Unique combinations: sums with one answer
Some Kakuro clues can only be made one way. Two cells that make 16 are 7 and 9, every time. Learn the short list and every grid opens up.
How it works
A run holds different digits. That rule does most of the work. The smallest total two cells can make is 1 + 2 = 3, and the biggest is 8 + 9 = 17.
Right at those edges there is no room to swap anything. 3 in two cells is 1 + 2. One step in, 4 is 1 + 3, because 2 + 2 repeats. The same goes at the top: 17 is 8 + 9, and 16 is 7 + 9. These are the unique combinations.
Every length has four of them: the two smallest totals and the two biggest. For three cells they are 6, 7, 23 and 24.
The table
Two cells: 3 = 1 + 2, 4 = 1 + 3, 16 = 7 + 9, 17 = 8 + 9.
Three cells: 6 = 1 + 2 + 3, 7 = 1 + 2 + 4, 23 = 6 + 8 + 9, 24 = 7 + 8 + 9.
Four cells: 10 = 1 + 2 + 3 + 4, 11 = 1 + 2 + 3 + 5, 29 = 5 + 7 + 8 + 9, 30 = 6 + 7 + 8 + 9. Five cells: 15, 16, 34 and 35.
Why it is a proof
List every set of different digits with the right length and total. If the list has one entry, the run uses exactly those digits. That is elimination over sets, and the rule that runs never repeat does the eliminating.
A two-cell 10 can be 1 + 9, 2 + 8, 3 + 7 or 4 + 6, but never 5 + 5. So 5 is out of both cells, even though the set is not unique.
Worth memorizing
Each length has four unique sums.
The two smallest totals and the two biggest.
Two cells: 3, 4, 16 and 17.
Three cells: 6, 7, 23 and 24.
Even sums that aren’t unique rule digits out.
A two-cell 10 never uses 5.
Practice
0 of 4 rightDown 16 in two cells. Which set does it use?
Which of these three-cell totals has exactly one set of digits?
Across 10 in two cells. Which digit can never be in it?
How many sets of digits make down 23 in three cells?