 6 11 44 . 40 7 . . . . . . . . . 37 . . . . . . . 22 39 . . . 30 . 48 . 33 6 24 28 4 9 . 12 25 10 2 17 11 18 . . . 34 . 42 46 36 21 5 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 . . . . . . 15 . . . . . . . . . 18 . 38 39 . . . . . . . . . 44 . . 7 . . . . . 13 . . . . . 32 . 10 11 . . . . 19 . 48 . . . . . . . 32 . . . . 5 . 6 . 49 . . . . . . . . 17 . 20 43 44 . . 24 29 . . . 7 . . 4 . . . . . 27 . 29 . . . . . 45 . 1 . . . . . 10 . . . . . 20 33 . 37 19 36 31 35 . . . . . . . 17 . 49 47 14 5 26 . 40 . 40 . . . . . . . . . 15 20 . . . . 17 8 . . 1 21 46 22 . 38 19 12 23 14 10 18 . . 25 49 11 47 . . 27 . 41 36 30 48 . . . . . . . . 36 . . 5 . . 47 . 35 3 . . . 2 48 30 9 . 31 37 29 27 32 25 39 . 49 11 13 . . 8 . 46 14 23 22 . 20 . . 18 45 . . . 49 . . 7 . . 4 . . 6 . . . . . 9 . 16 . 32 10 12 45 23 . 29 . 34 48 27 . . 28 19 22 37 36 20 41 8 31 33 39 13 46 25 . . . . . 25 . . . . . 19 28 . . . 49 . . 24 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5 8 . . . . . 29 . . . . 23 . 9 . . . . . . . 10 27 . . . . . . . . . . . . 40 . 22 . . . 26 14 . . . . 32 . 11 . . 48 . 4 23 . . . . . . . 46 . 37 28 . . . 38 . . . . 14 . . 33 31 . . 30 39 1 43 35 2 . . 12 13 . . . . . . . . 20 . 42 . . 3 . 29 . . 10 21 2 19 6 . . . 35 . 28 48 11 . 16 34 . . . . . . 41 47 7 37 27 32 5 33 1 . . . . . 28 . . . 24 36 33 . 13 42 17 . 1 4 . 14 34 43 . . 22 48 16 2 20 7 . . . 44 23 8 37 . . . 38 . 10 3 . 9 . . . . . 9 . . 48 . . 1 . 27 37 . . . 30 41 34 . . 2 . . . . 18 24 38 45 . 13 10 49 8 11 25 26 22 3 46 7 31 15 28 29 . 42 47 . . 3 39 . 23 . . . . 44 30 . 14 47 . 25 40 . . 33 4 45 . 7 13 . . 22 26 31 46 12 29 19 43 15 28 27 42 . . 18 . 24 34 . 8 36 . . 22 . . 38 44 . 41 . . . . . . . . . 40 34 36 . . . . 10 . 33 . . . . . 39 . . . . . 6 . 19 . . . . 16 . . . . . . . 33 10 . . 37 . . 49 29 . . . . 23 . 26 . . 18 . . 47 32 . 14 46 11 9 25 . . . 20 . . 7 16 . . . . . . . . 27 . . 16 . 35 9 . . . . . 33 8 5 . 14 . . . . . . . 36 . 4 . . . 22 . . 12 24 . 48 2 . . . 3 29 42 45 6 . 28 . 31 . 18 . . 23 . . . 48 . 25 7 . . . 10 . . . . . . 21 . 42 37 . 13 5 45 15 . . . 22 . 35 17 30 44 9 39 12 . . 24 . . . 13 . 25 . 24 . . 46 . 34 18 31 . 44 20 48 3 . 45 . 5 12 39 8 22 11 . 17 35 16 . 6 . . . 38 . 9 42 29 1 27 30 47 . 14 21 . . . . 43 4 . . . 2 23 . . 17 15 . 11 35 42 . . . 20 9 40 16 . 34 30 . 24 38 . 26 32 3 13 41 28 1 47 5 . . 25 46 8 . 22 . . . 5 . . . 6 . 39 14 . 19 . 30 . 17 9 22 32 . 48 3 41 . 24 27 13 36 . 37 1 44 33 42 46 10 . 12 8 49 11 . . 23 . . 35 38 . . . 31 11 8 . . 42 . . 44 5 21 . . 41 . . . . . . . 32 34 . 6 . . . . . . . . . 7 . . 18 . . . 19 16 . . . . 24 . . 26 3 40 . 31 . 13 . . 12 . . 42 . 35 22 27 . . 38 . . . 2 . . . . 17 8 10 . . . . 48 . 20 . . . . . 37 . . 21 . . . 20 13 . 3 14 . . 7 40 . . . 8 11 6 15 . . 44 18 23 . . . . 1 . . 4 . . . . . . 32 . . 12 38 . . 29 . . . 32 17 39 . . . . 20 15 48 4 25 . . . . . 40 . . . . 11 . 45 7 . 31 . . 33 24 27 26 2 . 1 28 . 46 . 23 10 43 44 34 6 . . 34 . 12 . . . . . 32 38 . 18 24 . . 3 13 29 . . 31 . 16 27 . 8 . . 23 30 . 36 28 22 45 6 44 19 43 10 14 5 . 41 46 15 2 . . 47 33 . . 29 . . 11 . 41 10 37 . 19 1 46 48 49 . . . 35 42 . . 14 18 7 15 44 2 . 45 9 38 36 5 . 34 . 24 4 . 17 22 13 32 . . 35 . . 2 25 . . . 33 28 27 36 . 23 43 . 16 . . 17 39 . 47 19 30 . 21 3 . 9 22 5 26 . 14 . 4 41 24 42 40 . . 18 1 . 7 . . 44 1 30 . 31 . 33 17 22 . . 39 . . 13 32 28 41 . . . 19 27 . . . . . 12 35 . . 7 . . . 43 34 . 25 . . . 9 . . 46 . . 9 32 . 41 49 . . 31 28 . 42 20 . . 15 . 39 48 17 23 . . 30 7 36 18 . . 47 26 34 . . . . 5 . . 8 . . . . . 24 2 11 . 10 . . 34 39 45 . 35 40 . . 26 4 20 . 24 42 49 . 25 . . . . . . . . 5 . 18 23 21 . . . . 15 . 2 . 47 6 . . . . 29 . . 46 42 . 47 3 . . 7 25 9 . 34 . 36 . 44 . . 10 . . 40 43 . . 1 . . 48 28 39 37 30 21 27 35 23 22 38 33 45 . 13 31 12 32 49 . . 28 . . 12 38 . . 10 . . 44 15 . . 5 43 18 16 3 49 . . . . 11 45 13 8 27 29 31 9 25 37 . 24 40 20 6 48 23 . 22 35 . 36 . . 26 . 35 . 11 . . 6 . . 1 41 . . 8 12 . 45 . 23 38 . 5 29 14 . 25 24 10 22 . 46 . . 16 . 44 39 31 . 49 27 18 43 7 33 20 4 . . 23 48 5 . . . 43 36 38 47 . 2 . . . 19 14 35 9 . 33 24 15 46 37 22 49 4 6 . . 40 16 32 18 12 . . 10 . 42 25 21 3 . . 30 . 20 29 . 46 44 16 . 30 . . . 21 . . . 22 . 19 . . . . 13 . 47 49 . . . . . 42 23 38 . . 34 . 27 . 36 . . . . 2 6 31 . 38 15 . 10 24 22 . 44 . 4 12 9 23 . 1 . . 2 5 . . . 11 41 . . . . 18 8 . 21 . 3 . . 29 . 40 33 . . . 37 . . 49 . . 25 . . 41 . 14 . 45 . 3 . 15 22 . 48 . 16 36 23 . . . 27 1 32 34 . . 43 . 6 4 7 . . . . . 49 21 28 . 26 35 . . 47 . . 42 21 . . 32 27 . 34 . 39 25 . 5 . 4 38 . 33 . . 31 . . 44 . 43 35 9 . . . 47 . 11 . . 17 10 24 37 . 28 46 . 48 19 1 8 . 11 37 . 33 . 12 . 2 . 24 31 . 49 . 18 26 45 . 3 32 . 22 14 38 . 7 17 . 41 . 5 . 28 . 15 1 39 46 13 . 35 4 36 . . 23 25 42 . 8 . 3 36 . . 43 18 . . . 13 16 . 10 . 49 41 . 28 5 4 37 46 2 . 26 17 . 45 19 1 48 34 . 44 30 . 12 25 9 21 . . . . 27 33 . 6 7 47 . . 34 . . 41 20 32 1 19 . 27 40 13 17 14 37 . 24 29 . 48 39 21 31 . 33 . 26 46 49 42 5 4 3 . 23 18 10 43 11 44 45 38 9

by **m_b_metcalf** » Wed Aug 22, 2018 12:15 pm

However, if I cheat a little, I can get fewer clues than 1050, namely 1021. (If anyone can tell me how I cheated, within 24 hours, I'll donate $10 to the Forum.) This puzzle is also SE ~5.

