A fresh Kakuro grid sits in front of you, every white cell empty. Your eye lands on a 27 running across five cells, so you start there. 9, 8, 6, 3, 1? 9, 7, 5, 4, 2? 8, 7, 6, 4, 2? All valid. There are eleven different digit sets that add to 27 in five cells, and nothing yet tells you which one is right. You pencil in a few, erase them, and start to wonder whether everyone else is just better at math.
They aren’t. They started somewhere else. In the corner of that same grid sits a little 4 over two cells, and that clue has exactly one answer. Kakuro solvers call these unique sums, and you should find every one of them before you touch anything else.
Understanding Unique Sums
The Math Behind the Magic
Two rules create unique sums: you can only use digits 1 through 9, and no digit can repeat within an entry. Together, those rules squeeze the extreme ends of the number range.
Take two cells. The smallest sum you can make is 1 + 2 = 3. The largest is 9 + 8 = 17. Near those edges you have almost no room to move. A clue of 3 has only one candidate pair. So does 17. Move one step toward the middle and the options open up. A 5 can be 1 + 4 or 2 + 3. A 10 can be 1 + 9, 2 + 8, 3 + 7, or 4 + 6.
That’s the whole idea. Clues near the bottom or top of what an entry can hold have one valid set of digits. Clues in the middle have several. Start at the edges, and you start with facts instead of possibilities.
Conceptis Puzzles calls an entry like this a “magic block”: a run of cells whose digits are fixed the moment you know its length and its sum. The name fits. Each one hands you its digits for free.
The Lows: 3 and 4
3 in two cells = {1, 2}. List every pair of digits that adds to 3: 1 + 2. That’s the only one. There’s no 0, so nothing else fits.
4 in two cells = {1, 3}. List every pair that adds to 4: 1 + 3 and 2 + 2. The no-repeat rule kills 2 + 2, which leaves 1 + 3.
+-------+-------+-------+ +-------+-------+-------+
| \3 | 1 2 | 1 2 | | \4 | 1 3 | 1 3 |
+-------+-------+-------+ +-------+-------+-------+
You know both digits the moment you see the clue. You don’t know which cell holds which digit, so both cells get both pencil marks.
The Highs: 16 and 17
17 in two cells = {8, 9}. The only pair that reaches 17 is 9 + 8. Nothing else gets that high.
16 in two cells = {7, 9}. The candidates are 9 + 7 and 8 + 8. The repeat rule kills 8 + 8, so it’s 7 and 9.
+-------+-------+-------+ +-------+-------+-------+
| \16 | 7 9 | 7 9 | | \17 | 8 9 | 8 9 |
+-------+-------+-------+ +-------+-------+-------+
The Mirror Principle
Mapping the low anchors to their high twins:
graph LR
A[Low Anchor: 3] -->|"{1, 2}"| C{Mirror Formula: 10 - X}
C -->|"{9, 8}"| D[High Anchor: 17]
B[Low Anchor: 4] -->|"{1, 3}"| C
C -->|"{9, 7}"| E[High Anchor: 16]
Notice the mirror. The high pairs are the low pairs flipped around 10: 1 becomes 9, 2 becomes 8, 3 becomes 7. So {1,2} for 3 mirrors to {9,8} for 17, and {1,3} for 4 mirrors to {9,7} for 16. Memorize two pairs and you get the other two free.
The same pattern holds for longer entries. Three cells have their own four unique sums (6, 7, 23, and 24), and so do four cells (10, 11, 29, and 30). Start with the two-cell set. Two-cell entries are everywhere on beginner grids, and four pairs are easy to hold in your head.
Practical Example: Finding the Anchors
Here’s a full beginner grid. Clue cells read down\across: 3\ is a down clue of 3, \4 is an across clue of 4, and 24\16 holds a down 24 and an across 16.
+-------+-------+-------+-------+-------+-------+
|#######| 3\ | 25\ |#######| 24\ | 17\ |
+-------+-------+-------+-------+-------+-------+
| \4 | | | 24\16 | | |
+-------+-------+-------+-------+-------+-------+
| \25 | | | | | |
+-------+-------+-------+-------+-------+-------+
|#######| 16\19 | | | | 3\ |
+-------+-------+-------+-------+-------+-------+
| \27 | | | | | |
+-------+-------+-------+-------+-------+-------+
| \17 | | | \4 | | |
+-------+-------+-------+-------+-------+-------+
Fourteen clues. Don’t do any arithmetic yet. Just scan.
Step 1: Scan for 3s and 4s over two cells. You’ll find four: the across 4 in the top-left, the down 3 above the first column, the across 4 in the bottom row, and the down 3 on the right edge.
Step 2: Pencil in their pairs. {1,2} for each 3, {1,3} for each 4.
Step 3: Scan for 16s and 17s over two cells. Four more: the across 16 and down 17 in the top-right, and the across 17 and down 16 in the bottom-left.
Step 4: Pencil in their pairs. {7,9} for each 16, {8,9} for each 17.
Bonus scan: three-cell anchors. The 24\16 cell’s down clue covers three cells. 24 in three cells is unique: {7,8,9}.
Mark the four corner cells with a star for now. Each one sits inside two anchors at once, and you’ll come back to them in a moment.
