The New Sudoku Players' Forum

[eleven]

Can Mr. RW solve the following puzzle step by step in detail? Please donot use the word like "easy to know".
If you can solve it without pencilmark, I beleive millions of player in the world must study your article as textbook.

it is out of question for me, that RW can solve this puzzle without pencilmarks. But why do you think, he should ? As i see it, it is just one of myriads uninteresting puzzles, which require a lot of time to solve manually, with or without pencilmarks. RW has already shown, that he is worth to be read by millions of players.

[Noumenon]

I found a new constraint I impose on myself while solving without pencil marks. I only allow myself to fill in the grid while maintaining rotational symmetry. Here's what I do:

Stage 1 -- If the puzzle is one which contains some asymmetry, I must first only allow myself to bring the puzzle back to a symmetric state one asymmetric cell at a time. For example if there is a clue in r1c2 but not in r9c8, I must find r9c8. (Of course if there's another asymmetry in the puzzle, I have a choice on which to find first.) For example r2c2 and r8c8, or r5c1 and r5c9.
Finally I reach a state where every clue cell has its rotationally symmetric "antipode" filled; it's now a symmetric puzzle.

Stage 2 -- Next, I only allow myself to fill in two cells at a time, they must be proven and antipodes. There is one exception; the center square (which is its own antipode), if unfilled -- r5c5. Note I am not allowed to fill it until stage one is complete.

This makes many of the duller puzzles more fun and reveals much about their construction.

[numpl_npm]

Postby gearss » Sat Dec 14, 2013 11:31 am

Code: Select all

 4 3 . | . . . | . 5 8
 1 . . | . 3 . | . 9 .
 6 7 . | . . 5 | 1 3 .
-------+-------+------
 . 1 . | 2 7 . | 9 8 .
 2 9 7 | . . . | 5 . 1
 8 4 6 | 5 9 1 | 3 2 7
-------+-------+------
 . . 1 | 6 . . | . 7 9
 9 2 . | . 8 . | . . 5
 7 6 . | . . . | . . 3


Code: Select all

 4 3 . | . . . | . 5 8
 1 . . | X 3 . | . 9 .
 6 7 A | A + 5 | 1 3 +
-------+-------+------
 . 1 . | 2 7 . | 9 8 .
 2 9 7 | B - B | 5 - 1
 8 4 6 | 5 9 1 | 3 2 7
-------+-------+------
 D D 1 | 6 E E | C 7 9
 9 2 . | . 8 . | o o 5
 7 6 . | . . . | C o 3


89 not in ++, so AA = 89.
38 not in --, so BB = 38.
28 not in ooo, so CC = 28.
4 not in DD, so 4 in EE and X=4.

[numpl_npm]

Postby gearss » Sat Dec 14, 2013 11:15 am
Puzzle 1

Code: Select all

*-----------*
|.2.|1.8|..5|
|1..|.76|.3.|
|7..|4..|...|
|---+---+---|
|..7|...|2..|
|48.|...|.59|
|..2|...|6..|
|---+---+---|
|...|..9|..1|
|.5.|74.|..6|
|6..|8.1|.7.|
*-----------*


This is a good example of the weakness of the npm method.
However, there is no problem at all.

Code: Select all

 . 2 X | 1 . 8 | 7 E 5
 1 . + | . 7 6 | . 3 .
 7 D + | 4 . . | . . .
-------+-------+-------
 . . 7 | . . . | 2 . B
 4 8 A | . . 7 | . 5 9
 . . 2 | . . . | 6 . 7
-------+-------+-------
 * 7 . | . . 9 | . . 1
 * 5 1 | 7 4 . | . . 6
 6 Y Z | 8 . 1 | . 7 .


++ = 58, ** = 28

Either A=3 or A=6, but in any case, X=4, Y=4, Z=9.
If A=3, then B=3, X=4, Y=4, Z=9.
If A=6, then D=6, E=6, X=4, Y=4, Z=9.

[numpl_npm]

Postby gearss » Fri Dec 20, 2013 5:26 am

Code: Select all

*-----------*
|.59|3.6|427|
|...|..2|..3|
|2..|.9.|51.|
|---+---+---|
|7..|1..|.3.|
|3..|.6.|...|
|.6.|923|7.4|
|---+---+---|
|.87|63.|..2|
|9..|2..|...|
|612|4.9|37.|
*-----------*


The details of step by step solution are as follows.