Mike

Code: Select all: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8 29 48 24 36 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 24 35 36 29 48 11 . . . . . . . 26 12 13 1 39 27 9 . . . . . . . 42 41 19 49 40 10 44 . . . . . . . 16 20 34 32 33 28 3 . . . . . . . 37 4 47 31 7 22 2 . . . . . . . . . . . . . . . . . . . . . 11 8 35 36 29 48 24 . . . . . . . 12 39 26 27 13 1 9 . . . . . . . 23 14 38 15 45 6 18 . . . . . . . 32 16 20 3 34 33 28 . . . . . . . 47 7 4 2 22 37 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 47 7 31 4 22 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 47 31 7 2 37 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 . . . . . . . 12 39 26 27 9 1 10 13 16 3 20 32 28 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 23 33 18 15 6 14 4 47 2 7 37 31 38 . . . . . . . . . . . . . . . . . . . . . 14 45 33 18 15 6 23 2 37 47 38 31 4 7 43 46 22 17 5 25 21 8 24 35 36 11 48 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 18 45 33 15 14 6 47 2 38 37 31 7 4 21 43 46 25 17 5 22 . . . . . . . . . . . . . 32 33 . 18 6 45 15 14 4 47 38 37 31 7 . . 17 43 22 5 21 25 24 . 30 11 48 36 8 40 41 42 19 44 49 29 . . . . 26 . . . . . 34 . 28 . . . . . . . . . . . . . . 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 35 36 11 . . 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 23 33 3 14 6 18 45 47 2 . 37 4 7 15 31 46 25 . 22 21 43 . . . . . . . 1 13 . . 32 20 16 . . . 14 6 . . . . 2 15 . 47 37 . 22 . 21 25 . . 24 35 30 11 . . 5 48 42 19 44 29 40 41 . . . 14 . 18 . . . . . . . . . . . . . 17 22 30 8 35 5 36 24 . 42 29 48 40 19 44 41 39 26 10 . 12 27 49 . . . . . . . . . . . . . . . . . . . 25 . 24 35 5 8 36 11 30 29 40 42 48 19 41 44 26 10 49 12 27 9 39 16 32 13 28 34 20 1 . . . . . . . . . . . . . . 5 30 8 11 35 36 24 41 42 48 40 19 44 29 10 12 26 49 27 39 9 . . . . . . 16 45 23 33 . . 6 3 15 2 7 37 38 4 . . . . . . . . . . . . . 28 . . . . . 18 . 3 . . . . . . . . . . . . . 46 . . . . . . . . . . . . . . . . . . 32 . 1 . . . 14 . . 23 . . . 47 . 4 . . . . . . . 21 . . . . . . 35 . . . . 41 44 . 19 10 9 12 49 . 26 . . . 45 23 . . . . . . . . . . . . 43 17 . . . . . . 11 . . . . . . 44 . . 26 10 49 12 39 9 19 27 13 . 34 1 16 32 . . . 24 35 8 . . . . . . . . . . . . . . . 1 16 13 27 . . 34 33 3 20 45 18 14 23 2 38 15 4 47 37 6 7 22 17 21 31 43 . . . . 43 46 21 . . . . . . . . . 29 36 41 44 . 48 49 39 10 19 9 26 12 13 1 27 16 . 34 32 23 33 . 14 45 . 20 6 38 37 4 15 . 2 . . . . . . . . . . . 10 9 26 32 13 27 16 28 34 1 . . . . . 23 14 38 15 6 47 37 4 2 46 22 31 . 43 17 7 25 30 11 8 5 24 . . . . 47 2 4 . 7 31 . . 22 17 . . 30 25 . 11 8 5 48 41 29 36 44 42 40 10 49 . 39 . 12 26 32 13 1 34 16 . 27 20 . 18 . 3 . 23 . . 37 . . . . . . . 24 . . . . . . . . . . . . . . . . . . . . 32 . . 13 . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . . . . . . 28 14 . 45 . . . . . 47 . . 31 7 43 46 21 . . 5 8 . 25 35 30 . 4 . 2 38 47 . . . 46 43 . 21 . . . . . . . . . 42 . . 40 . 41 49 19 44 26 . 39 . 13 . . . . . 9 . 3 . 45 20 23 33 . 34 . . 13 16 . . . 23 45 . 14 . . 15 18 . . . 6 7 . 31 . 21 22 . 5 25 17 35 8 . 30 29 48 36 . 42 . 11 44 . 12 39 . 26 10 . . 17 . . . . . . . 41 . 40 29 . . 44 26 . . . 27 32 1 9 34 13 16 . . 28 . 14 45 . 38 15 6 . . 4 18 . . . 43 7 46 . . 14 . 23 33 45 . . . . . . . . 22 31 37 46 21 43 7 25 . 5 17 8 . 24 48 36 11 42 40 41 29 10 49 19 39 26 12 44 . 1 . . 27 32 . . . . . . . . 9 27 32 . 1 . . 33 3 . 23 . . 20 6 2 15 18 . . . 31 7 . 46 21 43 22 30 5 25 24 35 8 17 11 48 40 41 36 42 29 . . . . . . 11 . . . . . . 49 . . . . . 32 . . 33 . . . 3 23 . . 14 . . . 15 . . . . . . . . . . . 17 30 5 . . . . . . . . . . 35 . . . . . . . . . . . . . . . . . . . . 13 16 . 1 . 20 28 . 33 45 34 . 6 47 2 . . . . . . . . . . . . . . . . . . . 34 33 . . 28 . . . . 47 . . 7 37 4 22 43 . 31 5 . 17 . . 24 . 8 36 41 42 . 29 48 . 47 14 38 . 2 18 . . 22 46 . 43 31 . . . . . . . 11 29 36 . 41 48 42 19 . . 10 . . 49 1 27 9 . 13 16 12 34 20 . 23 . . 3 . . . . . . 17 . . . . . . . . . 40 10 . 26 44 . 13 . . . 1 32 20 28 34 . 45 . 3 15 6 18 2 38 . 14 4 7 43 46 37 22 31 . 45 34 33 . 23 28 . . 38 2 6 . 15 31 7 4 22 43 46 . 17 30 25 21 24 5 35 . 11 . 29 41 42 48 49 19 44 26 . 39 . 12 . . . 9 . 1 . . . 13 1 32 9 . . 33 23 20 45 3 15 6 . . . 2 . 37 22 7 4 43 31 46 25 17 21 30 24 . 5 48 . . 42 29 41 8 40 . . 26 44 10 49 . 2 45 15 6 . . . . 31 22 37 46 7 . . . . 35 . . . . . . 42 . 29 . 40 . . . 10 . . . 12 . 1 . . . . 23 33 34 . 20 . 35 . 5 25 30 . . 8 . . . . 36 . . . . . . . . . . 39 32 . 13 . . . . 23 33 . . . . 38 . 2 . 47 . . . 4 . 7 . 23 16 3 20 . . . 14 . . 18 . . . . 47 . 46 22 4 . . . . . 25 30 11 8 . 48 42 29 36 19 . . . . . 41 39 9 . . . 1 27 . 32 39 . 27 13 . . . 3 33 28 23 20 . . 45 . . 38 14 4 31 37 . . 7 . . . 43 . 35 30 25 . 11 . 29 48 42 24 41 . 26 10 40 49 19 . . . . . . . . . . . 9 32 27 . 28 16 3 23 . 34 . 15 18 45 2 6 38 37 4 47 31 46 22 7 . 17 . . 5 35 43 . . 42 29 . 48 36 . 42 . 48 36 29 . 41 40 49 . 44 26 19 . . 39 1 . 13 12 34 . 28 16 23 20 33 . 14 . 15 2 38 6 7 37 4 . . . 47 43 17 . 30 21 5 25 . 46 . 31 7 22 4 43 . 5 . 17 35 25 . 11 24 48 42 29 8 40 49 . 41 . 19 10 9 12 . 1 . 13 27 20 28 34 33 3 . 16 45 18 . . . 15 6

by **hkociemba1** » Wed Aug 22, 2018 12:46 pm

I played a bit with your 49x49 pasting it into the mainwindow of my program (see software section) . It can be solved with basic methods and tuple sizes <=4. If I allow tuples of size 5, three more givens can be eliminated. You seem to try to eliminate the givens beginning with the upper left corner rowwise. I also observed that the number of givens which can be deleted is usually higher with this method compared to the method which randomly tries to eliminate givens from the grid (which is the standard behaviour of my program).

