there is absolutely no need for brute force
the (8-cage, 18-cell) White Room puzzle got featured on cracking the cryptic's youtube channel
here. congratulations to mith, you're youtube famous
(how do you know CTC?!) CTC finds a solution by hand in half an hour which is an impressive time, faster than what i did it in
the 7-cage 18-cell version is the same puzzle but rows 2&3 are swapped and the two 2-cell cages that add up to 5 and 6 are now replaced by an 11 cage. CTC actually doesn't use the fact that those two cages add to 5 and 6 until 18:35 into the video, and i can easily adapt his video solution to the 7-cage version of the puzzle. here is a summary below if you don't want to watch a half hour video. note that
rows 2 and 3 are swapped in the 7-cage version below if you want to directly compare it to his video solution
basics: A={124}, B={15} or {24}, C={1235}, D={689}, E={69} or {78}, F={89}, G={12}
r6c3=6, naked pairs 89 in c4, r7, r6, hidden pair 12 in b7
now we use the digit relabelling trick, first discovered by eleven
here and two more examples
here (there are many others, i think space used this a lot before he got banned)
we label 8,9 with a,b without knowing the order
any cell marked with x can't be 6+
- Code: Select all
. . . | . . . | . . .
. x . | . . . | x x .
. x . | . . x | x . .
------+-------+-------
. x . | . . x | x . .
. . . | . . . | . . .
. . 6 | a . . | . . b
------+-------+-------
x . a | b . . | . . 67
x . b | . . . | . . .
. . . | . . x | x . .
singles solve to here
- Code: Select all
. a . | . . . | b . .
. x . | . a b | x x .
b x . | . . x | x a .
------+-------+-------
a x . | . b x | x . .
. b . | . . . | a . .
. . 6 | a . . | . . b
------+-------+-------
x . a | b . . | . . 67
x . b | . . a | . . .
. . . | . . x | x b a
6 in c7 locked in b9 => r7c9=7 => b=8 and a=9
singles solve to here
- Code: Select all
6 9 . | . . . | 8 7 .
7 x . | . 9 8 | x x .
8 x . | . . x | x 9 .
------+-------+------
9 x 7 | . 8 x | x . .
. 8 . | . . 7 | 9 . .
. . 6 | 9 . . | 7 . 8
------+-------+------
x . 9 | 8 . 6 | . . 7
x 7 8 | . . 9 | 6 . .
. 6 . | . . x | x 8 9
cage C must contain 1 and 2, so combined with cage G, no 1 or 2 can appear anywhere else in columns 6 or 7 => r2c7=45, r2c8=12, 12 hidden pair in r6 => r6c58=12, 12 naked pair in r26c8, 12 hidden pair in b9 => r8c9=12
do the relabelling trick again, set y,z=1,2 in either order, do singles
- Code: Select all
6 9 . | . . . | 8 7 .
7 . . | . 9 8 | 45 y .
8 . . | . . . | . 9 .
------+-------+-------
9 . 7 | . 8 . | . . .
. 8 . | . . 7 | 9 . .
. . 6 | 9 y . | 7 z 8
------+-------+-------
y . 9 | 8 z 6 | . . 7
z 7 8 | . . 9 | 6 . y
. 6 . | . . y | z 8 9
this point is where CTC gets to at 18:35 in the video before he starts using the fact that the cage sums are 5,6 in r3,r4. this is where i adapt his solution path to the 7-cage version. there are more singles that he missed
- Code: Select all
6 9 . | y . . | 8 7 .
7 * . | . 9 8 | 45 y .
8 y . | . . . | . 9 .
------+-------+-------
9 z 7 | . 8 . | y . .
. 8 y | z . 7 | 9 . .
. . 6 | 9 y . | 7 z 8
------+-------+-------
y . 9 | 8 z 6 | . . 7
z 7 8 | . . 9 | 6 . y
. 6 . | . . y | z 8 9
y+z+r2c2=7 => r2c2=4 => r2c7=5 => r2c8=y=1 and z=2
cage C must be {1235} so r3c6=2, r4c6=5, r3c7=3
singles to here
- Code: Select all
6 9 . | 1 5 . | 8 7 .
7 4 . | . 9 8 | 5 1 .
8 1 5 | . . 2 | 3 9 .
------+-------+------
9 2 7 | . 8 5 | 1 . 3
. 8 1 | 2 . 7 | 9 . 5
. . 6 | 9 1 . | 7 2 8
------+-------+------
1 . 9 | 8 2 6 | 4 . 7
2 7 8 | . . 9 | 6 . 1
. 6 4 | . . 1 | 2 8 9
xy-wing r4c4,r6c6,r2c4 => r1c6=/=3 singles to the end
no need for brute force solving or anything ridiculous like that
it's one of the nicest applications of eleven's relabelling technique i've seen, and it's great that it can be used in a killer sudoku as well as normal sudokus
thanks to sclt for sending me the youtube link to the puzzle which i otherwise wouldn't have seen
-----------------------
for the original white room with 8 cages, i solved it by hand (without watching the video) with a different method than what CTC did, but my solution is not easily adaptable to the 7-cage version as his is. here is my solution
eliminate the possibility of {14} in the 5-cage. if the 5-cage is {14} then all the lower numbered cages instantly solve like this
- Code: Select all
. . . | . . . | . . .
. 2 . | . . 1 | 4 . .
. 4 . | . . . | 5 1 .
-------+--------+-------
. 1 . | . . 4 | 2 . .
. . . | . . . | . . .
. . 6 | 89 . . | . . 89
-------+--------+-------
12 . 89| 89 . . | . . 67
12 . 89| . . . | . . .
. . . | . . 2 | 1 . .
and you get a contradiction in r6 because the only place for both 1 and 2 is r6c5. so the 5-cage must be {23} and you get to here (r2c7 can't be 2 because of cage C and column 7)
- Code: Select all
. . . | . . . | . . .
. 14 . | . . 2 | 3 . .
. 124 . | . . . | 45 12 .
---------+--------+---------
. 24 . | . . 5 | 1 . .
. . . | . . . | . . .
. . 6 | 89 . . | . . 89
---------+--------+---------
12 . 89| 89 . . | . . 67
12 . 89| . . . | . . .
. . . | . . 1 | 2 . .
hidden singles in r6: r6c5=1 and r6c8=2 which resolves cage C
then do the same relabelling trick on 8,9 to get r7c9=7, r6c9=8 and solve the puzzle like a normal sudoku because all information from cages is represented - only thing needed from here is the same xy-wing
White Room was the name of my local hairdresser's in 2019... thanks to this puzzle for reminding me of that place, maybe i might give them another visit soon