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Kakuro Puzzles

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I've always been a fan of puzzles, riddles, and logic problems, from when I was a child, snatching sheets from newspapers, to now, when I'm away from the house and bored on my phone. Among my favorites were always the number puzzles, as well as the more logically-driven ones; crosswords and the like played too much on word/language knowledge for me while younger so I never got much into them beyond cryptograms, which I only started later on because my mom liked them and I miss her. However my bread and butter were always the number games, from the math-y 3x3s/4x4s to sudoku puzzles, and to two of my favorites: word math and kakuro. All of these puzzles, as well as the other myriad of fun puzzles, could use some nerd discussion on these forums I think, especially some of them who's intricacies aren't as well documented, easy to find, or give only basic knowledge with little discussion for the advanced levels, however this thread is about kakuro and those reasons wrt kakuro are partially why I wasted my time typing this, heh.


What is kakuro?

Kakuro is like a genetic deformity as a result of the improper crossbreeding of a crossword puzzle and sudoku. Given that, it maintains a lot of the rules you'd expect to find:

  • 1 digit per box, value of 1-9
  • Numbers cannot repeat in same immediate row/column (or same sum)
  • Clues are given in the form of vertical and horizontal numbers which all digits must add up to


Now given these rules, you can start to see how this becomes more of a logic issue than a math one, and the puzzles get much more in-depth both logically and mathematically than sudoku, though sudoku actually involves no math so any#>0 heh.


Now that's all well and good, but this isn't just an appreciation thread; I'd like to talk about some of the strategies behind playing the game and with any luck, hopefully I can learn some stuff from others here to help me solve puzzles more effectively. So with that, I'll start with some basics and then get into the fun stuff.


First a basic puzzle:

(Real Kakuro Easy 16)




So first things first let's just take a look at how the puzzle is laid out: the clues are written to be across and down, as with a crossword puzzle, so from their half of the diagonally divided squares their answers go either to the right or down. I'll get more into the Real Karuro app later, but the important thing to note is that it allows us to pencil/pen in numbers. Most of it should be self-explanatory but hopefully that short description will get us all looking at the puzzle layout the same.


So we understand the layout, the rules and the goal; now let's get into how to solve it.


So uuggghhhh ermagerd now we gotta do math, shoot me now. Good news though: it's not that bad and as with most both math-y and logical things there are patterns for us to find and utilize! Let's start with some easy squares, say, the 2x2 section in the top-left. Notice how small some of these numbers are, as well as the number of blocks? This is good news for us because it makes things easy. 3 can only be made up of 1 and 2, while 4 can only be made up of 1 and 3. Before you even begin to pencil in numbers you can see the discrepancy between the 2 clues wrt their shared square: the top-left-most square of the puzzle. One clue says no 2 while the other says no 3, with both agreeing on the 1, so the top-left square is a 1. This then means that the square below has to be a 2, and the square to the right has to be a 3. Looking at the clue of 6 with a 2 penned in beside it, it is clear that the last square of this 2x2 portion must be a 4.


This is an example of one of the most basic patterns we will use to solve these puzzles. Also, just as this holds true for the lower numbers, so does it hold true for the upper register. 17 divided into 2 squares can only be an 8 and a 9, while 16 into 2 can only be a 7 and a 9. There is a pattern to all numbers of squares per clue which we will use to start the puzzle off with the most info possible. In the process of this first, basic step we will also likely solve this entire puzzle. See if you can do it on your own; the rest of it will be in a spoiler after this next small section.


This is a listing of the numbers of squares and the clues that make for unique combinations of digits:

(Picture courtesy of Sourendu Gupta's Kakuro site; pic's source found here)




This list is your friend and while you will figure it out on your own, seeing it makes things much easier from the get-go. Notice how the closer the number of blocks is towards the center, (5) the more precarious the combinations become. In truth, 2 blocks are as easy as 8, 3 as 7, 4 and 6 ain't too bad, and 9 has all numbers in it. 5 tends to have the worst patterns, as it has the greatest difference between the lowest and highest numbers it can accommodate. For 8 squares, subtract the clue from 45 and you have your missing number; for 7 subtract the same, then divy that number amongst two possibilities to be left out, and so on. Also note other (negative or exclusive) combinations of numbers, such as that a 22 into 3 squares cannot have numbers 1-4 in any square, etc.


