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Let's do some more Kakuro! Today's puzzle is a tough one and it does involve some decent math, but it also had some really cool logic that I was able to put into play for it and it was fun for that, so I figure I'll share it:
It looks intimidating with the name and size, but for the shaping it's not as bad as it may seem :D For one, this shape can certainly be difficult depending on how vague the number hints are that you're given, but this is also not the most difficult shape and allows the puzzle to be split in half, which is nice. Secondly, this individual puzzle gives some decent numbers to start with before hitting an early brick wall, however using logic and pushing past that early wall leads to the entire rest of the puzzle falling into place, as always with no guessing needed. I try to avoid getting to the point of guessing and checking, however there is no denying that in some situations it can be faster to do so and when the goal is the lowest time that makes it an acceptable strategy to employ; I just try to stay away from it for the extra challenge and practice for myself, as well as for the love of doing these in the first place. Anyway, enough blabbing, on to solving it:
Spoiler the solution:Spoiler
The first thing we do when starting any of these is looking for gimmes and filling in notes for the easy stuff. I'm going to cover the notes in a separate section since that's a bit long-winded, but some squares will be important to have notes for. Also, some of these steps throughout the whole puzzle can be done in different orders.
To start with the gimmes:
There are 2 easy things to pen in off the bat, with a third falling into place after one of the first two is filled in. On the right side of the puzzle in the 26-down column there is a 2x2 block with hints 5-down, 8- and 4-across. 4x2 squares is 1/3, making the top square of the 5-down either of 2/4. 8x2 squares cannot contain a 4 so that makes it a 2/6 up top for the 8-across and a 3/1 for the 4-across below that.
Moving on to the bottom-right corner, a 16x2-across meets a 14x2-down. Given that a 16x2 must contain a 7/9 and a 14x2 cannot contain a 7, this gives us a 7/9 for the 16-across, with a 5 underneath the 9 to complete the 14-down. This 5 is a very important number to have figured out, as it will give us another set of numbers in the bottom-left, which when combined with some notes will set us up to break this puzzle wide open.
In the bottom-left we can begin to piece stuff together with that 5 we just got. To start, let's note that the 16x2-across atop this bottom-left corner has to be a 7/9. This is a gimme since the 11x3-down it intersects cannot contain a 9 since it needs at least a 1/2 in the other squares, which leaves a max of 8 for any of the three squares. We now know the two lower squares of the 16x3-down have to be 1/3 in some order. This gives us a 4/1 and a 2/3 as our possibilities for the 5-across, and a 5/7 above the 4/2 for the 9x2-down. Since we have that 5 from end of the previous paragraph, we can cross off the chain of 5/4/1 for the 9-down and 5-across respectively, which makes that read: (starting from the top square of the 9x2-down, to its bottom square, then right to the 5-across's right square) 7/2/3, which makes the middle square of this block, or of the 11x3-down, a 1.
From here we take notes:
There are a lot of notes to take; I'm going to try to run through most of the squares here, but I'll try to not cover some that aren't relevant. We'll start with a couple of the most important ones first though.
There are two places we can split this puzzle, with one being the 12x2-across on the far left side in the middle-top, and the other being its symmetrical equivalent on the far right side, middle-bottom, the 13x2-across; let's start with the 12x2-across. It's not pretty, but it's also not as bad as it seems. 12x2 can be 3/9, 4/8, or 5/7, so we'll fill these in and take out the 3/9 (note not the 9/3) since the 9 is already present from our efforts earlier lower in the 32-down that intersects the 12x2-across. Likewise across the puzzle, the 13x2-across can be 4/9, 5/8, or 6/7, however we can take out the 6/7 (not the 7/6 though) for the 7 being found lower in the 24-down intersecting the 13x2-across.