Code: Select all

 . 5 9 | 3 . 6 | 4 2 7
 . . . | . . 2 | . . 3
 2 . A | . 9 . | 5 1 B
-------|-------|-------
 7 . . | 1 . . | . 3 .
 3 . . | . 6 . | . . .
 . 6 . | 9 2 3 | 7 . 4
-------|-------|-------
 X 8 7 | 6 3 . | . E 2
 9 . C | 2 . . | . D .
 6 1 2 | 4 . 9 | 3 7 .
 

Either A=6 or B=6, but in any case, X=5.
If A=6, then C=3, X=5.
If B=6, then D=6, E=4, X=5.

With basic techniques only, to next figure.

Code: Select all

 D 5 9 | 3 . 6 | 4 2 7
 4 7 . | . . 2 | . 9 3
 2 3 . | 7 9 4 | 5 1 E
-------|-------|-------
 7 . . | 1 4 C | . 3 .
 3 . 4 | . 6 7 | . A .
 . 6 . | 9 2 3 | 7 B 4
-------|-------|-------
 5 8 7 | 6 3 1 | 9 4 2
 9 4 3 | 2 7 . | . 6 .
 6 1 2 | 4 X 9 | 3 7 .
 

Either A=8 or B=8, but in any case, X=8.
If A=8, then C=8, X=8.
If B=8, then D=8, E=8, X=8.

With basic techniques only, the rest can be solved.

[numpl_npm]

by gearss » Sat Dec 14, 2013 11:56 am

Code: Select all

 2 . 5 | 4 . . | 3 6 .
 . . 6 | . . 3 | . 4 2
 8 4 3 | 2 6 . | . 9 .
-------+-------+------
 5 6 . | . 4 2 | 7 1 3
 . 2 . | . 3 1 | 4 5 .
 1 3 4 | . 5 . | 2 8 .
-------+-------+------
 3 . . | . 2 6 | . 7 4
 4 . 2 | 1 . . | 6 3 .
 6 5 . | 3 . 4 | 9 2 .


Code: Select all

 2 . 5 | 4 . . | 3 6 .  Either A=5 or B=5.
 . . 6 | . . 3 | a 4 2  In any case, XY=57, so Z=1.
 8 4 3 | 2 6 X | Z 9 Y
-------+-------+------  If A=5, then a=5, so XY=57.
 5 6 . | . 4 2 | 7 1 3  If B=5, then XY=57.
 . 2 . | . 3 1 | 4 5 .
 1 3 4 | . 5 . | 2 8 .
-------+-------+------
 3 . . | A 2 6 | B 7 4
 4 . 2 | 1 . . | 6 3 .
 6 5 . | 3 . 4 | 9 2 .


Z is determined by X-wing.
However, even if Z is not found,
Another place determined by Z can be found.

Code: Select all

 2 . 5 | 4 . . | 3 6 .  Either A=5 or B=5.
 . . 6 | . . 3 | a 4 2  Then B=8 or B=5, thus X=1.
 8 4 3 | 2 6 . | . 9 . 
-------+-------+------  If A=5, then a=5, B=8.
 5 6 . | . 4 2 | 7 1 3
 . 2 . | . 3 1 | 4 5 .
 1 3 4 | . 5 . | 2 8 .
-------+-------+------
 3 . . | A 2 6 | B 7 4
 4 . 2 | 1 . a | 6 3 .
 6 5 . | 3 . 4 | 9 2 X


[numpl_npm]

Unsolveable #313 ( For details, see the following site. )
http://www.sudokuwiki.org/Weekly_Sudoku.asp?puz=313

With basic techniques, to next figure.

Code: Select all

 9 8 . | 7 . . | 6 . . 
 7 5 . | . . . | . 8 . 
 . . 6 | . . . | . . . 
-------|-------|-------
 . 4 . | . 3 . | . . . 
 . . 7 | 8 . . | . 5 . 
 . . . | . . . | 4 . . 
-------|-------|-------
 . . 9 | 6 . . | . 7 . 
 . 7 8 | 4 5 . | . 6 . 
 . . . | . 7 2 | 8 . 1 
-------|-------|-------
 

Code: Select all

 9 8 . | 7 . . | 6 . .  Either A=4, B=4 or C=4.
 7 5 . | . . . | . 8 .  In any case, X=6.
 . . 6 | . . . | . . . 
-------|-------|------- If A=4, then B=5, X=6.
 . 4 . | . 3 . | . . .  If B=4, then A=5, X=6.
 . . 7 | 8 . . | . 5 . 
 . . . | . . . | 4 . . 
-------|-------|-------
 . . 9 | 6 . . | . 7 . 
 . 7 8 | 4 5 . | . 6 . 
 A X B | . 7 2 | 8 C 1 
-------|-------|-------
 

If C=4, to next figure with basic techniques.