+-------+-------+-------+-------+-------+-------+
|#######| 3\ | 25\ |#######| 24\ | 17\ |
+-------+-------+-------+-------+-------+-------+
| \4 | * | 1 3 | 24\16 | 7 9 | * |
+-------+-------+-------+-------+-------+-------+
| \25 | 1 2 | | 7 8 9 | | 8 9 |
+-------+-------+-------+-------+-------+-------+
|#######| 16\19 | | 7 8 9 | | 3\ |
+-------+-------+-------+-------+-------+-------+
| \27 | 7 9 | | 7 8 9 | | 1 2 |
+-------+-------+-------+-------+-------+-------+
| \17 | * | 8 9 | \4 | 1 3 | * |
+-------+-------+-------+-------+-------+-------+
Nine anchors, fifteen cells narrowed down, and you haven’t made a single decision. That’s the foothold.
The starred cells are where it pays off. Take the top-left one. The across 4 says it’s 1 or 3. The down 3 says it’s 1 or 2. Only 1 satisfies both, so it’s a 1. The rest of each pair follows: the across 4 needs a 3 beside it, and the down 3 needs a 2 below it.
The other three corners work the same way:
- Top-right: across 16 {7,9} meets down 17 {8,9}. The only shared digit is 9.
- Bottom-left: across 17 {8,9} meets down 16 {7,9}. Again, 9.
- Bottom-right: across 4 {1,3} meets down 3 {1,2}. It’s 1.
+-------+-------+-------+-------+-------+-------+
|#######| 3\ | 25\ |#######| 24\ | 17\ |
+-------+-------+-------+-------+-------+-------+
| \4 | 1 | 3 | 24\16 | 7 | 9 |
+-------+-------+-------+-------+-------+-------+
| \25 | 2 | | 7 8 9 | | 8 |
+-------+-------+-------+-------+-------+-------+
|#######| 16\19 | | 7 8 9 | | 3\ |
+-------+-------+-------+-------+-------+-------+
| \27 | 7 | | 7 8 9 | | 2 |
+-------+-------+-------+-------+-------+-------+
| \17 | 9 | 8 | \4 | 3 | 1 |
+-------+-------+-------+-------+-------+-------+
Twelve cells solved, with nothing harder than “which digit appears in both lists?” That move has a name, cross-referencing, and it gets its own article next in this series.
The middle nine cells take more work, but the corners give you plenty to start from. Row 2 already has a 2 and an 8, so its middle three cells must sum to 25 − 10 = 15. Try finishing the grid yourself. The answer key is at the end of this article.
Do’s and Don’ts
Do:
- ✓ Memorize 3 = {1,2}, 4 = {1,3}, 16 = {7,9}, and 17 = {8,9} today. It takes five minutes and pays off on every grid you’ll ever solve.
- ✓ Pencil both digits into both cells when you don’t know the order yet.
- ✓ Run the full anchor scan before you commit to any digit.
Don’t:
- ✗ Assume you know the order. Knowing an entry is {1,3} doesn’t tell you which cell holds the 1. The crossing clue decides that.
- ✗ Mistake near-misses for unique sums. A 5 in two cells looks small enough, but it has two answers: {1,4} and {2,3}.
Troubleshooting Common Issues
“I found a 4, but it crosses a 20 across four cells. Neither 1 nor 3 helps.” You’ve hit a wide-open intersection. A 20 in four cells has twelve possible digit sets, and plenty of them contain a 1 or a 3, so the crossing tells you nothing yet. Leave the {1,3} pencil marks in place and move to a different anchor. As you fill in other cells along that 20, its options shrink, and eventually one of them will force the 1 or the 3 into place.
“I memorized the pairs but keep mixing up 16 and 17.” Use the mirror. 17 is the flip of 3 ({1,2} → {9,8}), and 16 is the flip of 4 ({1,3} → {9,7}). Or check the arithmetic directly: if your pair for 16 includes an 8, the other digit would have to be another 8, which is illegal.
Conclusion
Unique sums turn a wall of clues into a handful of known starting points. They’re the reason experienced solvers never start in the middle of the grid.
Key takeaways:
- A unique sum has only one valid set of digits for its entry length.
- For two-cell entries: 3 = {1,2}, 4 = {1,3}, 16 = {7,9}, 17 = {8,9}.
- These anchors are where every solve starts, and where they cross each other, you often get a definite digit right away.
Next steps: Knowing the digits is half the job. To learn which cell gets which digit, especially away from the corners, read the next article in the series.
Keep reading in this series:
- Next: How to Cross-Reference Intersecting Kakuro Clues
- Previous: Understanding Kakuro Rules and Grid Mechanics
Answer Key
+-------+-------+-------+-------+-------+-------+
|#######| 3\ | 25\ |#######| 24\ | 17\ |
+-------+-------+-------+-------+-------+-------+
| \4 | 1 | 3 | 24\16 | 7 | 9 |
+-------+-------+-------+-------+-------+-------+
| \25 | 2 | 1 | 9 | 5 | 8 |
+-------+-------+-------+-------+-------+-------+
|#######| 16\19 | 4 | 7 | 8 | 3\ |
+-------+-------+-------+-------+-------+-------+
| \27 | 7 | 9 | 8 | 1 | 2 |
+-------+-------+-------+-------+-------+-------+
| \17 | 9 | 8 | \4 | 3 | 1 |
+-------+-------+-------+-------+-------+-------+