Serg wrote:As fas as I know, nobody at this forum ever discussed 16 x 16 Sudoku solution grids enumeration.
Although our original method required a couple of hours of computer time, Guenter Stertenbrink and Kjell Fredrik Pettersen have subsequently developed a method which completes the entire calculation in less than a second!
coloin wrote:It’s been done
Paxmandaddy and kjflp did it
In spades
I’m on holiday in Tenerife can’t find the link !!!!
coloin wrote:Paxmandaddy and kjflp did it
Mathimagics wrote: counts require grid enumeration, of some kind, and the numbers are simply too huge for this to be feasible!
941
492
129 9*9*9*4*4*2*2*1*1 = 46656 [
46656 17972 6121 1848 443 96 24 2 1 PRODUCT = 1.93617E+22
46656 17972 6190 1879 426 96 18 4 1 PRODUCT = 2.87167E+22
46656 17972 6096 1680 392 88 14 2 1 PRODUCT = 8.29440E+21
46656 17972 6021 1784 359 63 4 0 0
16, 9, 4, 1
9 16, 1, 4
4, 1,16, 9
1 4, 9,16
16 16^4*9^4*4^4 = 110075314176
15 15^4*8.5^4*3.5^4 ~ 39656366213 [ it looks like this one would be constant ?]
14 14^4*7.5^4*2.5^4 ~ 4748071289
13 13^4*6.5^4*1.5^4 ~ 258102298
coloin wrote:? isnt VPT = 24^4,24^4,24^4,24^4 * 2 = 24^16 *2 = 24233149581890213117952 = 2e22
16 16^4*9^4*4^4 = 110075314176 1e11
15 15^4*8.5^4*3.5^4 ~ 39656366213 3e10
14 14^4*7.5^4*2.5^4 ~ 4748071289 5e9
13 13^4*6.5^4*1.5^4 ~ 258102298 2e8
12 12^4*5.5^4*1 18974736 2e7 60
11 11^4*4.5^4*1 6003725 6e6
10 10^4*3.5^4*1 1500625 1e6
9 9^4*2.5^4*1 256289 3e5 18
8 8^4*1.5^4*1 20736 2e4
7 7^4*1*1 2401 2e3
6 6^4*1*1 1296 1e3
5 5^4*1*1 625 6e2 24
4 4^4*1*1 256 3e2
3 3^4*1*1 81 8e2
2 2^4*1*1 16 1e2 24
1 1 1
-----
e80 * 6e5 = 6 e 85
red ed wrote:...... I worked out a while back that there are
T(0) = 1 way of laying down no digits at all
T(1) = 46656 ways of laying down all the 1s
T(2) = 838501632 ways of laying down all the 1s,2s
T(3) = 5196557037312 ways of laying down all the 1s,2s,3s
T(4) = 9631742544322560 ways of laying down all the 1s,2s,3s,4s
and then I ran out of patience, though it's obvious that the sequence ends
T(8) = 6670903752021072936960 ways of laying down all the 1s-8s
T(9) = 6670903752021072936960 ways of laying down all the 1s-9s
I was hoping at the time to see a nice rate of decay in the ratio of T(k)/T(k-1)
-- because there is a decreasing amount of freedom for the choice of 9-cell template as you move through the digits
-- but it wasn't looking very tidy so I gave up. My numbers could be wrong, though.
[EDIT: in fact, they were (thanks Condor for spotting this). I've corrected T(3) and probably T(4) now.]
46656
838501632 / 46656 = 17972
5196557037312 / 838501632 = 6197.4
9631742544322560 / 5196557037312 = 1853.5
46656 17972 6121 1848 443 96 24 2 1 PRODUCT = 1.93617E+22
46656 17972 6190 1879 426 96 18 4 1 PRODUCT = 2.87167E+22
46656 17972 6096 1680 392 88 14 2 1 PRODUCT = 8.29440E+21
46656 17972 6021 1784 359 63 4 0 0
I deleted 5 of the templates and the average grid solution count was 698919.2 [160800 - 2872800]
I deleted 4 of the templates and the average grid solution count was 1790.5 [96 - 10944]
I deleted 3 of the templates and the average grid solution count was 25.46 [6 - 264]
I deleted 2 of the templates and the average grid solution count was 2.29 [2,4 or 8]
T(1) = 46656 = 46656
T(2) = 838501632 / 46656 = 17972
T(3) = 5196557037312 / 838501632 = 6197.4
T(4) = 9631742544322560 / 5196557037312 = 1853.5
T(5) = * 3756379592285798400 / 9631742544322560 = * 390 = 698919.2/1790.5 templates can be added
T(6) = * 264073485337691627520 / 3756379592285798400 = * 70.3 = 1790.5/25.46 templates can be added
T(7) = * 2931215687248377065472 / 264073485337691627520 = * 11.1 = 25.46/2.29 templates can be added
T(8) = * 6712483923798783479930 / 2931215687248377065472 = * 2.29 = 2.29/1 templates can be added
T(9) = * 6712483923798783479930 / 6712483923798783479930 = 1
* ~ estimated
46656 * 17972 * 6197.4 * 1853.5 * 698919 = 6731825925247620121267 6.7e21
46656*17972*6197.4*1853.5*390*70.3*11.1*2.29*1 = 6712501929875011822517 6.7e21
+----+----+--
|12..|xxxx|
|....|1**x|
|....|***x|
|....|***x|
+----+----+--
|....|....|
|....|.1..|
|....|...2|
|....|....|
+----+----+-- example of 8 ways to insert a 2 in box 2
| | |