Code: Select all

 9 8 . | 7 . . | 6 . . 
 7 5 . | . . . | . 8 . 
 . . 6 | . . . | . . . 
-------|-------|-------
 . 4 . | . 3 . | . . . 
 . . 7 | 8 . . | . 5 . 
 . . . | . . . | 4 . . 
-------|-------|-------
 4 . 9 | 6 . . | . 7 . 
 . 7 8 | 4 5 . | . 6 . 
 . . . | 9 7 2 | 8 4 1 
-------|-------|-------


Code: Select all

 9 8 . | 7 . . | 6 Y .  Y<>3
 7 5 . | X c c | . 8 .  ab = XY
 a b 6 | . . . | . A . 
-------|-------|------- Because
 . 4 d | e 3 . | . B .  If 3 in ab, then
 . . 7 | 8 . . | . 5 .    X=3,
 . . d | e . . | 4 C .    Let y which is not 3 of ab, then
-------|-------|-------   y in dd and y in ee, so y not in BC, Y=y.
 4 . 9 | 6 . . | . 7 .  If 3 not in ab, then
 . 7 8 | 4 5 . | . 6 .    If X=a, Y=b.
 . . . | 9 7 2 | 8 4 1    If a in ee, Y=a, b in BC, X=b.
-------|-------|-------


Code: Select all

 9 8 . | 7 . . | 6 Y .  Y<>3
 7 5 . | X c c | . 8 .  ab = XY
 a b 6 | . . . | . A . 
-------|-------|------- Either A=3 or C=3.
 . 4 . | . 3 . | . B .  In any case, X=3.
 . . 7 | 8 . . | . 5 . 
 . . . | . . . | 4 C .  If 3 in ab, then X=3.
-------|-------|------- If 3 not in ab, then
 4 . 9 | 6 . . | . 7 .    cc = 6Y, YBC = ab9,
 . 7 8 | 4 5 . | . 6 .    so A=3, X=3.
 . . . | 9 7 2 | 8 4 1 
-------|-------|-------


Code: Select all

 9 8 . | 7 . . | 6 Y .  Y<>3
 7 5 . | 3 . . | . 8 .  ab = 3Y
 a b 6 | . . . | . . . 
-------|-------|------- Then, c=3, d=3, e=5, X=6
 . 4 . | . 3 . | . . . 
 . . 7 | 8 . . | . 5 . 
 . . . | . . . | 4 c . 
-------|-------|-------
 4 . 9 | 6 . . | . 7 . 
 . 7 8 | 4 5 . | . 6 . 
 e X d | 9 7 2 | 8 4 1 
-------|-------|-------


[numpl_npm]

Unsolveable #313 ( For details, see the following site. )
http://www.sudokuwiki.org/Weekly_Sudoku.asp?puz=313

Following the previous post.

Code: Select all

 9 8 . | 7 . . | 6 . .  6 is found.
 7 5 . | . . . | . 8 . 
 . . 6 | . . . | . . . 
-------|-------|-------
 . 4 . | . 3 . | . . . 
 . . 7 | 8 . . | . 5 . 
 . . . | . . . | 4 . . 
-------|-------|-------
 . . 9 | 6 . . | . 7 . 
 . 7 8 | 4 5 . | . 6 . 
 . 6 . | . 7 2 | 8 . 1 
-------|-------|-------
 

Code: Select all

 9 8 . | 7 . . | 6 . .  Either
 7 5 . | . . . | . 8 .    A=4 or A<>4 and
 . . 6 | . . . | . . .    B=3 or B=9.
-------|-------|-------
 . 4 . | . 3 . | . . .  In any case, X=4.
 . . 7 | 8 . . | . 5 . 
 . . . | . . . | 4 . .  If A=4, then X=4.
-------|-------|------- If B=3, then a=9, X=4.
 . . 9 | 6 . . | . 7 X 
 . 7 8 | 4 5 . | . 6 . 
 . 6 A | B 7 2 | 8 a 1 
-------|-------|-------
 

If A<>4 and B=9, to next figure.