From here let's go back to the side of the puzzle with the 12x2-across we just took notes for. Under that is a 2x2 block consisting of 15- and 6-across and a 9-down. Starting with either 15- or 6-across will net 4 numbers for each square's set of notes, but this will whittle down to 2 numbers for each square. These are a 7/8, 7/8 for the 15-across and a 4/5, 1/2 for the 6-across, which makes the 9-down a 7/8, 1/2. From here we can tell that the two squares on the left residing in the larger 24-down column will always add up to 12. We could also have found this by adding 15+6-9=12.
Moving on to the lower part of the puzzle, since we're on that side of the split anyway, there is a 16x2-across on the bottom middle-right which is an easy 7/9 to pencil into our notes. This gives us a 4/6 above the leftmost square of that for the 13x2-down. The 7x2-across and the 10x2-down above and to the right are a bit ugly, but 1-6 for the 7 across minus the specific combo of 2/5/5 (reading top-left, top-right, bottom-right) since the 10x2-down cant house a 5, as well as the combo of 4/3/7 since the bottom-right square shares a 7 in its 45-across row, gives us 1/3/5/6 for the top-right, 1/2/4/6 for the top-left, and 4/6/8/9 for the bottom square of the 10x2-down. This leaves a max of 7 for the middle square of this section between 13- and 10-down, or the middle square of the 15-down column.
The last piece in the bottom of the puzzle is in the right corner: the intersection of the 7x2-down and the 5x2-across. This can only be boiled down to three possibilities, notes will read: (starting at the top square of the 7, then the bottom square of the 7, and finally the rightmost square of the 5, reading all of these types of chains throughout this as a book) 3/4/6, 1/3/4, 1/2/4.
The top half of this puzzle only needs basic notes as well. Apologies that all this seems long-winded through text but taking these notes actually takes very little time to do. Starting with the right side for ease: the 5-across on the right middle-top cant have a 1 in the left square, so it's left with a 2/3/4, 1/2/3 respectively for the two squares. The 16x2-across above that in the far top-right is 7/9 either way, which makes the bottom square of the intersecting 10x2-down a 1/3. We can then boil down the middle square of this section (or the middle square of the 17-down column) to >=5 through math. From there go left, start with the 6-across on the top-middle of the puzzle, which boils that chain with the intersecting 11x2-down down to 1/2/4, 2/4/5, 6/7/9. The 14-down physically next to the 6-across is a good starting point for the set of notes to wrap up this little section, which do not boil down at all and leave you with a 5/6/8/9, 5/6/8/9, 2/3/5/6 for that chain intersecting with the 11x2-across below.
The final set of notes are in the top-left of this puzzle. The 12x2-down cannot contain a 9 in its top square because of the intersecting 9-across, which is all it will boil down. So that will read: (like a book, starting left square of the 9-across) 1/2/4/5/6, 3/4/5/7/8, and 4/5/7/8/9 for the bottom square of the 12x2-down. The nearby 8x2-down serves as a good starting point for its chain with the 11x2-across. That only boils down one notch with the 1 for the 8x2-down not being able to fit into the 11x2-across, which makes that read: 1/2/3/5/6, 2/3/5/6/7, 4/5/6/8/9. Congrats if you followed all of this long-winded section and apologies for both short-handing this in not listing off possibilities before saying what they boil down to and not having screenies to go with this bit.
We're now left with this:
This is where I hit the brick wall and sat for at least 5 minutes staring at the screen wondering what to do. (the time stamp is after I figured it out and decided to delete stuff to get back to here to take this screenie to draw on, before moving on and solving the rest of the puzzle) This was a rough one too because even splitting the puzzle in half, which we will do in a moment, only crosses off one of the 5 possibilities for each of the squares in both the 12- and 13-across, which is unfortunate. Additionally, looking into cumulative math stuffs like adding stuff up, seeing what numbers can go in what row/column based on math, etc, seems to lead nowhere, as does most algebraic approaches of trying to cut off sections from one-another or comparing squares against one-another. This is abysmal. However, in keeping observant as well as remembering the relationships we establish between squares, that what affects one square can directly affect another and being ready to change gears and positioning around the puzzle, we can find an interesting way to crack this.
