Interesting though these worked examples might be, they are not strictly on topic, which is asking for puzzle types, not puzzle solutions.
I would argue the solution is just as relevant as the problem in understanding a puzzle. It's especially important to help any readers who didn't get the puzzle.
It's also valuable as a beta test for how the problem is described to readers and by extension players. In this particular instance, Countyoungblood realized he left out part of the puzzle's description.
Great job on the buckets but youre not quite there on the grid. Each row left to right or right to left must be 15 and each column top to bottom or bottom to top must be 15.
9 + 5 + 1
4 + 3 + 8
2 + 7 + 6
My solution already accounted for this. If you check, all rows and columns add up to 15. You said as much originally:
Or lets say you have a 3x3 grid and you need to put numbers 1-9 on the grid using each only one time BUT each row and column must add up to 15.
And i didnt mention it since i wasnt expecting answers but additionally diagonals should add up to 15.
Diagonals might just happen naturally though..
Obviously I can't solve something you never mentioned! In that case, if I swap the 9 and 5 columns, I get one diagnal to equal 15.
5 + 9 + 1
3 + 4 + 8
7 + 2 + 6
The other equals 12, so not good enough. Diagnals do not happen naturally, it is another layer of challenge entirely.
The diagnal condition is a good way to up the difficulty of this puzzle, but you can also leave it out for an easier puzzle. You can do both as well, using the easier one first and building on it later with the new diagnal challenge.
The key to solving the non-diagnal puzzle is realizing 9 is what you want to start with, because there are only two ways to get the remaining 6 with two 1-9 numbers without repeating: 5+1 and 4+2. 3+3 won't work because it repeats. After starting with 9, the puzzle almost solves itself. With that in mind, a good hint for players is to emphasize the number 9 in some way, such as NPC dialogue. You could also show them the solution indirectly with objects in place of numbers on your map, like plants or stones.
Back to the
Diagnal Challenge, because of 9 there isn't leeway in re-ordering the numbers. However, I can swap rows and columns without affecting how they add up horizontally and vertically. This actually helps make the puzzle easier.
I can swap the middle row/column left or right / up or down. That's four movement possibilities. I can also swap the outer rows/columns with each other, but that won't affect how the diagnals add up. The key is the center number. It can't be too big. 9 only has two options to get to 15, whereas the center number needs at least four ways to get to 15.
9 needs 6 to equal 15.
6 = 5+1= 4+2.
Two options. Cannot be center number.
8 needs 7 to equal 15.
7 = 6+1 = 5+2 = 4+3.
Three options. Cannot be center number.
7 needs 8 to equal 15.
8 =
7+1 = 6+2 = 5+3.
Two options. Why? Because 7+1 repeats the original 7. Cannot be center number.
6 needs 9 to equal 15.
9 = 8+1 = 7+2 =
6+3 = 5+4.
Three options. Cannot be center number.
5 need 10 to equal 15.
10 = 9+1 = 8+2 = 7+3 = 6+4.
Four options! This can be the center number.
4 needs 11 to equal 15.
11 = 9+2 = 8+3 =
7+4 = 6+5.
Four options? No! 7+4 is invalid because it repeats the original 4. So only three options, which means it cannot be the center number.
3 needs 12 to equal 15.
12 =
9+3 = 8+4 = 7+5.
Two options.
2 needs 13 to equal 15.
13 = 9+4 = 8+5 = 7+6.
Three options.
1 needs 14 to equal 15.
14 = 9+5 = 8+6.
Two options.
This means there is literally only one candidate for the center number: 5.
4 + 3 + 8
9 + 5 + 1
2 + 7 + 6
All I had to do was swap the first row and middle row of my original solution, and everything fell into place.
The key hints are 5 in the center and starting with 9. It solves itself after that. Although 9 isn't the only two-option number, much to my surprise after working out all options. 7, 3, and 1 are all like 9 with only two options. In fact, all odd numbers except 5 have only two options. So technically any of those could work as a hint, but 9 might make it more obvious since it's the largest number.
Another interesting thing I realized is the amount of options a number has corresponds to where it is on the grid. All corner numbers are even, because they have three options. The odd numbers, who only have two options, can never be corner numbers, because the corners need three options for a horizontal, vertical, and one diagnal solution.
One last hint is that because the solution must always be 15, an odd number, all equations
must contain exactly
one or
three odd numbers. Even numbers can never add up to an odd number, and two odd numbers will always equal an even number.
By identifying patterns in the solution, we can point to them in our hints to the player. For example:
"My corners are even
my heart is odd
in every direction
Fifteen you must seek."
Yeah, doesn't rhyme, I leave that up to you.
What I like about this puzzle is that although it seems daunting at first, when you think through all your options using the rules as I just did, the answer becomes obvious. It's a great puzzle, but players may need a pencil and paper to visualize it.