Now, knowing this let's do this 'easy 16' puzzle:



So first we start by filling in what we got thus far:

I started arbitrarily with just the vertical clues; eventually you'll learn to cross-reference the horizontal ones at the same time but I would highly recommend taking a logical approach of focusing more on one over the other, when available, to start with. For example, assuming it's not a puzzle that throws a bunch of wtf numbers at me, I usually do the vertical clues first, and do a basic cross-reference of each box as I fill them in.




Ok so we got the vertical clues; let's do the horizontal ones, which will also leave us with 1 remaining possibility in more than a few squares. Remember also to look for places where numbers clearly don't fit, like the horizontal clue of 12 at the top of the puzzle, with a 1 and a 3 penciled in to one of the boxes: a 1 would require an 11 in the adjacent square so 12 in 2 squares must always be between 3-9 (it also can't be a 6, as that would require a second 6 to make 12). In this case, since only a 3 can go in one box that leaves us with a 9 in the adjacent square to the right and a 1 below. Knowing this let's cross out a 1 to the left of the 1 we just penned in, as well as the other 9s below the 9 we penned in.


Before I move on though, since simply cross-referencing the horizontal clues in reveals most of the numbers in the 29-down column, I want to use that 19-across as an example:

Notice how 2 squares are only 7 and 9? That means both that we can cross out 7 and 9 from all other squares in that row, which to be fair doesn't apply here, and that we can assume that 16 of the 19 will be in those 2 squares, leaving the other 2 squares with 3. 3 into 2 gives us a 1 and a 2, so even without cross-referencing in the horizontal clues below, we can eliminate the 5, 6, and 8 from the square directly below the 4 we penned in on the first step.




On the bottom-left, notice the vertical clue of 6: 3 can only go in one box.


Ok now let's do some quick clean-up, as well as fill in the boxes that the clean-up leaves with only 1 possibility and there prolly will be nothing left to this easy puzzle:




Well shit what do we do now? Good news: the clue of 19 will make for an easy solve, as 7+3=10, -19 leaves a difference of 9. With one box containing an 8 and a 9, we can cross off the 9 in favor for the 8, making the adjacent square a 1, then a 2 and a 9 on the bottom.


Ok so that's the easy stuff, and to be fair that knowledge, plus not being shy of doing lots of basic math on possibilities to whittle down some numbers, (and therein learning more of the patterns) will take you pretty far, but now let's get into the fun stuff that will take you much further than the basics can. Don't be afraid of 3x2 sections, or weird joinings of 2x2 and/or 3x2 sections; the hardest puzzles from my experience are shaped like squares made up of 1x-wide sections with 2x2s mixed in. Larger pieces just require a new pattern and keep a mind on how numbers fit together in rows/columns of each size.


This first piece of knowledge I figured out on my own, but it's actually just the inverse property of a far more advanced (what I'm gonna call a) trick involving comparing the clues against each other to find the difference in certain, strategic squares or groups. I originally posted this here, but I'll just quote the relevant part for convenience:





If the top 2 boxes are 8/4, then the bottom 2 are 9/5=14; likewise, if the top 2 are 9/3, then the bottom 2 are 8/6=14. Thus, those 2 squares always add up to 14. Combine that with the 1/2 and we have 17, leaving 18 over 3 squares. This knowledge will help us whittle things down for that row. Above that we have a similar situation, with a 5/8 or 7/6=13, +3+2 is 18, leaving 22 for that row, which means no number can be below a 5 thus eliminating the 1 from the top right-most box which is yet to be penned in. We now know the position of the 1 in the right-most column that is yet to be penned in.