Running the SAT-method on your puzzle on the other side reduces it to 1100 clues and the puzzle is minimal. I do not know to which SE rating this puzzle belongs now:

Code: Select all: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . . . . . . . . . . . . . . . . . . . 31 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 16 23 . 40 . 38 . 1 . . . 47 15 39 4 . . . . . . . . . . . . . . . . . . . . . . 13 36 . . . . . 46 . . . 40 . 15 4 21 . . 16 . 2 17 34 . 33 37 3 . . . . . . . . . . . . . . . 11 . . . . . . 42 40 49 14 15 16 . 39 6 20 . . 47 . . . 13 . 33 31 . 26 28 41 38 30 . 27 . . . . . . . . . . . . . . . . 34 4 15 . 47 . 30 39 2 25 9 46 . 21 . 36 . . 24 48 28 49 . 26 29 6 11 44 . 40 7 . . . . . . . . . 37 . . . . . . . 22 39 . . . 30 . 48 . 33 6 24 28 4 9 . 12 25 10 2 17 11 18 . . . 34 . 42 46 36 21 5 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 . . . . . . 15 . . . . . . . . . 18 . 38 39 . . . . . . . . . 44 . . 7 . . . . . 13 . . . . . 32 . 10 11 . . . . 19 . 48 . . . . . . . 32 . . . . 5 . 6 . 49 . . . . . . . . 17 . 20 43 44 . . 24 29 . . . 7 . . 4 . . . . . 27 . 29 . . . . . 45 . 1 . . . . . 10 . . . . . 20 33 . 37 19 36 31 35 . . . . . . . 17 . 49 47 14 5 26 . 40 . 40 . . . . . . . . . 15 20 . . . . 17 8 . . 1 21 46 22 . 38 19 12 23 14 10 18 . . 25 49 11 47 . . 27 . 41 36 30 48 . . . . . . . . 36 . . 5 . . 47 . 35 3 . . . 2 48 30 9 . 31 37 29 27 32 25 39 . 49 11 13 . . 8 . 46 14 23 22 . 20 . . 18 45 . . . 49 . . 7 . . 4 . . 6 . . . . . 9 . 16 . . 10 12 45 23 . 29 . 34 48 27 . . 28 19 22 37 36 20 41 8 31 33 39 13 46 25 . . . . . 25 . . . . . 19 28 . . . 49 . . 24 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5 8 . . . . . 29 . . . . 23 . 9 . . . . . . . 10 27 . . . . . . . . . . . . 40 . 22 . . . 26 14 . . . . 32 . 11 . . 48 . 4 23 . . . . . . . 46 . 37 28 . . . 38 . . . . 14 . . 33 31 . . 30 39 1 43 35 2 . . 12 13 . . . . . . . . 20 . 42 . . 3 . 29 . . 10 21 2 19 6 . . . 35 . 28 48 11 . 16 34 . . . . . . 41 47 7 37 27 32 5 33 1 . . . . . 28 . . . 24 36 33 . 13 42 17 . 1 4 . 14 34 43 . . 22 48 16 2 20 7 . . . 44 23 8 37 . . . 38 . 10 3 . 9 . . . . . 9 . . 48 . . 1 . 27 37 . . . 30 41 34 . . 2 . . . . 18 24 38 45 . 13 10 49 8 11 25 26 22 3 46 7 31 15 28 29 . 42 47 . . 3 39 . 23 . . . . 44 30 . 14 47 . 25 40 . . 33 4 45 . 7 13 . . 22 26 31 46 12 29 19 43 15 28 27 42 . . 18 . 24 34 . 8 36 . . 22 . . 38 44 . 41 . . . . . . . . . 40 34 36 . . . . 10 . 33 . . . . . 39 . . . . . 6 . 19 . . . . 16 . . . . . . . 33 10 . . 37 . . 49 29 . . . . 23 . 26 . . 18 . . 47 32 . 14 46 11 9 25 . . . 20 . . 7 16 . . . . . . . . 27 . . 16 . 35 9 . . . . . 33 8 5 . 14 . . . . . . . 36 . 4 . . . 22 . . 12 24 . 48 2 . . . 3 29 42 45 6 . 28 . 31 . 18 . . 23 . . . 48 . 25 7 . . . 10 . . . . . . 21 . 42 37 . 13 5 45 15 . . . 22 . 35 17 30 44 9 39 12 . . 24 . . . 13 . 25 . 24 . . 46 . 34 18 31 . 44 20 48 3 . 45 . 5 12 39 8 22 11 . 17 35 16 . 6 . . . 38 . 9 42 29 1 27 30 47 . 14 21 . . . . 43 4 . . . 2 23 . . 17 15 . 11 35 42 . . . 20 9 40 16 . 34 30 . 24 38 . 26 32 3 13 41 28 1 47 5 . . 25 46 8 . 22 . . . 5 . . . 6 . 39 14 . 19 . 30 . 17 9 22 32 . 48 3 41 . 24 27 13 36 . 37 1 44 33 42 46 10 . 12 8 49 11 . . 23 . . 35 38 . . . 31 11 8 . . 42 . . 44 5 21 . . 41 . . . . . . . 32 34 . 6 . . . . . . . . . 7 . . 18 . . . 19 16 . . . . 24 . . 26 3 40 . 31 . 13 . . 12 . . 42 . 35 22 27 . . 38 . . . 2 . . . . 17 8 10 . . . . 48 . 20 . . . . . 37 . . 21 . . . 20 13 . 3 14 . . 7 40 . . . 8 11 6 15 . . 44 18 23 . . . . 1 . . 4 . . . . . . 32 . . 12 38 . . 29 . . . 32 17 39 . . . . 20 15 48 4 25 . . . . . 40 . . . . 11 . 45 7 . 31 . . 33 24 27 26 2 . 1 28 . 46 . 23 10 43 44 34 6 . . 34 . 12 . . . . . 32 38 . 18 24 . . 3 13 29 . . 31 . 16 27 . 8 . . 23 30 . 36 28 22 45 6 44 19 43 10 14 5 . 41 46 15 2 . . 47 33 . . 29 . . 11 . 41 10 37 . 19 1 46 48 49 . . . 35 42 . . 14 18 7 15 44 2 . 45 9 38 36 5 . 34 . 24 4 . 17 22 13 32 . . 35 . . 2 25 . . . 33 28 27 36 . 23 43 . 16 . . 17 39 . 47 19 30 . 21 3 . 9 22 5 26 . 14 . 4 41 24 42 40 . . 18 1 . 7 . . 44 1 30 . 31 . 33 17 22 . . 39 . . 13 32 28 41 . . . 19 27 . . . . . 12 35 . . 7 . . . 43 34 . 25 . . . 9 . . 46 . . 9 32 . 41 49 . . 31 28 . 42 20 . . 15 . 39 48 17 23 . . 30 7 36 18 . . 47 26 34 . . . . 5 . . 8 . . . . . 24 2 11 . 10 . . 34 39 45 . 35 40 . . 26 4 20 . 24 42 49 . 25 . . . . . . . . 5 . 18 23 21 . . . . 15 . 2 . 47 6 . . . . 29 . . 46 42 . 47 3 . . 7 25 9 . 34 . 36 . 44 . . 10 . . 40 43 . . 1 . . 48 28 39 37 30 21 27 35 23 22 38 33 45 . 13 31 12 32 49 . . 28 . . 12 38 . . 10 . . 44 15 . . 5 43 18 16 3 49 . . . . 11 45 13 8 27 29 31 9 25 37 . 24 40 20 6 48 23 . 22 35 . 36 . . 26 . 35 . 11 . . 6 . . 1 41 . . 8 12 . 45 . 23 38 . 5 29 14 . 25 24 10 22 . 46 . . 16 . 44 39 31 . 49 27 18 43 7 33 20 4 . . 23 48 5 . . . 43 36 38 47 . 2 . . . 19 14 35 9 . 33 24 15 46 37 22 49 4 6 . . 40 16 32 18 12 . . 10 . 42 25 21 3 . . 30 . 20 29 . 46 44 16 . 30 . . . 21 . . . 22 . 19 . . . . 13 . 47 49 . . . . . 42 . 38 . . 34 . 27 . 36 . . . . 2 6 31 . 38 15 . 10 24 22 . 44 . 4 12 9 23 . 1 . . 2 5 . . . 11 41 . . . . 18 8 . 21 . 3 . . 29 . 40 33 . . . 37 . . 49 . . 25 . . 41 . 14 . 45 . 3 . 15 22 . 48 . 16 36 23 . . . 27 1 32 34 . . 43 . 6 4 7 . . . . . 49 21 28 . 26 35 . . 47 . . 42 21 . . 32 27 . 34 . 39 25 . 5 . 4 38 . 33 . . 31 . . 44 . 43 35 9 . . . 47 . 11 . . 17 10 24 37 . 28 46 . 48 19 1 8 . 11 37 . 33 . 12 . 2 . 24 31 . 49 . 18 26 45 . 3 32 . 22 14 38 . 7 . . 41 . 5 . 28 . 15 1 39 46 13 . 35 4 36 . . 23 25 42 . 8 . 3 36 . . 43 18 . . . 13 16 . 10 . 49 41 . 28 5 4 37 46 2 . 26 17 . 45 19 1 48 34 . 44 30 . 12 25 9 21 . . . . 27 33 . 6 7 47 . . 34 . . 41 20 32 1 19 . 27 40 13 17 14 37 . 24 29 . 48 39 21 31 . 33 . 26 46 49 42 5 4 . . 23 18 10 43 11 44 45 38 9

conserning your "cheat": No idea, nice.

by **m_b_metcalf** » Wed Aug 22, 2018 1:55 pm

hkociemba1 wrote:Running the SAT-method on your puzzle on the other side reduces it to 1100 clues and the puzzle is minimal. I do not know to which SE rating this puzzle belongs now:

I can solve your 1200-clue puzzle using techniques that are too expensive when generating puzzles for N>6. It has an SE rating of ~9.