To start, let's rewind a bit and split the puzzle in half. I chose to split it from the top (counting the numbers on the top side of the puzzle) rather than from the bottom simply because it looked easier to me at the time, but one could go off the bottom if they preferred. What we're going to do is maths, and we're gonna add up all of the-across totals and subtract them from all of the -down totals. How you chose to do this is up to you for your own head maths, I tend to add all of one up then subtract each number from the total of the first set as I go. As such, when writing the notes here I accidentally wrote "103" as "112." The math is actually correct I swears. I did fix it for this first screenshot but subsequent shots will erroneously read "115-112=12" because I'm good at thinking and brain.
So as I said, splitting the puzzle in half like this doesn't just give us the answer, and in fact it really doesnt seem to help much at all. As the picture states: the two squares underlined in red add up to 12, while the two in blue equal each other; they're married. While the 2 squares of the 12x2-across are in a relationship with each other in that they are codependent and add up to a whole, the one underlined in blue will always share the same digit as its soul mate across the board if you wanna get spicy with it; these squares are freaky and dirty like that... wait what was I typing about? This let's us cross off the 9/3 and the 9/4 for the 12x2-across and 13x2-across respectively. We have graduated from a guess having an 80% of being wrong to a 75% chance. Grats.
Now before we break our arms patting ourselves on the back for this, it's time to shift gears to cumulative stuff. Intersecting the 13x2-across from above is the 26x5-down. Normally 26x5 is bad news with 26 being too close to the median of sums to do much with 5 squares, but we can in fact look at this another way: as 26-7x2=19x3 squares, which still sucks but it is relevant given our set of numbers for the 13x2-across. Additionally, one of those squares (from the 5-across up top) is less than a 4, which means that we would need, even with a max of 9 in the second square, a minimum of 6 in the final square. (which has also been used making it a minimum of 7) This means that we can cross off the 4/5 for the left square of the 13x2-across.
With the blue-underlined, left square of the 13x2-across being reduced to a 7/8, we can safely say that the blue-underlined, right square of the 12x2-across has to also be a 7/8. Damn I love logic. Now you can update the notes for the other 2 squares of the 12- and 13-across, but honestly we can stop here and shift gears to this left side we're looking at now. Hey, 7/8 goes in two squares. You remember how we figured out that 2x2 box equaled 12 in the 32x5-down column? Good, me neither; this is one of the few times that won't even matter, because we just got something far more important:
32(x5squares)-(9+7+8)24=8x2, which makes the middle square of the 32x5-down column a 5 and thus the bottom square of that same column a 3. The 5 is in a relationship with all of its 2x2 block so that gives us a 7 above the 5, with an 8 up top. We can then fill in the left square of the 12x2-across, next to our new 8, with a 4. The 8 also gives us its soul mate; the left square of the 13x2-across is also an 8, and its partner is a 5 to complete the 13-across.
From here, combining the newest 5 in our repertoire, of the 13x2-across, with the 7 below it leaves us with 12x2 for the 24x4-down column. Looking at our notes, a 4 is the only thing that can fit in the bottom square since it has to be under 5 above 2, and 3 isnt possible from the first 5 we got way back when intersecting with the 7x2-down which in turn intersects with the 5-across intersecting with the 24x4-down we're trying to figure out. Thank goodness for notes. This makes the 24x4-down column read 5/7/8/4, which in turn makes the left square of the 5-across a 1 and the top square of the 7x2-down above that a 6.
Having that 6 in this row makes the top square of the 13x2-down (adjacent to the magic 3 we got earlier in the white square) a 4, which makes its partner below it a 9, and that 9's partner for the 16x2-across a 7. This only leaves a 2/9 for the last two squares of the huge 45-across, both of which can only go in one square. This makes the 15x3-down a 6/2/7 and the 10x2-down a 1/9, solving the bottom of the puzzle completely.
That picture is getting busy!
