Is Sumplete always analytically solvable?

Question

I am stuck here at Sumplete.

I have tried looking for numbers that must be used/must not be used using https://combinationsumcalculator.com/ and all of them are either not used in all combinations (when I try to achieve a value by adding) or used at least in one combination when I sum the whole row/column and try to subtract to reach the target value.

Some of my techniques so far:

• if the target value is low (e.g. 4, you can exclude all higher values).
• if you need something like 8 and you have 3, 4, 5 and 6 you can exclude 4 and 6 because they can never be combined with something that results in 8.
• exclude single odd values. If the target is 12 and you are sure about 6 (e.g, 2 + 4 or 6 alone) and one of the values you have is 3 and that is the only odd value, you can exclude that because you can never combine that with something that results in an even value

The goal of this game is to pick the numbers that add up to the target number outside of the grid.

Trivia: it is ChatGPT who "developed" this puzzle game, although it is more or less a copy of one or several existing games.

Answer 1

It depends on how much backtracking you still consider as analytically solvable.

For example, the first row here only has two options: 2+8 or 6+4.

Considering the case 2+8, now the third column only has two options, either take 8+1+1+6 or 5+5+6, either way 6 is selected.

This disables 9 in the last row, which in turn forces 5+4 in the 6th column, disabling the other 5 in the 4th row, forcing 8+1+1 on the third column.

From the initial 2+8 assumption in the first row, it forces 4 on the 4th column, disabling 3 in the last row, forcing 5 in the last row, forcing 3 in the last column, forcing 2 in the 7th row, forcing 1+1 in the first column.

Now second row has forces 1+8, while it should sum to 8.

Therefore our initial assumption of 2+8 in the first row is incorrect, and it should be 6+4. Then we can continue (e.g., disabling 4 in last column, which forces 1 in 5th row, etc.).

I was lucky to start with 2+8 instead of 6+4 which wouldn't lead to contradiction here, but we could always stop and assume the other path at any point to see if it leads to shorter contradiction.

But the idea I used here is to find some places where it could lead to lots of forcing in other places, and has only two options (so we can be sure that the other option is correct). The two options in the first row seems like a good candidate.

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Alternatively to the long chain above, we can consider the 5 in the 7th row. Assuming it is not selected, it forces 2+3 on 7th row and also forces 8 in the 3rd column. The forced 2 in the first column forces 1 in the second row, which makes 3rd row 1+8, a contradiction. So it means 5 is selected, and we can continue (e.g., it forces the other 5 in 3rd column to be selected as well, etc.). Now this one is a short chain, do you still accept this as solving it analytically?

Many grid puzzles require some form of multiple rows/columns deductions to solve it. For example in Sudoku there is XY wing which involves multiple rows and columns. And looking for which cells and rows and columns can move us forward is often part of the solving process.