Code: Select all

 9 8 . | 7 . . | 6 . .  Z<>4
 7 5 . | . . . | . 8 . 
 . . 6 | . . . | . . . 
-------|-------|-------
 . 4 . | . 3 . | . . . 
 . . 7 | 8 . . | . 5 . 
 . . . | . . . | 4 . . 
-------|-------|-------
 . . 9 | 6 . . | . 7 . 
 . 7 8 | 4 5 . | . 6 . 
 . 6 Z | 9 7 2 | 8 . 1 
-------|-------|-------
 

Code: Select all

 9 8 . | 7 . . | 6 . .  Z<>4
 7 5 . | . . . | . 8 . 
 A B 6 | . . . | . . .  Either 3 in AB or 3 not in AB.
-------|-------|------- In any case, X=4.
 . 4 . | . 3 . | . . . 
 . . 7 | 8 . . | . 5 . 
 . . . | . . . | 4 . . 
-------|-------|-------
 . . 9 | 6 . . | . 7 X 
 . 7 8 | 4 5 . | . 6 . 
 . 6 Z | 9 7 2 | 8 . 1 
-------|-------|-------
 

Code: Select all

 9 8 . | 7 e f | 6 E F  Z<>4
 7 5 . | C D D | . 8 . 
 A B 6 | g H H | . . .  If 3 in AB, then
-------|-------|-------   C=3,
 . 4 b | c 3 . | . d .    Let a which is not 3 of AB,
 . . 7 | 8 . . | . 5 .    a in bb and a in cc, so a not in dd,
 . . b | c . . | 4 d .    E=a, F=3, f=5, DD=6a, HH=89, e=4,
-------|-------|-------   g is not a of 12,
 . . 9 | 6 . . | . 7 X    EddY = ABg9 = 3ag9, Y=3, X=4.
 . 7 8 | 4 5 . | . 6 . 
 . 6 Z | 9 7 2 | 8 Y 1 
-------|-------|-------
 

Code: Select all

 9 8 . | 7 . . | 6 b .  Z<>4
 7 5 . | a D D | . 8 . 
 A B 6 | C . . | . . .  If 3 not in AB, then
-------|-------|-------   ab = AB,
 . 4 . | . 3 . | . c .    C=3, DD =6b, bccY = AB39,
 . . 7 | 8 . . | . 5 .    Y=3, X=4.
 . . . | . . . | 4 c . 
-------|-------|-------
 . . 9 | 6 . . | . 7 X 
 . 7 8 | 4 5 . | . 6 . 
 . 6 Z | 9 7 2 | 8 Y 1 
-------|-------|-------
 

[numpl_npm]

Unsolveable #313 ( For details, see the following site. )
http://www.sudokuwiki.org/Weekly_Sudoku.asp?puz=313

Following the previous post.

Code: Select all

 9 8 . | 7 . . | 6 . .  4 is found.
 7 5 . | . . . | . 8 .  And X=5.
 . . 6 | . . . | . . . 
-------|-------|-------
 . 4 . | . 3 . | . . . 
 . . 7 | 8 . . | . 5 . 
 . . . | . . . | 4 . . 
-------|-------|-------
 . . 9 | 6 . . | X 7 4 
 . 7 8 | 4 5 . | . 6 . 
 . 6 . | . 7 2 | 8 . 1 
-------|-------|-------
 

Code: Select all

 9 8 . | 7 . X | 6 b d  In any case,
 7 5 . | a c c | . 8 .    AB = ab, cc = 6b,
 A B 6 | . . . | . . .    acc = AB6 = 6bd, X=5
-------|-------|-------
 . 4 . | . 3 . | . . . 
 . . 7 | 8 . . | . 5 . 
 . . . | . . . | 4 . . 
-------|-------|-------
 . . 9 | 6 . . | 5 7 4 
 . 7 8 | 4 5 . | . 6 . 
 . 6 . | . 7 2 | 8 . 1 
-------|-------|-------
 