This is a very useful piece of info to have, but even that pales in comparison to what Mr. Gupta's site showed me:




The divide and conquer trick (and yeah I think the word "trick" sums it up well because of how stupidly easy it can make some select impossible-or-really-tough-seeming situations) can be used to massive effect in far more places than this guide even covers. Places such as these are great because when one box is being looked at, the total difference is equal to the number in that box, but who says you can't do the same for a place with 2 boxes? Or what about when used to find out the totals of 2 smaller sections of a row or column split by known digits? (Or digits who's total is known, for that matter). Or what about if you use it to split a puzzle in half? As you can see there are many, many uses for this trick which will compliment the math-y/logical diligence you've accumulated thus far. In case the example there didn't help enough, here is another example which may be of some use:




Let's just look at the red and blue sections for now:

 - Starting with the top-right, we have 17+3-(10+12)=(-)2. We can drop the negative because we are just focused on the number as a difference, so it's positive/negative relation to zero is not important. Notice what this does though: it compares the values of each of the squares against each other, except for one, which is different, ie the difference. So that red circle in-between the 12 and 22 in the top-right is a 2. Likewise, the blue circle to the right of that is 10, which isn't so important here but is important when we split rows/columns. In this case the 3 can only be a 1 or 2, which conflicts in the lower box with the 2 we just found via 'divide and conquer,' so that all pieces in on its own. 

 - The bottom-left red circle can be found in the same way: 3+4-(3+8)=4; the red circle is 4. The blue circle can only hold a 1 and a 3 for both of its squares. In this case as well, knowledge of the blue circle is redundant since the vertical 4 and horizontal 3 conflict, giving a 1 for that square and solving the section regardless.

 - Moving on to the green/orange and starting with the top-left: 15+14-(16+38)=25; the green circle adds up to 25 while the orange adds up to 13. Unfortunately this example sux and this does us no good yet here. Ironically this 2x2 is easily solvable by noticing the conflict between the vertical 16 and the horizontal 14: 16 into 2 is 7 and 9, but 14 into 2 cannot have 2 7s, so the conflicting square has to be a 9, making the square above a 7, with an 8 above and 5 below in the orange circle.

 - In the bottom-right we have 15+11-(17+38)=29; the green is 29 and the orange is 9. Now, this may not seem like it does us much good here, but I can tell you something else about that column: 38 into 6 boxes is a unique combination of 3 + 5-9. (3, 5, 6, 7, 8, 9) What can add up to 9 in that collection of numbers? Only 3 and 6, so the orange box is comprised of a 3 and a 6. Now we can solve that 2x2: the horizontal 15 cannot have a 3, so that's our 6 and the horizontal 11 gets the 3. This leaves a 9 and 8 respectively, which is also the numbers we need for our vertical 17. The rest does us no good without doing other stuff to the puzzle so I'll leave that example there in case you want to solve the rest on your own :) Start with the 2 remaining "real" 2x2s and work your way from there. Notice the vertical 21 and 22 clues.


As a final example of this method, I want to share this puzzle:




Look at the highlighted box and tell me if you think I can tell you what it is using this method. (don't really tell me that) Spoiler: I can. I can even tell you the value of the box on the left side of the puzzle, in the column of the clue valued at 29 and row of the clue valued 11 simply by using this complicated method. Joking aside it's actually very easy, if a bit tedious, but let's dive in using the same logic we used before.


So if I add all the rows and columns that encompass an area, while only leaving either a row or column with squares only they cover, I can find out how much those squares' totals are valued at. So math time; no fancy squigglies this time, but hopefully it'll be easy to follow along. Hey maybe the squigglies made the last pic look complicated, heh. Anyway, we will be targeting the two squares I mentioned from the top half of the puzzle. Given that, we will want to use this on the columns, so we will start counting with the rows. Note that as we do this some squares have been solved already; the best way to deal with this and not get confused is to simply subtract them from their respective clues. So the horizontal 21 becomes an 18, the horizontal 31 becomes a 27, the horzontal 18 becomes a 17, and the vertical 3 and 4 are both canceled out. Thus we have: 9+13+18+27+17+16[100]-(29+10+23+30+24)[116]=16; the two squares in question must add up to 16. Well only one combination of 2 squares equals 16, so 7 and 9 it is:




From here we can figure out what those two squares are stupidly easily. Look at the column for the vertical clue of 30 in the top-right. 7 and 9 in 2 squares means the top square is 8, plus 6 can only go in the second square anyway. Knowing the top square is an 8 makes the far top-right-most square a 9, which makes the square 2 squares below that a 7, and the middle one an 8. That right hand square of the horizontal clue of 16 being a 7 makes the left hand square a 9, which means that the first square in question from way back when has to be a 7, which in turn not only gives some nearby squares, but that also means that the other square in question from way back when must be a 9. Fml what just happened to this puzzle? Lol gg




From here some basic clean-up and a little math will take you the rest of the way. 


There are prolly some other things I'm forgetting to mention, but I'd highly recommend, if you are interested in these, both looking at Mr. Gupta's site and looking into the Real Kakuro app. I would love to stir some discussion on this as I'm sure there are still many strategies, tips, and tricks I do not understand or know of yet and which would help me on some of the more nefarious puzzles/designs.


Before I close I would also like to link another part of Mr. Gupta's tutorial which honestly I just do not understand beyond the first, basic part, lol:












Hopefully this will spark some interest/discussion that everybody interested, myself included, can learn from and maybe even introduce new people to something they may enjoy. Happy solving :)

Edited by Fonze

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Interest sparked, thanks for keeping me awake for longer than I should be, but I really gotta try at least an easy one real quick to see how this works in practice ;-)

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Pretty fun. Don't like to play on my phone though. Should look for a good website that you can actually mark, or did I miss the website because of too much stuff?

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Lol, glad it stirred your interest :)


I kinda forgot to talk about the Real Kakuro app, but it's on google play and available for android devices, maybe available for other devices but I have no idea. In any case, it is damn-near the perfect kakuro app, containing 5 difficulty levels and like 600 puzzles a piece for 3000 total or some rediculous number like that. The only things that could make it better to me would be the addition of multiple layers for penciling notes in, the ability to utilize algebraic equations in-game, and possibly a zoom feature since that writing would be reeeaaallly tiny on a phone screen, heh. Aside from that, which to be fair I never would have considered months ago, it's the perfect kakuro app and I don't see myself finishing it any time soon, though some of the expert level ones and one-or-two of the challenging level ones have murdered my soul, taking over an hour to solve. Those are best put down and come back to at the end of the page, (grouping of 20) lol. That said most of even the challenging ones have been <10 minutes iirc.


On that note it would be cool to have some casual competition for good blind times lol.


@Garrett I did link the app page on the google play website, but idk that it'll let you play in-browser and idk if that'll work easily on computer. Likely you'd need an android emulator or something I guess; idk. Prolly are some good websites out there to play on in-browser though; I never checked for them since I got this app.

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@FonzeWell, I forgot I have Bluestack, so I can use that. Talking about puzzle games, I really like a game called "Blendoku 2". Probably worth checking it out :D

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Oh man, I love puzzles. My main stuff is like crosswords, the stuff that comes out of those MIT mystery hunts/other puzzlehunts and shit like Stephen's Sausage Roll and Hexcells. This looks right up my alley though, I'll check it out.

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@Fonze I have that same app on my tablet, and I pay it almost every day. I also have an app called Epic Sudoku from Kristanix, which has a mode called Killer Sudoku, which you might like.

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Hell yeah empyre; what skill level do you play on? Do you have any additional tips, tricks, or strats to add? I'd love to hear about some other ways people approach these puzzles :)

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I only just recently found myself able to tackle the Extreme kakuro puzzles. I didn't read all of your long post, so I might be repeating some of it. I did one in a little over a half hour after I posted here earlier. I do keep the "cheat" sheet on, but I am referring to it less often them I used to. I like to start a puzzle by filling in the notes for all the ones with only one combination of 2 numbers: 3, 4, 16, and 17. That gives a nice head start. Then I do the ones with only 1 set of 3 numbers: 6, 7, 23, and 24.


Sometimes near the end, I get painted into a corner. Then I either try to memorize the state of the board or write it all down, and then make a guess on a number. If that leads to a contradiction, I know the guess was wrong. I only do this as a last resort.


