# Stage-1 Suduro Trick #1 Restricting Sums

The most important trick to solving the first stage of Suduro puzzles is to recognize that the each gray cell belongs to three restricted sets, the set of gray cells that share its column, the set of gray cells that share its row, and the set of gray squares that share its sub-grid. Each of the three sum clues that apply to each cell potentially restricts the set of numbers that might work in that cell, and as you solve for some of the cells, you gain further information, as you rule out numbers in the columns, rows, and sub-grids, and as you solve for portions of the sums. For example, consider the gray cell (with an asterisk) at the center of the following partial puzzle as example:

 ? ? ? 5 1 D a n T o w ? 10 ? 50 b u i l t m e ? ? ? ? ? ? ? 7 ? ? ? ? ? ? ? ? 14 * ? * ? ? ?

The number in the center square simultaneously belongs to three sets, the set of numbers that can add up to 14, in a two-cell sum (for the row clue), the set of numbers that can add up to 7 in a two-cell sum (for the column clue), and the set of numbers that can add up to 10 in a two-cell sum (for the sub-grid clue). Since the other cell in the row cannot be larger than 9, 5 is the smallest number consistent with the row sum of 14. Since the other cell in the column cannot be less than 1, 6 is the largest number consistent with the column sum of 7. So far, the row and column clues taken together restrict possible values for the center square to just the numbers 5 and 6. Finally, taking the sub-grid sum of 10, we know that neither gray cell in the center sub-grid can be a 5 because if one was a 5, the other would also have to be a 5 to make the center sub-grid gray cells sum to 10, but repeats are not allowed. Therefore, taking all three sum clues for the center cell, together, the only possible value for the center cell remaining is 6, and you could confidently fill in that square with no guessing the overlap of the three sets that the center cell must belong to contains just a single number.

This is very similar to the logic you probably already use for row and column clues in Kakuro, if you are an experienced Kakuro solver, but you must remember that with Suduro, each cell has three clues, thanks to the sub-grid sums, and you may find the necessary deductions with any combination of these three clues, with roughly equal likelihood. (I considered using just row and column clues for Suduro, when creating the rules for Suduro, to more closely mimic Kakuro, but there are two reasons I added sub-grid clues:

• Treating sub-grids just like rows and columns (having sum clues for sub-grids, too) is much more in the spirit of Sudoku, where there is a deep symmetry between rows, columns, and sub-grids.
• As it turns out, it is generally too hard to narrow the final answer to just one solution if you restrict only the sums of the rows and columns we really need those extra sub-grid clues to reach an answer to a complete Suduro, just as it is very difficult to construct a valid and fun Sudoku with a single solution without at least around 25 beginning numbers.)

Where rows, columns and sub-grids often contain more than 2 gray cells (3 is roughly average, in Suduro, at the beginning of the solution), it requires quite a lot of mental arithmetic to work out the allowed values, for each of the three sets, for each of the gray cells in the puzzle, until you get used to the recurring patterns. (The same is true for solving Kakuros.) Some people find it helpful to refer to tables of required and allowed values for potential sums. Here are the tables that work out these restrictions for you:

 2-Cell Sums Sum Restrictions 3 1,2 4 1,3 5 ≤4 6 ≤5,≠3 7 ≤6 8 ≤7,≠4 9 ≤8 10 ≠5 11 ≥2 12 ≥3,≠6 13 ≥4 14 ≥5,≠7 15 ≥6 16 7,9 17 8,9
 3-Cell sums Sum Restrictions 6 1,2,3 7 1,2,4 8 1,≤5 9 ≤6 10 ≤7 11 ≤8 12-18 any 19 ≥2 20 ≥3 21 ≥4 22 9,≥5 23 6,8,9 24 7,8,9
 4-Cell Sums Sum Restrictions 10 1,2,3,4 11 1,2,3,5 12 1,2,≤6 13 1,≤7 14 ≤8 15-25 any 26 ≥2 27 9,≥3 28 8,9,≥4 29 5,7,8,9 30 6,7,8,9
 5-Cell Sums Sum Restrictions 15 1,2,3,4,5 16 1,2,3,4,6 17 1,2,3,≤7 18 1,2,≤8 19 1,any 20-30 any 31 9,any 32 8,9,≥2 33 7,8,9,≥3 34 4,6,7,8,9 35 5,6,7,8,9

 6-Cell Sums Sum Restrictions 21 1,2,3,4,5,6 22 1,2,3,4,5,7 23 1,2,3,4,≤8 24 1,2,3,any 25 1,2,any 26 1,any 27-33 any 34 9,any 35 8,9,any 36 7,8,9,any 37 6,7,8,9,≥2 38 3,5,6,7,8,9 39 4,5,6,7,8,9
 7-Cell Sums Sum Restrictions 28 1,2,3,4,5,6,7 29 1,2,3,4,5,6,8 30 1,2,3,4,5,any 31 1,2,3,4,7,any 32 1,2,3,any 33 1,2,6,any 34 1,any 35 ≠5 36 9,any 37 4,8,9,any 38 7,8,9,any 39 3,6,7,8,9,any 40 5,6,7,8,9,any 41 2,4,5,6,7,8,9 42 3,4,5,6,7,8,9
 8-Cell Sums Sum Restrictions 36 All except 9 37 All except 8 38 All except 7 39 All except 6 40 All except 5 41 All except 4 42 All except 3 43 All except 2 44 All except 1
 9-Cell Sum Sum Restrictions 45 All