Code: Select all

 9 8 . | 7 . . | 6 b .  If A=4, then b=4 and
 7 5 . | a . . | . 8 .    a=B. Because
 A B 6 | . . . | . . .      If B in 12, then
-------|-------|-------       B in ee, B in cc, B not in dd, a=B.
 . 4 c | d 3 . | . e .      If B =3, then
 . . 7 | 8 . . | . 5 .        B in cc, B not in ee, g=B, a=B.
 . . c | d . . | 4 e . 
-------|-------|------- If A=3, then ab=AB.
 . . 9 | 6 . . | 5 7 4    Because B in 12 and
 . 7 8 | 4 5 . | . 6 .    If a=B, then
 . 6 . | f 7 2 | 8 g 1      f=A, A in cc, A not in ee, b=A.
-------|-------|-------   If a<>B, then
                             B in dd, B in cc, B not in ee,
                             b=B, g=A, a=A.
 

Code: Select all

 9 8 . | 7 . . | 6 b .  If A in 12, then ab=AB.
 7 5 . | a . . | . 8 .  Because
 A B 6 | . . . | . . . 
-------|-------|------- If a=A, then
 . 4 c | d 3 . | . e .    B in dd, B in cc, B not in ee, b=B.
 . . 7 | 8 . . | . 5 . 
 . . c | d . . | 4 e .  If a<>A, then
-------|-------|-------   A in dd, A in cc, A not in ee, b=A,
 . . 9 | 6 . . | 5 7 4    B in ee, B in cc, B not in dd, a=B.
 . 7 8 | 4 5 . | . 6 .   
 . 6 . | f 7 2 | 8 g 1 
-------|-------|-------
 

[numpl_npm]

Unsolveable #313 ( For details, see the following site. )
http://www.sudokuwiki.org/Weekly_Sudoku.asp?puz=313

This puzzle can be solved with one Exocet technique ( See http://www.sudokuwiki.org/Exocet ).

[numpl_npm]

Unsolveable #314 ( For details, see the following site. )
http://www.sudokuwiki.org/Weekly_Sudoku.asp?puz=314

Code: Select all

A   9 8 . | 7 6 . | 5 . . 
B   7 . . | . 5 . | . 9 . 
C   . . . | . . 9 | . . 8 
   -------+-------+------ 
D   5 9 . | . . 6 | . 8 . 
E   . . 4 | . 7 . | . . . 
F   . . . | 3 . . | . . 5 
   -------+-------+*------ 
G   4 6 . | . . 7 | . 5 . 
H   . . . | . 1 . | . . 4 
J   . . . | . . . | 2 . . 

    1 2 3   4 5 6   7 8 9


Postby Cenoman » Sunday 23-Sep-2018

WU#314 has a band exocet (1234)C45, A8, B2
* 4 cannot be a true base digit, as it false in both "mirror nodes" (A79, B13) ...
-4A8, -B2 (4 is also eliminated from the target cells which can contain only true base digits)

Code: Select all

 9 8 . | 7 6 . | 5 Y .  Let y which is not 4 in AA.
 7 X . | . 5 . | . 9 .    If 4 in AA, then X=4, Y=4.
 . . . | A A 9 | . . 8    But y<>123, so 4 not in AB.

 5 9 . | u u 6 | s 8 s  y<>1, y<>3 <-- y in ss and y in tt.
 . . 4 | . 7 . | . . .  y<>2 <-- y in uu and y in vv.
 . . . | 3 . . | . . 5 

 4 6 . | v v 7 | t 5 t 
 . . . | . 1 . | . . 4 
 . . . | . . . | 2 . . 


Code: Select all

 9 8 . | 7 6 . | 5 Y .  Let y which is not 4 in AA,
 7 X . | . 5 . | . 9 .    If 4 in X, then Y=y. So 4 not in X.
 . . . | A A 9 | . . 8    If 4 in Y, then X=y. So 4 not in Y.

 5 9 v | u u 6 | s 8 s  If 4 in X, then ...
 . . 4 | . 7 . | . . .    If y in 13, then y in ss and y in tt.
 . . . | 3 . . | . . 5    If y in 2, then y in ss.

 4 6 v | . . 7 | t 5 t  If 4 in Y, then ...
 . . . | . 1 . | . . 4    If y in 13, then y in vv.
 . . . | . . . | 2 . .    If y in 2, then y in vv.