The top is a bit more ugly than the bottom. For starters, let's reevaluate the 15- and 26-down columns and reestablish some totals for them, as well as solving for the sums of the leftmost and rightmost 3 squares of the long 45-across:
By subtracting the -acrosses from the -downs we can see that the leftmost 3 squares will always add up to 11. Likewise with the right side, the rightmost squares will always add up to 17. The middle 3 of the right side are 17, while the remaining bit of the 26-down is 11. Also the blue 8 in the top square of the 26-down (now 11x2-down) shouldn't be there, which also knocks out the 3 below it and the 2 to the right of that to add up to the 5-across. This will eventually be reflected in the screenies when it becomes relevant and when my brain finally processed it, heh.
From here let's cross stuff out and go back to notes. On the left, the lowest undefined square of the 15-down is 5 at a minimum. This means that we can cross off anything >=5, (or 4 given that's already used in this column) since that would leave us with nothing to put in the remaining square. This let's us cross off the 5/6, 3/4, 8/9 of the 9x2-across intersecting with the 12x2-down in the top-left. As with the top square of the 15-down here, we can also cross off anything >=4 in the middle square, which makes that a 1/2/3. The right square of the 11x2-across, directly above the puzzle-splitting 4 we found earlier for the 12x2-across, cannot contain a 9 for the situation of 11x3, and likewise it cannot be <=5 because the top two squares can only equal a 5 at maximum. This let's us cross off both the 5/9 from that square, which transfers left to crossing off 2/6 to complete the 11x2-across and then up to crossing off the 2/6 in the top square of the 8x2-down.
From here a quick look at the sum of the three leftmost squares of the 45-across (11) shows that the leftmost square of that row cannot be a 3, as that would require another 3 to be used with the available 5 in the third square to make up 11, which cannot work.
This in turn solves the rest of this block, with a 3 below the 5 to sum up to an 8-down and an 8 beside the 3 to complete the 11-across. The rightmost square of this block of 3 in the 45-across row cannot be a 5 now, so it must be a 4. This makes the square above it an 8 for 12-down, the square adjacent to the 8 a 1 for the 9-across, and finally the square below that a 2 to finally complete the 15-down.
Following that let's head to the far right. The three rightmost squares of the 45-across must add up to 17, as must the 17x3-down which is also in the middle of that block. Looking at the middle square of the block, the 5 had been used in the 45-across row already and you cant make an odd number of two odds and an even, which allows us to cross off both the 5/6 respectively for the middle square of this righthand block, which leaves a 7/9. That officially marries the 1/3 on the bottom of the 17x3-down and at the bottom of the 10x2-down. Combined with the fact that the top square of the 26-down can't be an 8, this leaves 7/9 for two of three squares for both the 17-down and (figured) 17-across (rightmost 3 squares of the 45-across). 7+9=16 we got a 1 in both married squares, which makes the top-right square a 9 to complete the 10x2-down and a 7 beside that to finish the 16x2-across. Beside the other married 1 is a 4 for the 5-across with a 7 above that to wrap up the 26-down we've been working on since before we split the puzzle. The final square (in the middle) is a 9 to sum up to the 17-down.
All that's left is the middle, let's cross off a few numbers we've used already for the remaining middle three squares of the 45-across row:
Gee I see a 6. Beside the 7, in the rightmost of the middle three squares of the 45-across, a 6 is all that remains as a possibility. This makes the square above that a 5 to finish the 11x2-down and the square adjacent to that one a 1 to complete the 6. Finally we are left with a 3/8 for the 45-across. The 14x2-down cannot contain a 3, so the top square here is an 8, making the square below that a 6 to add up to 14 and the one beside that a 5 to equate to 11. Lastly, the 9-down can be completed with the final digit of the 45-across, a 3, making it read down: 1/3/5.
For any who have solved along: congrats! For any who have read along, thanks for reading; I hope this intrigues you and you find yourself giving these a shot one day as they are a ton of fun! Especially so when you can push past the mathy stuff and get into the logic of it all; it's super interesting to solve these and explore the logical relationships between squares based on the shaping and sums of the puzzle. It's my own personal little soap opera. Cheers y'all stay with it.