Could you apply your SAT program to reduce my 1021-clue puzzle further? I don't know whether it's minimal or not.

Thanks,

Mike

by **hkociemba1** » Wed Aug 22, 2018 6:03 pm

hkociemba1 wrote:Could you apply your SAT program to reduce my 1021-clue puzzle further? I don't know whether it's minimal or not.

Using tuples up to size 6 I can reduce to 1013. This takes onle a few minutes (if you have a Windows system you could try it yourself

). Using SAT takes longer, about 20-30s for each test of a removed clue. With 1021 clues this takes some time....

Edit: I now reduced with tuples up to size 15. I could reduce to 995 clues. Solving this 995 clue puzzle uses hidden tuples of size 7 24 times during the solving process, solving time is 1.3 s. Higher tuples than 7 do not occur.

Code: Select all: +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 35 8 29 48 24 36 11 | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 8 24 35 36 29 48 11 | . . . . . . . | 26 12 13 1 39 27 9 | | . . . . . . . | 42 41 19 49 40 10 44 | . . . . . . . | 16 20 34 32 33 28 3 | . . . . . . . | 37 4 47 31 7 22 2 | . . . . . . . | | . . . . . . . | . . . . . . . | 11 8 35 36 29 48 24 | . . . . . . . | 12 39 26 27 13 1 9 | . . . . . . . | 23 14 38 15 45 6 18 | | . . . . . . . | 32 16 20 3 34 33 28 | . . . . . . . | 47 7 4 2 22 37 31 | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | 2 47 7 31 4 22 37 | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 38 47 31 7 2 37 4 | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 12 39 26 27 9 1 10 | 13 16 3 20 32 28 34 | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 45 23 33 18 15 6 14 | 4 47 2 7 37 31 38 | . . . . . . . | | . . . . . . . | . . . . . . . | 14 45 33 18 15 6 23 | 2 37 47 38 31 4 7 | 43 46 22 17 5 25 21 | 8 24 35 36 11 48 30 | . . . . . . . | | . . . . . . . | . . . . . . . | . . . . . . . | 23 18 45 33 15 14 6 | 47 2 38 37 31 7 4 | 21 43 46 25 17 5 22 | . . . . . . . | | . . . . . . 32 | 33 . 18 6 45 15 14 | 4 47 38 37 31 7 . | . 17 43 22 5 21 25 | 24 . 30 11 48 36 8 | 40 41 42 19 44 49 29 | . . . . 26 . . | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . . 34 . 28 . | . . . . . . . | . . . . . . 38 | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 5 35 36 11 . . 24 | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . 28 | 23 33 3 14 6 18 45 | 47 2 . 37 4 7 15 | 31 46 25 . 22 21 43 | | . . . . . . . | 1 13 . . 32 20 16 | . . . 14 6 . . | . . 2 15 . 47 37 | . 22 . 21 25 . . | 24 35 30 11 . . 5 | 48 42 19 44 29 40 41 | | . . . 14 . 18 . | . . . . . . . | . . . . . 17 22 | 30 8 35 5 36 24 . | 42 29 48 40 19 44 41 | 39 26 10 . 12 27 49 | . . . . . . . | | . . . . . . . | . . . . . 25 . | 24 35 5 8 36 11 30 | 29 40 42 48 19 41 44 | 26 10 49 12 27 9 39 | 16 32 13 28 34 20 1 | . . . . . . . | | . . . . . . . | 5 30 8 11 35 36 24 | 41 42 48 40 19 44 29 | 10 12 26 49 27 39 9 | . . . . . . 16 | 45 23 33 . . 6 3 | 15 2 7 37 38 4 . | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . . . . . . | . . . . . 28 . | . . . . 18 . 3 | . . . . . . . | . . . . . . 46 | . . . . . . . | . . . . . . . | | . . . . 32 . 1 | . . . 14 . . 23 | . . . 47 . . . | . . . . . . 21 | . . . . . . 35 | . . . . 41 . . | 19 10 9 12 49 . 26 | | . . . 45 23 . . | . . . . . . . | . . . 43 17 . . | . . . . 11 . . | . . . . 44 . . | 26 10 49 12 39 9 19 | 27 13 . 34 1 16 32 | | . . . 24 35 8 . | . . . . . . . | . . . . . . . | 1 16 13 27 . . 34 | 33 3 20 45 18 14 . | 2 38 15 4 47 37 6 | 7 22 17 21 31 43 . | | . . . 43 46 21 . | . . . . . . . | . 29 36 41 44 . 48 | 49 39 10 19 9 26 12 | 13 1 27 16 . 34 32 | 23 33 . 14 45 . 20 | 6 38 37 4 15 . 2 | | . . . . . . . | . . . . . . . | 32 13 27 16 28 34 1 | . . . . . . 14 | 38 15 6 47 37 4 2 | 46 22 31 . 43 17 7 | 25 30 11 8 5 24 . | | . . . 47 2 4 . | 7 31 . . 22 17 . | . 30 25 . 11 8 5 | 48 41 29 36 44 42 40 | 10 49 . 39 . 12 26 | 32 13 . 34 16 . 27 | 20 . 18 . 3 . 23 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . 37 . . . . | . . . 24 . . . | . . . . . . . | . . . . . . . | . . . . . . 13 | . . . . . . . | . . . . . . . | | . . 11 . . . . | . . . . . . . | . . . . . . . | . . . 28 14 . 45 | . . . . . 47 . | . 31 7 43 46 21 . | . 5 8 . 25 35 30 | | . 4 . 2 38 47 . | . . 46 43 . 21 . | . . . . . . . | . 42 . . 40 . 41 | 49 19 44 . . 39 . | 13 . . . . . 9 | . 3 . 45 20 23 33 | | . 34 . . 13 16 . | . . 23 45 . 14 . | . 15 18 . . . 6 | 7 . 31 . 21 22 . | 5 25 17 35 8 . 30 | 29 48 36 . 42 . 11 | 44 . 12 39 . 26 10 | | . . 17 . . . . | . . . 41 . . 29 | . . 44 26 . . . | 27 32 1 9 34 13 16 | . . 28 . 14 45 . | 38 15 6 . . 4 18 | . . . 43 7 46 . | | . 14 . 23 33 45 . | . . . . . . . | 22 31 37 46 21 43 7 | 25 . 5 17 8 . 24 | 48 36 11 42 40 41 29 | 10 49 19 39 26 12 44 | . 1 . . 27 32 . | | . . . . . . . | 9 27 32 . 1 . . | 33 3 . 23 . . 20 | 6 2 15 18 . . . | 31 7 . 46 21 43 22 | 30 5 25 24 35 8 17 | 11 48 40 41 36 42 29 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . . . . . 11 | . . . . . . 49 | . . . . . 32 . | . 33 . . . 3 23 | . . 14 . . . 15 | . . . . . . . | . . . . 17 30 5 | | . . . . . . . | . . . 35 . . . | . . . . . . . | . . . . . . . | . . . . 16 . 1 | . 20 28 . 33 45 34 | . 6 47 2 . . . | | . . . . . . . | . . . . . . . | . . 34 33 . . 28 | . . . . 47 . . | 7 37 4 22 43 . 31 | 5 . 17 . . 24 . | 8 36 41 42 . 29 48 | | . 47 14 38 . 2 18 | . . 22 46 . 43 31 | . . . . . . . | 11 29 36 . 41 48 42 | 19 . . 10 . . . | . 27 9 . 13 . 12 | 34 20 . 23 . . 3 | | . . . . . . . | . . . . . . . | . . 40 10 . 26 44 | . . . . . 1 32 | 20 28 34 . 45 . 3 | 15 6 18 2 38 . 14 | 4 7 43 46 37 22 31 | | . 45 34 33 . 23 28 | . . 38 2 6 . 15 | 31 7 4 22 43 46 . | 17 30 25 21 24 5 35 | . 11 . 29 41 42 48 | 49 19 44 26 . 39 . | 12 . . . 9 . 1 | | . . . 13 1 32 9 | . . 33 23 20 45 3 | 15 6 . . . 2 . | 37 22 7 4 43 31 46 | 25 17 21 30 24 . 5 | 48 . . 42 29 41 8 | 40 . . 26 44 10 49 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . 2 45 15 6 . . | . . 31 22 37 46 7 | . . . . 35 . . | . . . . . . . | . 40 . . . 10 . | . . 12 . . . . | . . 23 33 34 . 20 | | . 35 . . . 30 . | . 8 . . . . 36 | . . . . . . . | . . . 39 32 . . | . . . . 23 33 . | . . . 38 . 2 . | 47 . . . 4 . 7 | | . 23 16 3 20 . . | . 14 . . 18 . . | . . 47 . 46 22 4 | . . . . . 25 30 | 11 8 . 48 42 29 36 | 19 . . . . . 41 | 39 9 . . . 1 27 | | . 32 39 . 27 13 . | . . 3 33 28 23 20 | . . 45 . . 38 14 | 4 31 37 . . 7 . | . . 43 . 35 30 25 | . 11 . 29 48 42 24 | 41 . 26 10 40 49 19 | | . . . . . . . | . . . . 9 32 27 | . 28 16 3 23 . 34 | . 15 . 45 2 6 38 | 37 4 47 31 46 22 7 | . 17 . . 5 35 43 | . . 42 29 . 48 36 | | . 42 . 48 36 29 . | 41 40 49 . 44 26 19 | . . 39 1 . 13 12 | 34 . 28 16 23 20 33 | . 14 . 15 2 38 6 | 7 37 . . . . 47 | 43 17 . 30 21 5 25 | | . 46 . 31 7 22 4 | 43 . 5 . 17 35 25 | . 11 24 48 42 29 8 | 40 49 . 41 . 19 10 | 9 12 . 1 . 13 27 | 20 28 34 33 3 . 16 | 45 18 . . . 15 6 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ 995 givens, 20002 candidates(pencilmarks).