[numpl_npm]

Unsolveable #314 ( For details, see the following site. )
http://www.sudokuwiki.org/Weekly_Sudoku.asp?puz=314

Code: Select all

A   9 8 . | 7 6 4 | 5 . . 
B   7 . . | . 5 . | . 9 . 
C   . 4 5 | . . 9 | . . 8 
   -------+-------+------ 
D   5 9 . | . . 6 | . 8 . 
E   . . 4 | . 7 . | . . . 
F   . . . | 3 . . | . . 5 
   -------+-------+------ 
G   4 6 . | . . 7 | . 5 . 
H   . . . | . 1 . | . . 4 
J   . . . | . . . | 2 . . 

    1 2 3   4 5 6   7 8 9


Postby Cenoman » Sunday 23-Sep-2018

WU#314 has a band exocet (1234)C45, A8, B2

1. Eliminations from the exocet pattern: ...
** 6 is eliminated from B3 as non-base digit in the second cell of a mirror node
(first cell already locked with a non-base digit 7B1) => -6B3

Code: Select all

 9 8 . | 7 6 4 | 5 Y y  If X=a, then Y=b, y=a, x=b.
 7 X x | . 5 . | . 9 . 
 . 4 . | a b 9 | . . 8  If Y=a, then
-------+-------+------    either b not in ss or b not in tt,
 5 9 u | . . 6 | s 8 s    so b in uu, and X=b, x=a, y=b.
 . . 4 | . 7 . | . . . 
 . . . | 3 . . | . . 5  And either X=a or X=b.
-------+-------+------  Xx=ab=Yy.
 4 6 u | . . 7 | t 5 t 
 . . . | . 1 . | . . 4 
 . . . | . . . | 2 . . 


[Madolite]

Pencil Marks are nothing more than a helpful tool if you have difficulties creating pencil marks in your mind's eye and remembering them all. I can imagine that people with photographic memory will never need a single pencil mark, no matter how hard a puzzle is. I don't have photographic memory, but I can memorize quite a few numbers, so I rarely, if ever, need pencil marks on Very Easy thru Medium puzzles. Some easier Hard puzzles may not need too many pencil mark, but that's pushing it for me. I'm exceptionally good at abstract visualization, which is possibly why I suck horribly at visualizing the concrete (cause my brain has specialized in top-down thinking, rather than bottom up).

I'm just conjecturing here, but it might be some food for thought, idk.

[StrmCkr]

Pencil Marks are nothing more than a helpful tool if you have difficulties creating pencil marks in your mind's eye and remembering them all. I can imagine that people with photographic memory will never need a single pencil mark, no matter how hard a puzzle is. I don't have photographic memory, but I can memorize quite a few numbers, so I rarely, if ever, need pencil marks on Very Easy thru Medium puzzles. Some easier Hard puzzles may not need too many pencil mark, but that's pushing it for me. I'm exceptionally good at abstract visualization, which is possibly why I suck horribly at visualizing the concrete (cause my brain has specialized in top-down thinking, rather than bottom up).

I'm just conjecturing here, but it might be some food for thought, idk.

i have a photo memory and there is a point where marking pencil marks is absolutely required to even make head way on puzzles in 8~9+ ER rating, there is simply too much data considering the 27 sectors and 729 constraints that need satisfied on the extreme end of the solving world and most of them require following massive chains of multi-digits on the grid to prove 1 elimination.

take this current hardest puzzle - try solving it with out using a guess and test method ps if you do succumb to trial and error this is a back door size 3 puzzle which requires 3 different clues added to the puzzle at the same time to reach a single solution. :ie massive sub sub nets per elimination derived

gl

Code: Select all

 . . . | . . . | . 3 9
 . . . | . . 1 | . . 5
 . . 3 | . 5 . | 8 . .
-------+-------+------
 . . 8 | . 9 . | . . 6
 . 7 . | . . 2 | . . .
 1 . . | 4 . . | . . .
-------+-------+------
 . . 9 | . 8 . | . 5 .
 . 2 . | . . . | 6 . .
 4 . . | 7 . . | . . . 

[eleven]

Hm, this thread was sureley not intended to show how puzzles can be solved without pencilmarks, which most machine solvers can't do without "guessing" (backtracking).
If you solved it with pencilmarks on paper, i don't think, that this is the place to announce that.

@Madolite: Yes, it depends strongly on personal skills and favors, what is possible without pencilmarks. But of course it can be trained and RW gave great hints.
E.g. numpl_npm has his own method, and shows astonishing results (at least if they were achieved without the help of programs).