KAKURO
Neal Christensen
KAKURO
KAKURO
RULES
1. Fill in each cell
with a digit from 1
to 9
2. The sum of the
digits in any entry
must be equal to
the clue at the left
(for rows) or top
(for columns) of the
entry.
3. No digits may be
duplicated in an
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Kakuro’s can be any size. Most that I do are about the size
shown here. (14 x 14)
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IMPORTANT KNOWLEDGE
1. Any group of cells has a minimum and maximum
possible value, and these values have just one set of
digits.
For instance, a group of three cells cannot contain
a value less than 6 (or more than 24) which
correspond to the three smallest digits 1+2+3 = 6
(or largest digits 7+8+9 = 24)
2. The values minimum+1 and maximum-1 likewise
only have one solution: 7 = 1+2+4 (and 23 = 6+8+9)
(You might notice that all the smallest values
correspond to the triangle numbers 3, 6, 10, 15, 21,
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No. of Cells
Minimum
Min + 1
Maximum
Max - 1
2
3 = 1+2
4 = 1+3
17 = 8+9
16 = 7+9
3
6 = 1+2+3
7 = 1+2+4
24 = 7+8+9
23 = 6+8+9
4
10 = 1+2+3+4
11 = 1+2+3+5
30 = 6+7+8+9
29=5+∑7…9
5
15 = ∑1…5
16=∑1…4 + 6
35 = ∑5…9
34=4+∑6…9
6
21 = ∑1…6
22=∑1…5 + 7
39 = ∑4…9
38=3+∑5…9
7
28 = ∑1…7
29=∑1…6 + 8
42 = ∑3…9
41=2+∑4…9
8
36 = ∑1…8
37=∑1…7 + 9
44 = ∑1…9
43=1+∑3…9
9
45 = ∑1…9
45 = ∑1…9
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We will solve this simple puzzle
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Find cells that can have only one possible value
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Find cells that can have only one possible value
At the intersection of a
row and column
1. If the largest possible
digit in one of the
numbers equals the
smallest possible digit
in the other number,
enter it in the
intersecting cell
2. If the two sets of digits
have only one digit in
common, enter it in
the intersecting cell.
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Find cells that can have only one possible value
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Fill in cells which are the only unknown digit in a number
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ill in cells which are the only unknown digit in a number
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More cells which can only have one value
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More cells which can only have one value
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More cells which can only have one value
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More cells which can only have one value
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Enter ‘possibilities’ into cells
When the possibilities for cells is small
enough (my limit is 4, your mileage may
differ) enter them into the cells. [These
are sometimes called ‘pencil marks’.]
You may find that only one set of
possibilities gives you an acceptable
answer
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More cells which can only have one value
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More cells which can only have one value
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More cells which can only have one value
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More cells which can only have one value
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23=6+8+9; The largest of four digits adding to 12 is 6
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23=6+8+9; The largest of four digits adding to 12 is 6
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23=6+8+9; 29=5+7+8+9
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23=6+8+9; 29=5+7+8+9
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You cannot have a digit repeated, e.g. 11=1+5+5
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You cannot have a digit repeated, e.g. 11=1+5+5
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You cannot have a digit repeated, e.g. 15=1+1+2+3+8
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You cannot have a digit repeated, e.g. 15=1+1+2+3+8
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Insert ‘pencil marks’ (possibilities) wherever possible
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Insert ‘pencil marks’ (possibilities) wherever possible
In this simple puzzle the
possibilities I have entered
correspond to the only
possible entries.
In more difficult puzzles an
entire range of possibilities
may be entered, some of
which will not be used.
For instance, at this point,
the lower left row totaling
to 15 can only contain the
digits 1, 2, 3, 4, 5. But
earlier in the puzzle, in the
upper right corner, the
column totaling 12 could
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The highlighted entry can’t contain 4, since 4 is not in 11
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It is trivial to complete this quadrant of the puzzle
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1 = 1+2+3+5; 22 = 5+8+9 or 6+7+9. Only 5 is in common
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1 = 1+2+3+5; 22 = 5+8+9 or 6+7+9. Only 5 is in common
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The smallest digit in 12 is 3, the largest left in 11 is 3
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It is trivial to complete from here.
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The Completed Puzzle
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Other Solving Techniques
Filled areas:
In this example, we can deduce
that the square outside of the
blank 2x2 area shown must be
a 3, we can do this using the
filled area technique.
Firstly, add together all the
"across" clues (4+3=7).Then
add together all the "down"
clues (4+6=10). Now work out
the difference between those
totals (10-7=3) and that will be
the value of the square which is
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Other Solving Techniques
Naked doubles and triples etc.
As in Soduko, some of the cells for an entry
may contain naked doubles etc. When you
notice this, those digits can’t appear in any of
the other cells of the entry.
In Kakuro, you can have ‘naked or hidden
singles’. When a digit that you know is part of
the answer appears in only one of the cells,
perhaps among other possibilities in that cell, it
must be the entry for that cell.
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Other Solving Techniques
Sometimes,
when you have a
number with n
cells and all cells
have known
possibilies, the
minimum (or
maximum)
possible sum
equals the clue.
In this case you
can enter every
digit of the
number.
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Other Solving Techniques
Suppose you have an entry of n cells, and you
have possibilities in n-1 of the cells.
You can find the range of possibilities in the
empty cell. The minimum possible value is the
clue minus the sum of maximum values in all
the other cells. Likewise, the maximum value
is the clue minus the sum of of minimum
values in the other cells.
Sometimes this results in just one value, or the
allowed values dictated by the intersecting row
or column with the empty square will restrict
the digit to a single value.
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Kakuro Web Sites
http://comicskingdom.com/conceptis-kakuro
http://www.kakuro.com/index.php#daily
http://www.kakuro-online.com/
SUMDOKU
SUMDOKU
A sumdoku puzzle is a standard sudoku puzzle
overlaid with with a ken-ken puzzle
1. Standard Sudoku rules apply. (note, however, that
you may be given a lot fewer clues. In fact a
sumdoku puzzle may be completely empty of
clues.
2. Standard Ken-Ken rules apply Except digits may
not be repeated within a cage.
3. Only addition is used for the cage sums
http://puzzles.comicskingdom.com/games/sudokusum/