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What goes in the second cell?
Level 5 · Outside clues and specials
How to solve X-sums sudoku
A clue gives the sum of the first X cells, where X is the first digit from that side.
In X-sums sudoku, a clue outside the grid looks along its row or column. The first digit you meet, X, tells you how many cells to add, and the clue is the sum of those X digits.
The first digit counts itself. A 3 at the front means you add the 3 and the next two digits.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Clues outside the grid give the sum of the first X digits, where X is the first digit in that direction.
- X-sums.
Worth memorizing
- The first digit counts itself. A 3 at the front means add three digits including the 3.
- A clue of 1 means the first digit is 1.
Solving tips
A clue of 1
The first digit is part of the sum, so the smallest possible clue is 1, with a 1 at the front. No other first digit can total 1.
45 means a 9
A 9 at the front means you add the whole row, which is always 45. Eight digits can total 44 at most, so a clue of 45 always means a 9 comes first.
Test each first digit
For a clue of 10, the first digit can be 2 (then 8), 3 (then two digits making 7) or 4 (then 1, 2 and 3). A 5 fails, since 5 plus four other digits is at least 15.
Small clues cap the first digit
A first digit of 4 needs three more different digits, so the clue is at least 4 + 1 + 2 + 3 = 10. A clue of 6 means the first digit is 2 or 3.
A worked example: a clue of 7
We name cells by row and column, so row 1, column 1 is r1c1. Row 1 has an X-sums clue of 7 on its left and a clue of 45 on its right.
- Test each possible first digit for the 7. A 1 at the front gives a sum of just 1, which is too small.
- A 2 at the front means you add two digits, the 2 and one more. 2 + 5 = 7 works.
- A 3 at the front means three digits that add to 7, so the other two add to 4. The only pair is 1 + 3, which repeats the 3. So 3 fails. A 4 or higher needs at least 4 + 1 + 2 + 3 = 10, which is too big.
- So r1c1 is 2 and r1c2 is 5.
- Now the right side. A clue of 45 means the first digit from that side is 9, so r1c9 is 9. The row reads 2, 5, six open cells, then 9.
- The 45 says nothing else about the row, since every full row adds to 45. Its only job is placing the 9. The six open cells hold 1, 3, 4, 6, 7 and 8, and the columns will decide their order.
Practice
Reference table
| First digit (X) | Cells added | Smallest clue | Largest clue |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 2 | 3 | 11 |
| 3 | 3 | 6 | 20 |
| 4 | 4 | 10 | 28 |
| 5 | 5 | 15 | 35 |
| 6 | 6 | 21 | 39 |
| 7 | 7 | 28 | 42 |
| 8 | 8 | 36 | 44 |
| 9 | 9 | 45 | 45 |
Going deeper
Clues that fix the first digit
Several clues allow only one first digit. A clue of 1 needs a 1. Clues of 3, 5 and 7 need a 2, and they also fix the second digit as 1, 3 or 5. An 8 is the only first digit for 43 and 44, and 45 needs a 9.
Two clues can't happen at all, 2 and 4. A 2 at the front plus one more digit is at least 3, and it reaches 4 only with a second 2. A 3 at the front already reaches at least 6.
Large clues work from the far end. The first X digits plus the rest of the row add to 45. So with an 8 first, a clue of 44 leaves a 1 in the last cell, and 43 leaves a 2 there.
Clues on both ends of a line
When a row has clues on both sides, compare the two first digits. If they add to 9 or less, the two groups don't overlap, and the cells between them add to 45 minus both clues.
If the first digits add to more than 9, the groups share cells. Add the two clues and subtract 45, and you get the sum of the shared cells. A 6 on the left and a 5 on the right share two cells, since 6 + 5 = 11 and the row has 9.
Clues in the middle of the range
Mid-size clues allow several first digits. A clue of 10 allows 2, 3 or 4 at the front, and a clue of 15 allows 3, 4 or 5. For each option, check the table's smallest and largest clue for that first digit, and strike any option the clue falls outside.
Then look at the line itself. With a 4 at the front, a clue of 10 needs 1, 2 and 3 in the next three cells. If any of those digits is already placed further along the row, the 4 fails. Digits in the edge cell's column can knock out options too, since they limit what the first digit can be.
Write the surviving first digits as center marks in the edge cell. Each time one option drops out, recheck the cells it would have covered.
Common mistakes
Reading from the wrong end
Each clue reads from its own side. A clue on the right of a row starts counting at the rightmost cell, and a clue below a column starts at the bottom.
Fixing the length too early
The number of cells in the sum depends on the first digit. Until you know X, don't mark combinations as if the group had a set length.
Missing what a clue says about the edge
A 1 at the front makes the clue 1. So any clue above 1 means the edge cell isn't 1, and any clue below 45 means it isn't 9.
In mixed-rule puzzles
X-sums pair well with sandwich clues, since both read a line from the edge and both care about 1s and 9s. They also work with thermometers and killer cages that touch the edge. A thermometer bulb on the edge keeps that cell low, and a low first digit keeps the X-sum short. In mixed puzzles, pin down the first digit on each clued edge early, because every other deduction on that line depends on it.
What the video shows
- The first digit says how many cells to add. The clue is their total.
- The first digit is 3, so add the first three digits.
- 3 + 2 + 5 = 10, the clue.
Questions solvers ask
How do X-sums work in sudoku?
Take the first digit from the clue's side and call it X. The clue is the sum of the first X digits in that row or column, counting X itself.
Does the first digit count in X-sums?
Yes. If the first digit is 4, you add it and the next three digits.
What does a clue of 45 mean in X-sums?
The first digit is 9, so the sum covers the whole row or column. A full row always adds to 45.
What is the difference between X-sums and sandwich sudoku?
Sandwich clues add the digits between the 1 and the 9. X-sums clues add the first X digits, where X is the digit nearest the clue.
What is the smallest X-sums clue?
1, from a 1 at the front. The next smallest is 3, from 2 + 1. Clues of 2 and 4 can't happen.
Can an X-sums clue be more than 45?
No. The largest sum is the whole row or column, which happens with a 9 at the front. A full line always adds to 45.
What does an X-sums clue of 44 mean?
The first digit is 8. The first eight cells add to 44, so the last cell of that row or column is 1.
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