The New Sudoku Players' Forum

by **JPF** » Tue Aug 11, 2020 8:55 pm

Each box of a sudoku grid may have x clues with 0 ≤ x ≤ 9; it makes 10 possibilities.
Therefore, there are 10^9 = 1,000,000,000 possible ways to fill in the 9 boxes.

Here is an example - 27 clues:

Code: Select all: 246 302 901

How many are non equivalent (ed) ?

The answer is 14,068,450.
Here is the breakdown by the total number of clues (n) in the grid:

Code: Select all: n 0 1 1 1 2 3 3 7 4 16 5 32 6 68 7 126 8 238 9 422 10 729 11 1211 12 1973 13 3097 14 4764 15 7132 16 10451 17 14959 18 21007 19 28877 20 38998 21 51682 22 67332 23 86222 24 108680 25 134776 26 164649 27 198125 28 234987 29 274727 30 316784 31 360248 32 404243 33 447599 34 489165 35 527695 36 562042 37 591010 38 613688 39 629268 40 637191 41 637191 42 629268 43 613688 44 591010 45 562042 46 527695 47 489165 48 447599 49 404243 50 360248 51 316784 52 274727 53 234987 54 198125 55 164649 56 134776 57 108680 58 86222 59 67332 60 51682 61 38998 62 28877 63 21007 64 14959 65 10451 66 7132 67 4764 68 3097 69 1973 70 1211 71 729 72 422 73 238 74 126 75 68 76 32 77 16 78 7 79 3 80 1 81 1 ---------- 14068450

For instance, for n=4, here are the 16 possibilities:

Code: Select all: 000 000 000 000 000 000 000 000 004 013 022 112 000 000 000 000 001 001 001 001 012 021 030 111 000 000 000 000 001 002 002 011 120 020 110 011 000 001 001 001 011 001 010 010 101 110 101 200

Now, if boxes with only 0,1 or 9 clues are considered, the ed-number of ways to fill in the grid is 438 out of 3^9 = 19683
Here they are:

Hidden Text: Show

Code: Select all: 000000000 000000001 000000009 000000011 000000019 000000099 000000111 000000119 000000199 000000999 000001010 000001011 000001019 000001090 000001091 000001099 000001110 000001111 000001119 000001190 000001191 000001199 000001990 000001991 000001999 000009090 000009091 000009099 000009110 000009111 000009119 000009190 000009191 000009199 000009990 000009991 000009999 000011011 000011019 000011099 000011101 000011109 000011111 000011119 000011199 000011901 000011909 000011911 000011919 000011999 000019091 000019099 000019109 000019111 000019119 000019190 000019191 000019199 000019909 000019910 000019911 000019919 000019990 000019991 000019999 000099099 000099111 000099119 000099199 000099909 000099911 000099919 000099999 000111111 000111119 000111199 000111999 000119119 000119191 000119199 000119991 000119999 000199199 000199919 000199999 000999999 001001110 001001111 001001119 001001190 001001191 001001199 001001990 001001991 001001999 001009190 001009191 001009199 001009990 001009991 001009999 001010100 001010101 001010109 001010111 001010119 001010199 001010900 001010901 001010909 001010911 001010919 001010999 001011110 001011111 001011119 001011190 001011191 001011199 001011900 001011901 001011909 001011910 001011911 001011919 001011990 001011991 001011999 001019190 001019191 001019199 001019900 001019901 001019909 001019910 001019911 001019919 001019990 001019991 001019999 001090900 001090901 001090909 001090910 001090911 001090919 001090990 001090991 001090999 001091110 001091111 001091119 001091190 001091191 001091199 001091901 001091909 001091910 001091911 001091919 001091990 001091991 001091999 001099190 001099191 001099199 001099909 001099910 001099911 001099919 001099990 001099991 001099999 001110110 001110111 001110119 001110190 001110191 001110199 001110990 001110991 001110999 001111111 001111119 001111190 001111191 001111199 001111990 001111991 001111999 001119119 001119190 001119191 001119199 001119990 001119991 001119999 001190191 001190199 001190910 001190911 001190919 001190990 001190991 001190999 001191191 001191199 001191911 001191919 001191990 001191991 001191999 001199199 001199919 001199990 001199991 001199999 001990990 001990991 001990999 001991991 001991999 001999999 009009990 009009991 009009999 009011910 009011911 009011919 009011990 009011991 009011999 009019990 009019991 009019999 009090900 009090901 009090909 009090911 009090919 009090999 009091910 009091911 009091919 009091990 009091991 009091999 009099990 009099991 009099999 009110110 009110111 009110119 009110190 009110191 009110199 009110990 009110991 009110999 009111111 009111119 009111190 009111191 009111199 009111990 009111991 009111999 009119119 009119190 009119191 009119199 009119990 009119991 009119999 009190191 009190199 009190910 009190911 009190919 009190990 009190991 009190999 009191191 009191199 009191911 009191919 009191990 009191991 009191999 009199199 009199919 009199990 009199991 009199999 009990990 009990991 009990999 009991991 009991999 009999999 011101110 011101111 011101119 011101190 011101191 011101199 011101990 011101991 011101999 011109190 011109191 011109199 011109910 011109911 011109919 011109990 011109991 011109999 011111111 011111119 011111199 011111901 011111909 011111911 011111919 011111999 011119191 011119199 011119901 011119909 011119910 011119911 011119919 011119990 011119991 011119999 011199199 011199901 011199909 011199911 011199919 011199999 011901990 011901991 011901999 011909911 011909919 011909990 011909991 011909999 011911911 011911919 011911999 011919919 011919991 011919999 011999999 019109990 019109991 019109999 019111910 019111911 019111919 019111990 019111991 019111999 019119990 019119991 019119999 019190901 019190909 019190911 019190919 019190991 019190999 019191910 019191911 019191919 019191990 019191991 019191999 019199990 019199991 019199999 019909990 019909991 019909999 019910919 019910999 019911911 019911919 019911990 019911991 019911999 019919919 019919990 019919991 019919999 019990991 019990999 019991991 019991999 019999999 099909990 099909991 099909999 099911911 099911919 099911999 099919991 099919999 099999999 111111111 111111119 111111199 111111999 111119191 111119199 111119991 111119999 111199199 111199919 111199999 111999999 119119991 119119999 119191911 119191919 119191999 119199991 119199999 119991991 119991999 119999999 199919991 199919999 199999999 999999999

Obviously, some patterns can't have valid puzzles: 000019119 for example.

Code: Select all: 000 019 119

More to come.

JPF

by **Serg** » Wed Aug 12, 2020 5:39 pm

Hi, JPF!

JPF wrote:Here is the breakdown by the total number of clues (n) in the grid:

Code: Select all
n 0 1 1 1 2 3 3 7 4 16 5 32 6 68 7 126 8 238 9 422 10 729 11 1211 12 1973 13 3097 14 4764 15 7132 16 10451 17 14959 18 21007 19 28877 20 38998 21 51682 22 67332 23 86222 24 108680 25 134776 26 164649 27 198125 28 234987 29 274727 30 316784 31 360248 32 404243 33 447599 34 489165 35 527695 36 562042 37 591010 38 613688 39 629268 40 637191 41 637191 42 629268 43 613688 44 591010 45 562042 46 527695 47 489165 48 447599 49 404243 50 360248 51 316784 52 274727 53 234987 54 198125 55 164649 56 134776 57 108680 58 86222 59 67332 60 51682 61 38998 62 28877 63 21007 64 14959 65 10451 66 7132 67 4764 68 3097 69 1973 70 1211 71 729 72 422 73 238 74 126 75 68 76 32 77 16 78 7 79 3 80 1 81 1 ---------- 14068450

I confirm these results. Well done!

Serg

The New Sudoku Players' Forum

Distribution of clues in boxes

Distribution of clues in boxes

Re: Distribution of clues in boxes