Need help learning advanced kakuro strategies (Full Version)

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Fonze -> Need help learning advanced kakuro strategies (10/10/2020 2:21:58 AM)

Hello all. I've recently joined this forum hoping to connect with others who enjoy kakuro puzzles as I do. Unfortunately the amount of us seems to be fairly small and thus I don't have many outlets to turn to when looking to expand my knowledge and skill repertoire when doing these puzzles. I'm well-versed in many skills needed to do these, up to and including taking the difference of rows and columns to find the values or relationships between (groups of) squares, but I still find myself struggling on some puzzles, so I know there is more for me to learn. Unfortunately most online posts and blogs I've seen offer only a select few bits of insight beyond the rudimentary level solving techniques and the rest are research papers on programming a solver, which is a bit beyond my current understanding, as I am not very familiar with coding and the logic that goes into that; thus this post.

Context aside, here are some puzzles that show where my skill level stops:


This first one I think is a good example for how small the unsolved portion is; while it is trivial to guess-and-check the solution I struggle to find the logical way to deduce the correct solution without guessing. I've gone as far as to figure out the relationships between each of the 4 major "groupings" here but nailing down their values proves to be out of my current understanding, or maybe this bit of information is unrelated/unresolvable:


A lot of information can be gleaned from this, however this is also one of the many cases where this information at the least needs to be teamed up with something else. I cant see a way to cross-reference any of these pieces together to get the solution, which is apparently that x=18 and y=14. Again, even this info may not solve the puzzle, which is the real goal here, and this info could be totally irrelevant in my lack of understanding of finding the next actual step.


This second one is a good example because it is such a common shape for being very difficult, in large part due to the fact that one can only cleanly cut the puzzle to find the difference between columns and rows two ways, due to the diagonal sections running through the center between sides. Also, it only leaves sections of 3 for that, so one really cannot compare one single square against anything else.

I hope somebody here will see this and help me further my learning of these amazing puzzles; thanks in advance and happy puzzling :)

kimbro84 -> RE: Need help learning advanced kakuro strategies (10/12/2020 8:41:47 AM)

Your work is awesome! I found a couple tricks that worked, these are probably basic but you never know!

I've learned any connecting squares can't have repeating number

So if I have 4 squares

7 9 789

7 9 7 9

Then the top right corner HAS to be an 8. No matter if the squares are touching or are at opposite corners. So long as they have touching connectors, there can't be repeats. Another example:

7 9 789

7 9 789

The 2nd column will have to have an 8. I have found this useful for helping with removing possibilities from other cells.

Another thing I found out the hard way was making groups of 7 between 32 and 40 simpler. Groups of 8 are pretty cut and dry as is the 41,42 in 7, but the groups of 7 between 32-40 were really challenging me. Here's an example, let's use 39:

45-39 = 6. So there are 2 missing squares (because it only has 7 cells). The two missing squares, if there, would bring our total to 45. They have to equal six (45-39). So one of these groups below will NOT be found in the answer:

The answer must contain 9,8,7,6,3 and then 5,1 OR 4,2.

Here's an example for 34:
45-34=11. So the two 'missing squares' have to total 11.

I'll have to have
Answer will be ONLY three of these groupings and a '1'.

This really helps me when I have cell clues that are high and low because you can only have three high and three low. I've tried using this with groups of 5,6 but there were simply so many options once you are talking about 3 missing digits for most of the super hard puzzles. They throw in 25 in 5 and 32 in 6 and it's just too many options!

The other helpful things I learned:
17 in 5 always has a 321
18 in 5 always has a 21
19 in 5 always has a 1
20 in 5 doesn't have to have a 1. 6,5,4,3,2
21 in 5 has to have at least one cell with a 7+.
a 16 in three has to have at least a 7+,
a 19 has to have at least an 8+
a 25 in 6 has to have a 2 and a 1,
a 26 in 6 has to have a 1,
a 27 is the first in 6 that doesn't have to contain a 1 (7,6,5,4,3,2)

You prob know these easier ones 30 in 7 has a 5,4,3,2,1 then either a 9,6 or 8,7. Then 31 in 7 has the 4,3,2,1 and then either 9,7,5 or 8,7,6.

I do the big puzzle book so these smaller combinations are not something I'm used to. Sorry I don't have any guidance on the ones you posted. I'd love to hear some of your strategies. I see a lot of your grouping and relationships have all been tested out. YOu're definitely a step ahead of me! I'm disappointed there aren't more difficult/expert tips out there. Would sure love it! Keep up the good work.