The SAT-method still is running...

by **hkociemba1** » Thu Aug 23, 2018 4:46 am

Reducing with SAT had finished now. THe resulting minimal puzzle has 991 clues.

Code: Select all: +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 35 8 29 48 24 36 11 | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 8 24 35 36 29 48 11 | . . . . . . . | 26 12 13 1 39 27 9 | | . . . . . . . | 42 41 19 49 40 10 44 | . . . . . . . | 16 20 34 32 33 28 3 | . . . . . . . | 37 4 47 31 7 22 2 | . . . . . . . | | . . . . . . . | . . . . . . . | 11 8 35 36 29 48 24 | . . . . . . . | 12 39 26 27 13 1 9 | . . . . . . . | 23 14 38 15 45 6 18 | | . . . . . . . | 32 16 20 3 34 33 28 | . . . . . . . | 47 7 4 2 22 37 31 | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | 2 47 7 31 4 22 37 | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 38 47 31 7 2 37 4 | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 12 39 26 27 9 1 10 | 13 16 3 20 32 28 34 | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 45 23 33 18 15 6 14 | 4 47 2 7 37 31 38 | . . . . . . . | | . . . . . . . | . . . . . . . | 14 45 33 18 15 6 23 | 2 37 47 38 31 4 7 | 43 46 22 17 5 25 21 | 8 24 35 36 11 48 30 | . . . . . . . | | . . . . . . . | . . . . . . . | . . . . . . . | 23 18 45 33 15 14 6 | 47 2 38 37 31 7 4 | 21 43 46 25 17 5 22 | . . . . . . . | | . . . . . . 32 | 33 . 18 6 45 15 14 | 4 47 38 37 31 7 . | . 17 43 22 5 21 25 | 24 . 30 11 48 36 8 | 40 41 42 19 44 49 29 | . . . . 26 . . | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . . 34 . 28 . | . . . . . . . | . . . . . . 38 | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 5 35 36 11 . . 24 | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . 28 | 23 33 3 14 6 18 45 | 47 2 . 37 4 7 15 | 31 46 25 . 22 21 43 | | . . . . . . . | 1 13 . . 32 20 16 | . . . 14 6 . . | . . 2 15 . 47 37 | . 22 . 21 25 . . | 24 35 30 11 . . 5 | 48 42 19 44 29 40 41 | | . . . 14 . 18 . | . . . . . . . | . . . . . 17 22 | 30 8 35 5 36 24 . | 42 29 48 40 19 44 41 | 39 26 10 . 12 27 49 | . . . . . . . | | . . . . . . . | . . . . . 25 . | 24 35 5 8 36 11 30 | 29 40 42 48 19 41 44 | 26 10 49 12 27 9 39 | 16 32 13 28 . 20 1 | . . . . . . . | | . . . . . . . | 5 30 8 11 35 36 24 | 41 42 48 40 19 44 29 | 10 12 26 49 27 39 9 | . . . . . . 16 | 45 23 33 . . 6 3 | 15 2 7 37 38 4 . | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . . . . . . | . . . . . 28 . | . . . . 18 . 3 | . . . . . . . | . . . . . . 46 | . . . . . . . | . . . . . . . | | . . . . 32 . 1 | . . . 14 . . 23 | . . . 47 . . . | . . . . . . 21 | . . . . . . 35 | . . . . 41 . . | 19 10 9 12 49 . 26 | | . . . 45 23 . . | . . . . . . . | . . . 43 17 . . | . . . . 11 . . | . . . . 44 . . | 26 10 49 12 39 9 19 | 27 13 . 34 1 16 32 | | . . . 24 35 8 . | . . . . . . . | . . . . . . . | 1 16 13 27 . . 34 | 33 3 20 45 18 14 23 | 2 38 15 4 47 37 6 | 7 22 17 21 31 43 . | | . . . 43 46 21 . | . . . . . . . | . 29 36 41 44 . 48 | 49 39 10 19 9 26 12 | 13 1 27 16 . 34 32 | 23 33 . 14 45 . 20 | 6 38 37 4 15 . 2 | | . . . . . . . | . . . . . . . | 32 13 27 16 28 34 1 | . . . . . . 14 | 38 15 6 47 37 4 2 | 46 22 31 . 43 17 7 | 25 30 11 8 5 24 . | | . . . 47 2 4 . | 7 31 . . 22 17 . | . 30 25 . 11 8 5 | 48 41 29 36 44 42 40 | 10 49 . 39 . 12 26 | 32 13 . 34 16 . 27 | 20 . 18 . 3 . 23 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . 37 . . . . | . . . 24 . . . | . . . . . . . | . . . . . . . | . . . . . . 13 | . . . . . . . | . . . . . . . | | . . 11 . . . . | . . . . . . . | . . . . . . . | . . . 28 14 . 45 | . . . . . 47 . | . 31 7 . 46 21 . | . 5 8 . 25 35 30 | | . 4 . 2 38 47 . | . . 46 43 . 21 . | . . . . . . . | . 42 . . 40 . . | 49 19 44 26 . 39 . | 13 . . . . . 9 | . 3 . 45 20 23 33 | | . 34 . . 13 16 . | . . 23 45 . 14 . | . 15 18 . . . 6 | 7 . 31 . 21 22 . | 5 25 17 35 8 . 30 | 29 48 36 . 42 . 11 | 44 . 12 39 . 26 10 | | . . 17 . . . . | . . . 41 . . 29 | . . 44 26 . . . | 27 32 1 9 34 13 16 | . . 28 . 14 45 . | 38 15 6 . . 4 18 | . . . 43 7 46 . | | . 14 . 23 33 45 . | . . . . . . . | 22 31 37 46 21 43 7 | 25 . 5 17 8 . 24 | 48 36 11 42 40 41 29 | 10 49 19 39 26 12 44 | . 1 . . 27 32 . | | . . . . . . . | 9 27 32 . 1 . . | 33 3 . 23 . . 20 | 6 2 15 18 . . . | 31 7 . 46 21 43 22 | 30 5 25 24 35 8 17 | 11 48 40 41 36 42 29 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . . . . . 11 | . . . . . . 49 | . . . . . 32 . | . 33 . . . 3 23 | . . 14 . . . 15 | . . . . . . . | . . . . 17 30 5 | | . . . . . . . | . . . 35 . . . | . . . . . . . | . . . . . . . | . . . 13 16 . 1 | . 20 28 . 33 45 34 | . 6 47 2 . . . | | . . . . . . . | . . . . . . . | . . 34 33 . . 28 | . . . . 47 . . | 7 37 4 . 43 . 31 | 5 . 17 . . 24 . | 8 36 41 42 . 29 48 | | . 47 14 38 . 2 18 | . . 22 46 . 43 31 | . . . . . . . | 11 29 36 . 41 48 42 | 19 . . . . . . | 1 27 9 . 13 16 12 | 34 20 . 23 . . 3 | | . . . . . . . | . . . . . . . | . . 40 10 . 26 44 | . . . . . . 32 | 20 28 34 . 45 . 3 | . 6 18 2 38 . 14 | 4 7 43 46 37 22 31 | | . 45 34 33 . 23 28 | . . 38 2 6 . 15 | 31 7 4 22 43 46 . | 17 30 25 21 24 5 35 | . 11 . 29 41 42 48 | 49 19 44 26 . 39 . | 12 . . . 9 . 1 | | . . . 13 1 32 9 | . . 33 23 20 45 3 | 15 6 . . . 2 . | 37 22 7 4 43 31 46 | 25 17 21 30 24 . 5 | 48 . . 42 29 41 8 | 40 . . 26 44 10 49 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . 2 45 15 6 . . | . . 31 22 37 46 7 | . . . . . . . | . . . . . . . | . 40 . . . 10 . | . . 12 . . . . | . . 23 33 34 . 20 | | . 35 . 5 . 30 . | . 8 . . . . 36 | . . . . . . . | . . . 39 32 . 13 | . . . . 23 33 . | . . . 38 . 2 . | 47 . . . 4 . 7 | | . 23 16 3 20 . . | . 14 . . 18 . . | . . 47 . 46 22 4 | . . . . . 25 30 | 11 8 . 48 42 29 36 | 19 . . . . . 41 | 39 9 . . . 1 27 | | . 32 39 . 27 13 . | . . 3 33 28 23 20 | . . 45 . . 38 14 | 4 31 37 . . 7 . | . . 43 . . 30 25 | . 11 . 29 48 42 24 | 41 . 26 10 40 49 19 | | . . . . . . . | . . . . 9 32 27 | . 28 16 3 23 . 34 | . 15 . 45 2 6 38 | 37 4 47 31 46 22 7 | . 17 . . 5 35 43 | . . 42 29 . 48 36 | | . 42 . 48 36 29 . | 41 40 49 . 44 26 19 | . . 39 1 . 13 12 | 34 . 28 16 23 20 33 | . . . 15 2 38 . | 7 37 . . . . 47 | 43 17 . 30 21 5 25 | | . 46 . 31 7 22 4 | 43 . 5 . 17 35 25 | . 11 24 48 42 29 8 | 40 49 . 41 . 19 10 | 9 12 . 1 . 13 27 | 20 28 34 33 3 . 16 | 45 18 . . . 15 6 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ 991 givens, 20070 candidates(pencilmarks).

