You’ve spotted the 16 in the corner of a Kakuro grid. Two cells, so it’s 7 and 9. You pencil “7 9” into both cells and feel good about it for about four seconds. Then you notice the 17 next to it is “8 9” in both cells too, and the 24 below is “7 8 9” in all three, and the top corner of the grid now looks like a phone keypad.
Knowing the combination is half the job. The other half is knowing which digit goes in which cell, and the clue itself can’t tell you that. The clue running the other way through the same cell can. Every white cell sits in exactly one across entry and one down entry, and the digit in it has to satisfy both. Read the two together and most of those pencil marks never need to be written.
The Power of Opposites
High Meets Low
Every entry has a ceiling and a floor, set by its clue and its length:
- Highest possible digit = clue − (the smallest digits that could fill the other cells)
- Lowest possible digit = clue − (the largest digits that could fill the other cells)
A 14 in four cells has a ceiling of 14 − (1 + 2 + 3) = 8. A 23 in three cells has a floor of 23 − (9 + 8) = 6.
Now cross a high clue with a low one:
17 across (2 cells): {8, 9}
14 down (4 cells): ceiling = 14 − (1 + 2 + 3) = 8
Shared cell: {8, 9} ∩ {1 … 8} = 8
The 9 can’t go in the shared cell because the 14 can’t hold a 9 anywhere. So the shared cell is 8, the 17’s other cell is 9, and the 14 down has also been decided: the only four-cell set for 14 that contains an 8 is {1, 2, 3, 8}. One glance placed two digits and settled a clue with five possible sets.
Push the gap further and the intersection vanishes. A 16 in two cells {7, 9} crossing a 10 in four cells {1, 2, 3, 4} has no digit in common. A valid puzzle will never contain that crossing. If you think you see one, recount the cells, because one of the clues isn’t what you think it is.
Low Meets Low
Two small clues crossing each other work just as fast, because they compete for the same few digits. A 3 in two cells is {1, 2}. A 4 in two cells is {1, 3}. Where they cross, the only digit both allow is 1.
The high end mirrors it: 16 {7, 9} crossing 17 {8, 9} shares only 9. Both appear constantly in grid corners, where an across entry and a down entry start from the same clue cell.
Eliminating Candidates
Most crossings won’t hand you a single digit. They’ll trim a list, and trimming adds up.
The “Too Big” Rule
If a low clue crosses a looser one, the low clue’s ceiling caps the shared cell. Take a 10 in two cells, which could be {1, 9}, {2, 8}, {3, 7}, or {4, 6}. Cross it with a 6 in three cells, which is {1, 2, 3}. The shared cell can’t be bigger than 3, so it takes the small half of a pair (1, 2, or 3) and the 10’s other cell takes the matching big half (9, 8, or 7). The {4, 6} pair is gone, and the other cell has lost every candidate below 7.
The “Too Small” Rule
The reverse is easier to miss. A big digit in a modest entry eats the budget for every other cell in it. If the remaining cells can’t reach their own minimums, the big digit doesn’t fit.
Say a 17 runs down four cells, and the bottom cell can only be 6 or 7. Try a 9 at the top. The middle two cells would need to total 17 − 9 − 6 = 2 at most, and two different digits can’t add up to less than 1 + 2 = 3. So the top cell is not 9. That’s the exact move the practical example below needs.
Common Intersection Cheat Sheet
These crossings show up in almost every grid. Learn the first block and you’ll place digits in corners without thinking.
| Across | Down | Shared cell | Notes |
|---|---|---|---|
| 3 {1,2} | 4 {1,3} | 1 | Forced |
| 4 {1,3} | 7-in-3 {1,2,4} | 1 | Forced |
| 17 {8,9} | 16 {7,9} | 9 | Forced |
| 16 {7,9} | 23-in-3 {6,8,9} | 9 | Forced |
| 3 {1,2} | 6-in-3 {1,2,3} | 1 or 2 | Needs a third clue |
| 4 {1,3} | 6-in-3 {1,2,3} | 1 or 3 | Needs a third clue |
| 17 {8,9} | 24-in-3 {7,8,9} | 8 or 9 | Needs a third clue |
| 16 {7,9} | 24-in-3 {7,8,9} | 7 or 9 | Needs a third clue |
| 4 {1,3} | 16 {7,9} | none | Can’t appear in a valid puzzle |
The table works in both directions. Swap across and down and the answers stay the same.
High Sum vs. Moderate Sum
Moderate clues don’t force digits, but magic blocks still trim them. A 16 {7, 9} crossing a 12 in two cells, which could be {3, 9}, {4, 8}, or {5, 7}:
- 9 in the shared cell works, paired with 3.
- 7 in the shared cell works, paired with 5.
- The {4, 8} pair has nothing to offer the 16, so it’s gone.
The 12’s other cell drops from six candidates to two: {3, 5}.
Practical Example: Forcing a Corner
Here’s a grid with a corner where a 17 across meets a 24 down. Clue cells read down\across.
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######| 24\ | 16\ |#######|#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| 7\17 | | |#######| 17\ | 21\ |
+-------+-------+-------+-------+-------+-------+-------+
| \21 | | | | \16 | | |
+-------+-------+-------+-------+-------+-------+-------+
| \9 | | |#######| 3\8 | | |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| \9 | | | |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| \8 | | |#######|
+-------+-------+-------+-------+-------+-------+-------+
Step 1: Compare the candidate lists. The 17 across is {8, 9}. The 24 down is {7, 8, 9}. The corner cell sits in both, so it’s 8 or 9. The 7 is out.