Fonze -> RE: Need help learning advanced kakuro strategies (10/12/2020 7:34:09 PM)

Hey thanks for the reply! Agreed that's it's disappointing that there are not more expert tips and such out there, but I suppose part of the fun of puzzles is also figuring out the strategies needed, heh. I guess if we felt like we had it all figured out there would be no point beyond time-trials to solving these :)

One case-in-point is that first tip you've shared: that wasn't something I really put together until a couple months ago, that being that there cannot be multiple valid solutions (as with the 7/9 7/9/8 2x2 block you used as an example), so any cases which fall into that can be crossed off. Actually while doing research before coming here, I found it to be a little funny that several of the research papers I've been looking through (and mostly failing to understand) have termed kakuro puzzles with multiple solutions to be not well-formed; the salt is real, though I also find that I share this view too so I guess heh. I think that's a good sort-of meta approach for a clue to use though and actually it's a very frequently useful one, which is cool.

I also really like the second tip you shared of finding the inverse of what can go in a sum; I find myself using that quite often as well, as finding what doesn't go in a sum therefore tells what does go into it, so reducing the inverse down to their "magic numbers/blocks" really helps. I too need to further memorize the combinations of some of the tougher sets of numbers and some of the stuff you've posted are not things I've explicitly added to my memory (typically I just "figure them out" as I analyze sums, though that is also slow and requires more thought when cross-referencing with other sums) so these will be good to add to my repertoire; thanks :D

I could be reading a bit too far into what you've typed but I don't think I've missed something like these for these puzzles; if your post was a clue I'm sorry to say that it may have flown over my head, though I do appreciate the info to think on and the conversation about this nerd hobby ;p


ORIGINAL: kimbro84
I do the big puzzle book so these smaller combinations are not something I'm used to.

This made me lol because my mind is in the gutter. Seriously tho I havent really expanded out into the big puzzles; I recently tried one of the larger ones (I think it was like 13x23? certainly bigger than the 9x9's I'm used to), dubbed by some website as one of the toughest kakuro puzzles around (for no real reason than to see where I stack up to something of the sort, as I've mostly been sheltered in my one kakuro app, which doesn't have big puzzles), and failed miserably as one might expect, haha. The big puzzles seem to be very punishing on mistakes as you may not realize that you've made a mistake until far later when stuff finally connects. Rough times heh, but they do seem to be interesting. I have been seeing a lot of recommendations for the nasty kakuro books or something similarly named and I did also download the kakuro app from this site, so maybe I should give some of the larger ones a more fair shot soon.

Again, thanks for the reply; I appreciate the conversation.

kimbro84 -> RE: Need help learning advanced kakuro strategies (10/12/2020 8:11:52 PM)

No advice on your specific puzzles. But it wasn't for lack of effort! I printed them out and tried everything I could think of with boxing and adding etc but man...those small puzzles don't give much! Here's an example of the bigger puzzle book I'm on. I applaud you for not having to write things down. My columns of the book are usually filled with random notes, I can't do the app puzzles without a notepad handy lol. So I guess we all have our own strength. I did compare the big and small and the smaller puzzles use much more of the boxing/adding/subtracting than the bigger puzzles. Which is probably why I'm terrible at that strategy. If you want a different type of challenge the conceptis books on Amazon are all really good, the book in the pic is definitely the most challenging I've ever done. I also bought Kakuro U and it's another level of difficulty, kind of like the ones you posted but larger size.

kimbro84 -> RE: Need help learning advanced kakuro strategies (10/12/2020 8:18:57 PM)

One of my favorites, but most challenging


kimbro84 -> RE: Need help learning advanced kakuro strategies (10/12/2020 8:30:45 PM)

Here is just an image in case you were curious about the larger puzzles.

Can you explain more how you use the x and y and some of your strategies?

Fonze -> RE: Need help learning advanced kakuro strategies (10/12/2020 10:06:29 PM)

Ah ok, I wasn't sure if I was just missing something I should have seen, heh. Thanks anyway :) Also, I should note that I'm by no means an expert with these puzzles. I have gone down the rabbit hole of enjoying them, but there is certainly still lots for me to learn :)

As for writing stuff down, I do actually do that, as is the case for the second screenshot. My notes typically take the form of a screenshot that I doodle on, which is kinda helpful in a way that paper notes could never really match. You actually might find some good use out of taking a picture of your puzzles and doodling on them. A lot of times I just write stuff down to remember later some cool puzzle, or occasionally to share as like a write-up for solving a puzzle, but a lot of times I also write stuff down because it is simply a lot to remember and just the act of writing something down can help with remembering the info quickly later. On top of all that though, it is also probably largely due to the small size of the puzzles I have been doing compared to what you do. These puzzles are small enough for me to try to push myself to remember stuff, so I haven't really been faced with the situation of puzzle after puzzle of needing notes like that, as I likely would be if going through the book you posted. When I get there, I'll prolly have a notebook too lol.