by **m_b_metcalf** » Thu Aug 23, 2018 9:41 am

hkociemba1 wrote:Reducing with SAT had finished now. THe resulting minimal puzzle has 991 clues.

An excellent result! The two reduced puzzles that you've produced are too hard for my solver, implying an SE rating of ~10. The second is thus most likley the smallest and hardest 49x49 ever published.

More details on the 'cheating' at 2.15 CET.

Regards,

Mike

by **m_b_metcalf** » Thu Aug 23, 2018 12:15 pm

m_b_metcalf wrote:However, if I cheat a little, I can get fewer clues than 1050, namely 1021. (If anyone can tell me how I cheated, within 24 hours, I'll donate $10 to the Forum.)

The question was whether a random 49x49 sudoku could be generated with fewer that 1050 clues. The puzzle I posted is far from being random. Its solution grid is a permutation (values, rows, columns, bands) of the 49x49 MC (Most Canonical) grid. (I have a program that can generate n^2 x n^2 MC [Edit: actually canonical but, at the time of writing, not MC] grids, randomized or not, of arbitrarily large size.) That this puzzle has a 'fishy' structure is better revealed if it is renumbered, as below.

Although the 991-clue puzzle that was derived from it is formally extremely difficult, knowledge of the underlying structure probably allows it to be solved easily by hand; consider, for instance, the likely value at r14c9, or at r49c41.

Regards,

Mike

Code: Select all: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 4 5 6 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5 1 6 3 4 7 . . . . . . . 8 9 10 11 12 13 14 . . . . . . . 15 16 17 18 19 20 21 . . . . . . . 22 23 24 25 26 27 28 . . . . . . . 29 30 31 32 33 34 35 . . . . . . . . . . . . . . . . . . . . . 7 2 1 6 3 4 5 . . . . . . . 9 12 8 13 10 11 14 . . . . . . . 36 37 38 39 40 41 42 . . . . . . . 25 22 23 28 24 26 27 . . . . . . . 31 33 30 35 34 29 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 31 33 32 30 34 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 31 32 33 35 29 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . . . . . . . 9 12 8 13 14 11 20 10 22 28 23 25 27 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 36 26 42 39 41 37 30 31 35 33 29 32 38 . . . . . . . . . . . . . . . . . . . . . 37 40 26 42 39 41 36 35 29 31 38 32 30 33 43 44 34 45 46 47 48 2 5 1 6 7 4 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 42 40 26 39 37 41 31 35 38 29 32 33 30 48 43 44 47 45 46 34 . . . . . . . . . . . . . 25 26 . 42 41 40 39 37 30 31 38 29 32 33 . . 45 43 34 46 48 47 5 . 49 7 4 6 2 19 16 15 17 21 18 3 . . . . 8 . . . . . 24 . 27 . . . . . . . . . . . . . . 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1 6 7 . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 36 26 28 37 41 42 40 31 35 . 29 30 33 39 32 44 47 . 34 48 43 . . . . . . . 11 10 . . 25 23 22 . . . 37 41 . . . . 35 39 . 31 29 . 34 . 48 47 . . 5 1 49 7 . . 46 4 15 17 21 3 19 16 . . . 37 . 42 . . . . . . . . . . . . . 45 34 49 2 1 46 6 5 . 15 3 4 19 17 21 16 12 8 20 . 9 13 18 . . . . . . . . . . . . . . . . . . . 47 . 5 1 46 2 6 7 49 3 19 15 4 17 16 21 8 20 18 9 13 14 12 22 25 10 27 24 23 11 . . . . . . . . . . . . . . 46 49 2 7 1 6 5 16 15 4 19 17 21 3 20 9 8 18 13 12 14 . . . . . . 22 40 36 26 . . 41 28 39 35 33 29 38 30 . . . . . . . . . . . . . 27 . . . . . 42 . 28 . . . . . . . . . . . . . 44 . . . . . . . . . . . . . . . . . . 25 . 11 . . . 37 . . 36 . . . 31 . 30 . . . . . . . 48 . . . . . . 1 . . . . 16 21 . 17 20 14 9 18 . 8 . . . 40 36 . . . . . . . . . . . . 43 45 . . . . . . 7 . . . . . . 21 . . 8 20 18 9 12 14 17 13 10 . 24 11 22 25 . . . 5 1 2 . . . . . . . . . . . . . . . 11 22 10 13 . . 24 26 28 23 40 42 37 36 35 38 39 30 31 29 41 33 34 45 48 32 43 . . . . 43 44 48 . . . . . . . . . 3 6 16 21 . 4 18 12 20 17 14 8 9 10 11 13 22 . 24 25 36 26 . 37 40 . 23 41 38 29 30 39 . 35 . . . . . . . . . . . 20 14 8 25 10 13 22 27 24 11 . . . . . 36 37 38 39 41 31 29 30 35 44 34 32 . 43 45 33 47 49 7 2 46 5 . . . . 31 35 30 . 33 32 . . 34 45 . . 49 47 . 7 2 46 4 16 3 6 21 15 19 20 18 . 12 . 9 8 25 10 11 24 22 . 13 23 . 42 . 28 . 36 . . 29 . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . 25 . . 10 . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . . . . . . . . 27 37 . 40 . . . . . 31 . . 32 33 43 44 48 . . 46 2 . 47 1 49 . 30 . 35 38 31 . . . 44 43 . 48 . . . . . . . . . 15 . . 19 . 16 18 17 21 8 . 12 . 10 . . . . . 14 . 28 . 40 23 36 26 . 24 . . 10 22 . . . 36 40 . 37 . . 39 42 . . . 41 33 . 32 . 48 34 . 46 47 45 1 2 . 49 3 4 6 . 15 . 7 21 . 9 12 . 8 20 . . 45 . . . . . . . 16 . 19 3 . . 21 8 . . . 13 25 11 14 24 10 22 . . 27 . 37 40 . 38 39 41 . . 30 42 . . . 43 33 44 . . 37 . 36 26 40 . . . . . . . . 34 32 29 44 48 43 33 47 . 46 45 2 . 5 4 6 7 15 19 16 3 20 18 17 12 8 9 21 . 11 . . 13 25 . . . . . . . . 14 13 25 . 11 . . 26 28 . 36 . . 23 41 35 39 42 . . . 32 33 . 44 48 43 34 49 46 47 5 1 2 45 7 4 19 16 6 15 3 . . . . . . 7 . . . . . . 18 . . . . . 25 . . 26 . . . 28 36 . . 37 . . . 39 . . . . . . . . . . . 45 49 46 . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 10 22 . 11 . 23 27 . 26 40 24 . 41 31 35 . . . . . . . . . . . . . . . . . . . 24 26 . . 27 . . . . 31 . . 33 29 30 34 43 . 32 46 . 45 . . 5 . 2 6 16 15 . 3 4 . 31 37 38 . 35 42 . . 34 44 . 43 32 . . . . . . . 7 3 6 . 16 4 15 17 . . 20 . . 18 11 13 14 . 10 22 9 24 23 . 36 . . 28 . . . . . . 45 . . . . . . . . . 19 20 . 8 21 . 10 . . . 11 25 23 27 24 . 40 . 28 39 41 42 35 38 . 37 30 33 43 44 29 34 32 . 40 24 26 . 36 27 . . 38 35 41 . 39 32 33 30 34 43 44 . 45 49 47 48 5 46 1 . 7 . 3 16 15 4 18 17 21 8 . 12 . 9 . . . 14 . 11 . . . 10 11 25 14 . . 26 36 23 40 28 39 41 . . . 35 . 29 34 33 30 43 32 44 47 45 48 49 5 . 46 4 . . 15 3 16 2 19 . . 8 21 20 18 . 35 40 39 41 . . . . 32 34 29 44 33 . . . . 1 . . . . . . 15 . 3 . 19 . . . 20 . . . 9 . 11 . . . . 36 26 24 . 23 . 1 . 46 47 49 . . 2 . . . . 6 . . . . . . . . . . 12 25 . 10 . . . . 36 26 . . . . 38 . 35 . 31 . . . 30 . 33 . 36 22 28 23 . . . 37 . . 42 . . . . 31 . 44 34 30 . . . . . 47 49 7 2 . 4 15 3 6 17 . . . . . 16 12 14 . . . 11 13 . 25 12 . 13 10 . . . 28 26 27 36 23 . . 40 . . 38 37 30 32 29 . . 33 . . . 43 . 1 49 47 . 7 . 3 4 15 5 16 . 8 20 19 18 17 . . . . . . . . . . . 14 25 13 . 27 22 28 36 . 24 . 39 42 40 35 41 38 29 30 31 32 44 34 33 . 45 . . 46 1 43 . . 15 3 . 4 6 . 15 . 4 6 3 . 16 19 18 . 21 8 17 . . 12 11 . 10 9 24 . 27 22 36 23 26 . 37 . 39 35 38 41 33 29 30 . . . 31 43 45 . 49 48 46 47 . 44 . 32 33 34 30 43 . 46 . 45 1 47 . 7 5 4 15 3 2 19 18 . 16 . 17 20 14 9 . 11 . 10 13 23 27 24 26 28 . 22 40 42 . . . 39 41