Very likely these tables look confusing at first, so let me walk through them and explain what they say. Numbers in bold and underlined are required for that sum. Therefore, for example, 1 and 3 are both absolutely required for a legal sum of two numbers equaling 4, from the second row of the first table. (2 and 2 are not allowed because duplicates are not allowed.) Plain-text restrictions (not underlined, not in bold text) are further restrictions that apply to all the numbers in the sum. For example, any number participating in a two-number sum of 5 must be less than or equal to 4 (≤4), according to the third row in the first table, and all numbers in a two-number sum of 10 must not be equal to 5 (≠5) (else youd need duplicates). The restriction any simply means that any number can participate in that sum. Where you have both bold and underlined restrictions, and other restrictions as well, we know some numbers precisely, but not all. For example, the three-cell sum of 8 (from the third row of the second table) must contain a 1, but all we know about the other two numbers in the sum is that they are ≤5 (and leave a sum of 8, of course).

Even understanding how to read these tables, they may still seem intimidating, but there are tricks to get past this. First, youll almost never find 6 or more cells in a sum in Suduro, so you can pretty well ignore the second row of tables. (These are useful in Kakuro, though, where such many-cell sums come up more often.) Also, you can learn to quickly derive the most useful restrictions in your head if you remember just a few facts and rules:

1. The lowest two and highest two sums for any number of cells is completely restricted. The lowest and highest sums come from counting up from 1 or down from 9. The second-lowest and second-highest sums are just the same as the lowest and highest, except that you skip one when you reach the last number.
2. Knowing the highest and lowest sums for the first few cell counts is very useful see the table below.
3. When working out the highest or lowest a number can be, use this trick: For low sums (nearer the lowest value possible), imagine that the sum for all cells except one is the lowest value possible, and find the value that completes the sum this is the highest possible value. For high sums (nearer the highest value possible), imagine that the sum for all cells except one is the highest value possible, and find the value that completes the sum this is the lowest possible value. For example, a 3-cell sum of 21 is near the highest possible (24). The highest possible sum of two of those cells (see the table, below) is 17, so 4 (21-17) is the lowest possible number you can find in a 3-cell sum of 21. Each of the three cells must be ≥4, just as the 3-Cell sums indicates for the sum of 21, but all you really need to remember to deduce this is that the highest sum for 2 cells is 17, 4 less than 21.

Here is the short table of highest and lowest possible sums that makes the above rules easy to apply remember these 8 numbers, and then you can work out the rest of what you need quickly in your head:

 # of Cells Lowest Sum Highest Sum 2 3 17 3 6 24 4 10 30 5 15 35

So far, Ive shown how to reduce the possible values for certain cells to just a single value at the very beginning of solving stage 1, before you have found any other values, and this is important, because getting started is sometimes the hardest part. Once you fill in a few numbers, though, the sum clues become even more useful. For example, consider the following partly solved partial puzzle:

 ? ? ? 5 1 D a n T o w 8 ? ? 50 b u i l t m e ? ? ? 6 ? ? ? ? ? ? ? ? ? ? ? ? 3 b ? a ? * ? ? ?

Using Trick 2 (see here), we find the 3 in the cell shown. Previously, based on the column clue shown, we knew that cell a was 1, 2, or 3, and based on the sub-grid clue shown, we knew that all gray cells in that sub-grid were ≤5, and that at least one of the cells was a 1. Once we fill in the 3, however, we can find the exact values of a and b fairly easily. One approach is to recognize that the two unknown cells must sum to 5 (8-3), and that since one of these cells must be a 1, the other must be a 4. Further, since a can be only a 1, 2, or 3 (based on the column clue), a must be 1, and b must be 4. Even if we forgot that one of the cells in that sub-grid must be a 1, we could quickly deduce the fact, since a+b=5, leaving just the combinations (1,4) and (2,3), but they cannot be 2 and 3, since we already have a 3 in the sub-grid. (Quickly finding the pairs of numbers that add to something is easy: for numbers less than 11, start with 1 and another number (for example, (1,8) for the sum of 9), and work your way inward (for example, for the sum of 9, find the pairs (1,8), (2,7), (3,6), (4,5)). For sums greater than 10, start with a pair using 9 and work your way inward (for example, for the sum of 12, find the pairs (3,9), (4,8), (5,7)).

Sometimes, you have to follow a chain of reasoning to deduce the value in a single square at the beginning of a puzzle. For example, below, note that a must be 8 or 9, since the two gray squares in that row sum to 17. This means, in turn, that b must be 1 or 2, since a+b=10, according to the center sub-grid clue. Therefore, c must be 5 or 6 to satisfy the row clue, b+c=7, but c may not be 6, since c+d=12, and cd, which just leaves c=5. (This also allows us quickly to work out that d=7, b=2, a=8, among others.)

 ? ? ? 5 1 D a n T o w ? 10 ? 50 b u i l t m e ? ? ? ? ? ? ? ? ? ? 12 ? ? ? ? ? 17 a 7 b c ? d ? ?