I can't read the puzzle you posted, but I can at least see its structure; it looks similar to the one I tried and failed at recently. I have read some good things about that book though, so I'll definitely have to look into it, as well as Kakuro U, which I hadn't heard of.

Wrt using the x and y in strategies, it's not something I have a lot of practice with in terms of it being the crux of finding the solution. Most of the time by the time I get desperate enough to attempt it I have already found a much easier path to the solution, so I'm not sure how necessary it is in the first place. It also may come down to just an abstraction of the easier math and logic. I first got the idea from this website which unfortunately shows its use in an entirely too simplistic manner for it to be of any realistic use:

It is still an interesting way of looking at these puzzles; maybe it's just a different perspective. The unfortunate example from this website is essentially just an abstraction of the basic adding and subtracting rows/columns, but I guess that's partly the point of the method perhaps. Sometimes it can help me establish mins/maxes for stuff, other times it just makes it easier to see and remember the relationships between squares and groupings when viewed as equations (pieces that can be picked up and interchanged), though that could be down to the way my personal memory works. As you said, we all have different strengths and different strokes :)

I don't think this is a good example; iirc this line of logic never went anywhere with this puzzle, or at least not without more info elsewhere, but here is an old screenshot I have on hand which shows I guess it working in the sense of easily showing the relationships between groupings:
Of course I say "working" here but this was another puzzle that unfortunately I simply had to give up on eventually and guess and check my way to the end. Still, the algebraic approach does allow us to clearly see that if we take the min/max of "B," we can simply plug that range into the equation and get our mins/maxes for other groupings (note "A" in this screenshot is the combination of the two lowest groups highlighted in red in the right column while "C" is only the top group; I didn't make this one to share heh). The key would just be narrowing them down or otherwise finding one of their values, which would give you the rest of them in a clear-to-think/see manner. The problem that arises in most situations though is that by the time you have this info, you have other, easier/faster methods to solve the remainder. This was one reason I mentioned in the OP that it may not exactly be relevant or neccesarily part of the solution. It's also definitely something I could use more work with to learn better, as I could be missing some aspect of the algebraic approach that makes it work better.

Hopefully that helps to put it into perspective.

kimbro84 -> RE: Need help learning advanced kakuro strategies (10/13/2020 3:43:44 AM)

I know this is silly but how did you get the '8' in the very first part?

Fonze -> RE: Need help learning advanced kakuro strategies (10/13/2020 5:15:39 AM)

No it's not silly at all. Apologies if the weird way I highlighted stuff here made things less clear; on other screenies I've shared in like write ups I'll usually use a different color to show the rows vs columns so it's more clear what boxes are actually being compared to what, and which set of numbers (top and right here vs bottom and left from the other perspective) are being used.

I'm assuming you mean the 8 from the A-8=B with this: short answer adding/subtracting rows/columns. Long answer if you add the 3 relevant rows in the top: (41-6)+9+12=56 and subtract from that the 3 relevant columns on the right: 10+(43-3)+14=64 -> 56(rows)-64(columns)=8 in favor of the columns, thus making the 4 bottom unknown squares of the rightmost 43-column (so not counting the known 3) 8 higher than the 2 unknown leftmost squares of the top 41-row.

I'm not too well versed on the names for the different techniques and of course some have multiple names, but isn't this boxing?

kimbro84 -> RE: Need help learning advanced kakuro strategies (10/13/2020 7:16:06 AM)

Yep! It all makes much more sense now. Thanks! And that sounds like a good term for it too. I'm going to start doing the smaller puzzles, thanks for the motivation.

P.S. If you come upon any more strategies would love to see a post!

Happy Kakuro-ing!

Ahlyis -> RE: Need help learning advanced kakuro strategies (1/8/2021 9:56:56 PM)


ORIGINAL: kimbro84

I've learned any connecting squares can't have repeating number

So if I have 4 squares

7 9 789

7 9 7 9

Then the top right corner HAS to be an 8. No matter if the squares are touching or are at opposite corners. So long as they have touching connectors, there can't be repeats.

This is faulty logic. This makes the assumption that the puzzle has a unique solution. I NEVER use this when solving a puzzle. I always look for numbers which I can prove must be correct through pure logic instead of assumptions about how well the puzzle is formed.

You are of course free to solve these however you want to. But I feel strongly that this particular "trick" is not a valid method for solving puzzles. This includes Kakuro and others where similar logic could be used.

Any puzzle where I can see a section that I "know" will be a certain way because otherwise it would have multiple solutions will remain unsolved until I can find logic that PROVES the puzzle has a unique solution.

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