by **hkociemba1** » Thu Aug 23, 2018 3:42 pm

m_b_metcalf wrote: Its solution grid is a permutation (values, rows, columns, bands) of the 49x49 MC (Most Canonical) grid.

Can you explain how your most canonical grid is constructed? The grids I used to derive the formula are that what in my program is called "Default Grid". These also are canonical grids but they seem to be built differently because the number of clues left is bigger:

Code: Select all: +-------+-------+-------+ | 1 2 3 | 4 5 6 | 7 8 9 | | 4 5 6 | 7 8 9 | 1 2 3 | | 7 8 9 | 1 2 3 | 4 5 6 | +-------+-------+-------+ | 2 3 4 | 5 6 7 | 8 9 1 | | 5 6 7 | 8 9 1 | 2 3 4 | | 8 9 1 | 2 3 4 | 5 6 7 | +-------+-------+-------+ | 3 4 5 | 6 7 8 | 9 1 2 | | 6 7 8 | 9 1 2 | 3 4 5 | | 9 1 2 | 3 4 5 | 6 7 8 | +-------+-------+-------+ +-------------+-------------+-------------+-------------+ | 1 2 3 4 | 5 6 7 8 | 9 10 11 12 | 13 14 15 16 | | 5 6 7 8 | 9 10 11 12 | 13 14 15 16 | 1 2 3 4 | | 9 10 11 12 | 13 14 15 16 | 1 2 3 4 | 5 6 7 8 | | 13 14 15 16 | 1 2 3 4 | 5 6 7 8 | 9 10 11 12 | +-------------+-------------+-------------+-------------+ | 2 3 4 5 | 6 7 8 9 | 10 11 12 13 | 14 15 16 1 | | 6 7 8 9 | 10 11 12 13 | 14 15 16 1 | 2 3 4 5 | | 10 11 12 13 | 14 15 16 1 | 2 3 4 5 | 6 7 8 9 | | 14 15 16 1 | 2 3 4 5 | 6 7 8 9 | 10 11 12 13 | +-------------+-------------+-------------+-------------+ | 3 4 5 6 | 7 8 9 10 | 11 12 13 14 | 15 16 1 2 | | 7 8 9 10 | 11 12 13 14 | 15 16 1 2 | 3 4 5 6 | | 11 12 13 14 | 15 16 1 2 | 3 4 5 6 | 7 8 9 10 | | 15 16 1 2 | 3 4 5 6 | 7 8 9 10 | 11 12 13 14 | +-------------+-------------+-------------+-------------+ | 4 5 6 7 | 8 9 10 11 | 12 13 14 15 | 16 1 2 3 | | 8 9 10 11 | 12 13 14 15 | 16 1 2 3 | 4 5 6 7 | | 12 13 14 15 | 16 1 2 3 | 4 5 6 7 | 8 9 10 11 | | 16 1 2 3 | 4 5 6 7 | 8 9 10 11 | 12 13 14 15 | +-------------+-------------+-------------+-------------+

by **m_b_metcalf** » Thu Aug 23, 2018 3:59 pm

I hope this explains it:
[Edit: make the grids for odd N MC]

Code: Select all: sudoku(1, :) = (/ (i, i = 1, N**2) /) select case(mod(N, 2)) ! ! FOR EVEN N (which does not generate an MC (Most Canonical) grid) case(0) ! ! Fill the second row with a copy of the first in reverse order. ! sudoku(2, :) = sudoku(1, N2:1:-1) row = 2 ! ! Fill each pair of rows of the remainder of the first row of boxes (chute) ! with a copy of the previous two rows, circularly shifted right by N+N ! columns. ! do add_rows = 1, N/2 - 1 sudoku(row+1:row+2, :) = cshift(sudoku(row-1:row, :), -2*N, dim=2) row = row +2 end do ! ! Fill the second row of boxes (chute) with a copy of the first, circularly ! shifted right by N columns. ! sudoku(row+1:row+N, :) = cshift(sudoku(row-N+1:row, :), -N, dim=2) row = row + N ! ! Fill all the remaining rows of boxes (chutes) two at a time with a copy ! of the two previous rows of boxes (chutes). ! do add_rows = 1, N/2 - 1 sudoku(row+1:row+2*N, :) = sudoku(row-2*N+1:row, :) ! ! In each of the copied pairs of rows of boxes (chutes), divide the N columns ! in each box into two equal groups, and for each group circularly shift ! the columns one position right. ! do spin = 1, N2, N/2 sudoku(row+1:row+2*N, spin:spin+N/2-1) = & cshift(sudoku(row+1:row+2*N, spin:spin+N/2-1), -1, dim=2) end do row = row + 2*N end do ! ! FOR ODD N (which gives an MC grid) case(1) ! ! Fill each row of the remainder of the first row of boxes (chute) ! with a copy of the previous row, circularly shifted leftt by N columns. ! row = 1 do add_rows = 2, N row = row + 1 sudoku(row, :) = cshift(sudoku(row-1, :), N) end do ! ! Fill all the remaining rows of boxes (chutes) with a copy ! of the previous row of boxes (chute), shifted leftt by 1 column. ! do add_rows = 1, N - 1 sudoku(row+1:row+N, :) = cshift(sudoku(row-N+1:row, :), 1, dim=2) row = row + N end do end select

and examples:

Code: Select all: 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 1 2 3 7 8 9 1 2 3 4 5 6 2 3 4 5 6 7 8 9 1 5 6 7 8 9 1 2 3 4 8 9 1 2 3 4 5 6 7 3 4 5 6 7 8 9 1 2 6 7 8 9 1 2 3 4 5 9 1 2 3 4 5 6 7 8

Code: Select all: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1 16 15 14 13 12 11 10 9 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 4 3 2 1 16 15 14 13 12 11 10 9 8 7 6 5 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 12 11 10 9 8 7 6 5 4 3 2 1 16 15 14 13 2 1 4 3 6 5 8 7 10 9 12 11 14 13 16 15 15 16 13 14 11 12 9 10 7 8 5 6 3 4 1 2 10 9 12 11 14 13 16 15 2 1 4 3 6 5 8 7 7 8 5 6 3 4 1 2 15 16 13 14 11 12 9 10 14 13 16 15 2 1 4 3 6 5 8 7 10 9 12 11 3 4 1 2 15 16 13 14 11 12 9 10 7 8 5 6 6 5 8 7 10 9 12 11 14 13 16 15 2 1 4 3 11 12 9 10 7 8 5 6 3 4 1 2 15 16 13 14

Code: Select all: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 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by **m_b_metcalf** » Thu Aug 23, 2018 4:58 pm

hkociemba1 wrote:Can you explain how your most canonical grid is constructed? The grids I used to derive the formula are that what in my program is called "Default Grid". These also are canonical grids but they seem to be built differently because the number of clues left is bigger:

I now see that my code does not actually produce the MC, but an automorph. As it was written 12 years ago I have no idea where the algorithm came from. To be investigated. However, that doesn't explain why my grids result in fewer clues!

by **hkociemba1** » Thu Aug 23, 2018 8:10 pm

Thanks for the explanation. I will try to implement the algorithm. Now it's my riddle: Using your canonical grid can you explain how I generated this puzzle with only 834 clues? It is solvable with only singles and box-line interaction (and yet not minmal), so you should be able to verrify it.