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######| 24\ | 16\ |#######|#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| 7\17 | 8 9 | 8 9 |#######| 17\ | 21\ |
+-------+-------+-------+-------+-------+-------+-------+
| \21 | | 7 8 9 | 7 9 | \16 | | |
+-------+-------+-------+-------+-------+-------+-------+
| \9 | | 7 8 9 |#######| 3\8 | | |
+-------+-------+-------+-------+-------+-------+-------+
Step 2: Use the neighbor. The 17’s other cell sits on top of the 16 down, which is {7, 9}. {8, 9} against {7, 9} leaves only 9. So the 17 reads 8, 9, and the corner is 8.
Step 3: Let it cascade. The 16 down has 9 on top, so 7 below. The 24 down still needs a 7 and a 9 for its last two cells. The middle cell sits in the 21 across, which already has that 7, so it takes the 9 and the bottom cell takes the 7. The 9 across now needs 2 beside that 7, and the 7 down needs 5 above the 2. Check the 21 across: 5 + 9 + 7 = 21.
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######| 24\ | 16\ |#######|#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| 7\17 | 8 | 9 |#######| 17\ | 21\ |
+-------+-------+-------+-------+-------+-------+-------+
| \21 | 5 | 9 | 7 | \16 | | |
+-------+-------+-------+-------+-------+-------+-------+
| \9 | 2 | 7 |#######| 3\8 | | |
+-------+-------+-------+-------+-------+-------+-------+
Step 4: The right side needs the “too small” rule. The 16 across is {7, 9}. The 3 down is {1, 2}, so the 8 across at the bottom has 7 or 6 beside it. Now look at the 17 running down four cells:
+-------+-------+-------+-------+-------+-------+-------+
| \21 | 5 | 9 | 7 | \16 | 7 9 | 7 9 |
+-------+-------+-------+-------+-------+-------+-------+
| \9 | 2 | 7 |#######| 3\8 | ? | |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| \9 | 1 2 | ? | |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| \8 | 1 2 | 6 7 |#######|
+-------+-------+-------+-------+-------+-------+-------+
If its top cell were 9, the two ? cells would need 17 − 9 − 6 = 2 or less. Impossible. So the top is 7, which makes its neighbor 9. The bottom can no longer be 7, so it’s 6, which puts 2 beside it and 1 above that. The two ? cells now total 17 − 7 − 6 = 4, so they’re {1, 3}. The 9 across already has a 1, so its ? is 3 and its last cell is 5. The upper ? is 1, and the 8 across finishes with 7. The 21 down checks out: 9 + 7 + 5 = 21.
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######| 24\ | 16\ |#######|#######|#######|
+-------+-------+-------+-------+-------+-------+-------+
|#######| 7\17 | 8 | 9 |#######| 17\ | 21\ |
+-------+-------+-------+-------+-------+-------+-------+
| \21 | 5 | 9 | 7 | \16 | 7 | 9 |
+-------+-------+-------+-------+-------+-------+-------+
| \9 | 2 | 7 |#######| 3\8 | 1 | 7 |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| \9 | 1 | 3 | 5 |
+-------+-------+-------+-------+-------+-------+-------+
|#######|#######|#######| \8 | 2 | 6 |#######|
+-------+-------+-------+-------+-------+-------+-------+
No guesses, and the only arithmetic harder than the clues themselves was one subtraction.
Best Practices and Tips
Do:
- ✓ Read the crossing entry before writing any candidates in a cell.
- ✓ Hunt for clashes between a high clue and a low one. Those are the fastest eliminations on the board.
- ✓ Work out the ceiling and floor of a long entry once and write them by the clue.
Don’t:
- ✗ Write out every candidate for a 20 in five cells before checking what its crossings allow. That clue has six possible sets, and most cells will drop to two or three candidates as soon as you look across.
- ✗ Trust a pencil mark that was written before its crossing entry changed. Re-check it.
Troubleshooting Common Issues
A shared cell has no candidates left.
- Cause: Usually a miscounted entry length. A 16 in two cells is {7, 9}. A 16 in three cells has eight possible sets.
- Fix: Recount both entries, then recompute their ceilings and floors.
A candidate you allowed breaks the entry later.
- Cause: You worked out the ceiling from the clue alone and ignored a digit already placed in the entry. If a 4 already sits in a four-cell 14, the other three cells share 10 and can’t use another 4, so their ceiling is 10 − (1 + 2) = 7, not 8.
- Fix: Subtract placed digits first, then work out the ceiling from what’s left.
Two candidates survive every crossing.
- Cause: Intersections alone can’t settle that cell.
- Fix: Move on and come back. If the whole grid stalls, try the 45 rule or the deductions in Eliminating Guesswork.
Conclusion
Key takeaways:
- Every cell answers to two clues. The digit has to fit both.
- High-against-low and low-against-low crossings place digits on sight.
- Ceilings and floors (“too big” and “too small”) trim candidates even when nothing is forced.
- Cross-referencing is faster than listing full combinations, and it leaves fewer pencil marks to misread.
Next steps: Copy the cheat sheet table onto the back of your magic-block card. In your next puzzle, place every corner digit before writing a single pencil mark.
Keep reading in this series:
- Previous: Mastering the 45 Rule in Kakuro
- Next: Eliminating Guesswork in Kakuro: Parity and Logic
- Back to basics: How to Cross-Reference Intersecting Clues