Code: Select all: +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | 48 45 . . 39 . 25 | 1 20 42 41 27 34 11 | . 36 . . . . . | . . . . . . . | . . . . . . . | . 9 21 35 14 17 8 | 40 3 7 37 33 43 28 | | 3 . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 2 38 . . 47 23 . | . . . . . . . | | . . . . . . . | 31 47 38 23 . 10 6 | . 30 . . . . . | 49 19 13 22 . 12 26 | 44 48 29 4 . . . | 3 40 . . 43 33 . | 16 5 18 15 24 32 46 | | 36 49 12 22 19 26 13 | 8 14 9 17 30 21 35 | . . . . . . . | . . . . . . . | . . . . . . . | . 39 4 29 25 45 44 | . . . . . . . | | . . 10 6 . 31 . | . 13 19 49 36 . . | . . . . . . . | . . . . . . . | . . . . . . . | 5 . . . . . . | . . . . . . . | | . 24 15 46 16 18 32 | . . . . . . . | . 2 . . . . . | . . . . . . . | . . . . . . . | . . 12 22 . . 26 | . . . . . . . | | . . . . . . . | 7 43 40 33 3 37 28 | . . 45 . 25 . 39 | 24 16 32 46 5 15 18 | . . . . . . . | 27 42 . . 20 41 . | 38 2 31 10 23 47 6 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | 38 10 32 5 31 24 6 | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 16 18 13 36 46 15 49 | . . . . . . . | | 16 15 13 36 18 49 46 | 17 29 44 4 39 14 30 | . . . . . . . | . . . . . . . | . . . . . . . | 19 26 25 48 22 12 45 | . . . . . . . | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | 33 11 1 34 42 43 3 | . . . . . . . | 10 31 6 5 38 32 24 | . . . . . . . | . . . . . . . | 7 40 23 47 37 28 2 | | . . . . . . . | . . . . . . . | . . . . . . . | 15 18 46 36 16 13 49 | 45 19 48 25 26 22 12 | . . . . . . . | 31 38 24 32 10 6 5 | | 40 37 47 2 7 23 28 | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | 41 35 8 21 9 20 27 | . . . . . . . | . . . . . . . | . . . . . . . | 39 44 14 30 29 4 17 | 1 42 33 43 34 11 3 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . 6 38 23 10 . | . 36 . 13 18 . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | 37 3 33 43 1 28 40 | . . 25 . 48 . . | 32 24 5 16 31 46 15 | . . . . . . . | 8 . . . 27 20 . | 23 7 10 6 47 2 38 | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 7 . . . 2 47 . | . . . . . . . | | 18 13 22 19 49 12 36 | 21 30 17 14 44 35 9 | . . . . . . . | . . . . . . . | . . . . . . . | 26 45 29 39 48 25 4 | . . . . . . . | | 31 32 46 16 24 15 5 | . . . . . . . | . 7 . . . . . | . . . . . . . | . . . . . . . | . 49 22 19 . . 12 | . . . . . . . | | . . . . . . . | . 2 . 47 . 6 38 | . 44 . . . . . | 13 49 36 19 . 22 12 | 4 26 39 29 45 . . | 1 . . . 3 43 . | 24 31 15 46 32 5 16 | | 26 25 . . . . 48 | 34 27 41 20 8 11 42 | . 18 . . . . . | . . . . . . . | . . . . . . . | 44 17 35 9 30 14 21 | 33 1 37 28 43 3 40 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | 47 . 31 . . 5 . | . 49 22 19 13 . 12 | . . . . . . . | . . . . . . . | . . . . . . . | 32 . . . 24 . . | . . . . . . . | | . . . . . . . | 5 23 6 . . 31 10 | . 14 . . . . . | 19 22 49 12 . 26 48 | 30 25 . 44 . . . | 43 28 . 37 33 40 . | 46 32 36 18 16 24 15 | | 25 39 . 4 29 . 45 | 3 41 11 42 20 1 34 | . 13 . . . . . | . . . . . . . | . . . . . . . | 14 35 8 21 17 9 27 | 28 43 2 7 40 33 37 | | . . . . . . . | 2 33 28 40 43 7 37 | . . 39 4 45 . 29 | 16 46 24 15 32 18 36 | . . . . . . . | 20 11 . 34 41 42 . | 6 47 5 31 38 23 10 | | . 16 18 15 46 36 . | . . . . . . . | . 47 38 . . . . | . . . . . . . | . . . . . . . | . . 26 . . . 48 | . . . . . . . | | 13 19 26 12 22 48 49 | 27 17 35 9 14 8 21 | . . . . . . . | . . . . . . . | . . . . . . . | 25 29 44 4 . 39 30 | . . . . . . . | | 43 . . . . . 33 | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . 6 . 10 23 38 . | . . . . . . . | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | 49 22 48 26 12 25 . | 20 9 21 35 17 27 8 | . . . . . . . | . . . . . . . | . . . . . . . | 45 4 30 44 39 29 14 | . . . . . . . | | 45 29 30 44 4 . 39 | 43 42 34 11 41 3 1 | . 49 . . 19 . . | . . . . . . . | . . . . . . . | 17 21 27 8 9 35 20 | 37 33 47 2 28 40 7 | | . . 36 18 15 13 . | . . . . . . . | . 23 6 . 38 . . | . . . . . . . | . . . . . . . | . . 48 . 19 . 25 | . . . . . . . | | 33 28 . . . . 40 | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . 10 5 31 . . . | . . . . . . . | | . . . . . . 9 | 47 40 37 28 33 2 7 | 30 . 29 44 39 . 4 | 46 15 16 18 24 36 13 | . . . . . . . | 41 34 3 1 42 11 . | 10 23 32 5 6 38 31 | | . 6 . . . 32 38 | . . 12 22 . 48 26 | . . . . . . . | . . . . . . . | . . . . . . . | 24 . . . 16 46 . | . . . . . . . | | 41 11 . . . . 42 | 32 . 10 . . 5 31 | . 17 . . . . . | 22 12 . 26 . 48 25 | 14 45 . . . . . | 33 37 2 7 40 28 47 | 15 24 13 36 46 16 18 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | . . . . 28 . . | . 29 . . . . . | 18 36 15 13 46 49 19 | 39 . 25 45 48 12 26 | . . . . . . . | 5 6 16 24 31 10 32 | | 6 31 24 32 5 16 10 | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 46 36 49 13 15 18 19 | . . . . . . . | | 46 18 49 13 36 19 15 | 9 4 30 44 29 17 14 | . . . . . . . | . . . . . . . | . . . . . . . | 22 48 45 25 12 26 39 | . . . . . . . | | 28 7 23 47 2 38 37 | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | . . . . . . . | 42 21 27 8 35 41 20 | . . . . . . . | . . . . . . . | . . . . . . . | 29 30 17 14 4 44 9 | 3 11 40 33 1 34 43 | | . . . . . . . | 40 34 3 1 11 33 43 | . . . . . . . | 31 5 10 32 6 24 16 | . . . . . . . | . . . . . . . | 2 28 38 23 7 37 47 | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ | 10 5 16 24 32 46 31 | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | 15 13 19 49 18 36 22 | . . . . . . . | | . . . . . . . | . 7 47 2 37 . . | . 4 . . . . . | 36 13 18 49 . 19 22 | 29 12 45 39 25 . 48 | 34 . . . . 3 . | 32 . 46 16 5 31 24 | | . . . . . . . | 11 8 20 27 21 42 41 | . 15 . . . . . | . . . . . . . | . . . . . . . | 4 14 9 17 44 30 35 | 43 34 28 40 3 1 33 | | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | | 15 36 19 49 13 22 18 | 35 44 14 30 4 9 17 | . . . . . . . | . . . . . . . | . . . . . . . | 12 25 39 45 26 48 29 | . . . . . . . | | . . . . . . . | 28 1 43 3 34 40 33 | . . . . 26 . . | 5 32 31 24 10 16 46 | . . . . . . . | . . . . . . . | 47 37 6 38 2 7 23 | | 37 2 38 23 47 6 7 | . 18 . . 15 . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | . . . . . . . | +----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+----------------------+ 834 givens, 32834 candidates(pencilmarks).

by **m_b_metcalf** » Fri Aug 24, 2018 7:54 am

m_b_metcalf wrote:
hkociemba1 wrote:I now see that my code does not actually produce the MC, but an automorph. As it was written 12 years ago I have no idea where the algorithm came from. To be investigated. However, that doesn't explain why my grids result in fewer clues!

Following some archaeological research, it is clear that at that time the algorithms were intended to produce canonical but not necessarily MC grids (the task at the time was to produce huge grids up to 1000 x 1000). It turns out that two simple changes to the code for odd N result in MC grids being produced. For even N the changes would be more extensive and I'll not bother. My posts upthread has been modified accordingly.

Regards,

Mike

by **hkociemba1** » Fri Aug 24, 2018 10:31 am

m_b_metcalf wrote:Following some archaeological research, it is clear that at that time the algorithms were intended to produce canonical but not necessarily MC grids

I do not know the definition of "MC grid". Can you give it to me or give me a link in the forum where it is defined/discussed? The search function did not reveal anything to me.

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Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)

Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)

Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)

Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)

Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)

Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)

Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)

Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)

Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)

Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)

Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)

Re: giant sudoku's (16x16, 25x25, 36x36 .... 100x100)