I just skimmed your post and it goes into some rather deep theory. There are some other strategies I have worked out besides the ones I mentioned, but they are hard to explain because I don't have the right terminology. They aren't as sophisticated as the spreadsheet method.

Edited by Empyre : Added empty lines between paragrahs to make it easier to read.

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Ah ok, interesting. I'm about in the same boat then; I've only attempted 13 expert puzzles and finished 10, with times ranging from under 8 to over 90. I don't use the hint feature, since that only accomplishes the basics of what I do automatically already anyway, plus it's a bit confusing to look at. Ironically, when I first started using this program and hit the first couple puzzles that wound up taking over an hour to complete, using the hint feature never actually helped me, lol.


More recently I've been trying to stay away from guessing and checking too, attempting to stick only to logical deductions, but I need to figure out or learn an actual system to compare the conflicts between separate rows and columns to avoid having to do this for the extreme cases.


Truth be told though I don't understand the spreadsheet method as much as I'd like to; I think that is one of the things that could take these expert level puzzles down to the average times I get on other difficulties.


If you want to try to detail out some other strategies you've picked up on I'm sure the terminology won't hinder understanding too much :) and I'll prolly add it the OP, along with any other stuff I think of. Next puzzle I do where I split a row/column I'll grab a screenie of to use as an example and add that to the OP as well.


As a note to others: while yes, there is some arithmetic to be done with these, they are far more logic problems than math ones, and the logical strategies to solve these are much more rewarding than sudoku for the extra layer of depth within the puzzle's design. 

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The one I did earlier today was number 7. The last Challenging puzzle I did was number 31, and I am keeping Easy, Medium, and Hard even with each other, currently at 228 each.


Let's say, for example, that there is a row with 2 squares that add to 7 and a column with 2 squares that add to 13. the square where they overlap can only have 4, 5, and 6. The other square in the 7 row must be 1, 2, and 3, an the other square in the 13 column must be 7, 8, and 9. then I check the "cheat" sheet and see that the column where I put 1, 2, and 3 has no 2 in it, so I remove the 2, and the 5 that would have added to that 2 to make 7, and the 8 that would have added to that 5 to make 13. Then, I notice that there is no 7 in the row with that square that now has 7 and 9, so it must me 9, and 4 below it to add to 13, and 3 next to that to add up to 7. It is much easier to do it than to describe it.


The rest of my strategies are even harder to describe. I do logic to it until done. I guess I am just getting better at it with experience, and so can any of you.

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Bit of a bump for this thread, but I figured this was cool enough to share; despite being close to the end this puzzle is still proving to be tricky:


Looking at the long, vertical column on the left side of the puzzle with clue 37:




Honestly brute force on the 2 squares with only 2 options would get you to the end, but this is also logically solvable without guesses:




On a random note, I need to stop using the red color on these puzzles, so idk if that is legible, but I'll retype it out here:


 - From the clue itself and number of boxes it fits into, we know it is missing 2 numbers that add up to 8. (45-37=8; 7 boxes used so 2 numbers missing) This leaves the possible combinations of numbers that are not present as: 1,7; 2,6; or 3,5. On the flip side, 4, 8 and 9 must all be present.

 - We have a 2 in the column at the top; this means that there must be a 6 in the column. If my notes are at all to be trusted, then all even numbers are present, and none of them are in the red box. Speaking of the red box: from strategies outlined in previous posts here it should be clear that it can always only equal 12, even without notes on the possible answers. (16+7-11=12)

 - Math time: 37-(2+12)=23. Remember all even numbers are present and 2 has been used already; 23-(4+6+8)=5. This means both 5 and 3 must be present, while 1 and 7 are not used at all.

 - This gives us a 3,9 for the red box straight away, as well as solves the rest of the puzzle.




Below the red box we get an immediate reduction of possibilities to 6/8, which leaves 7/9 for the 15-across. That gives 2 squares with 7/9 for 16-20=4 in the middle square. Below the 6,8 we were reduced to a 4/5; this 4 adjacent to it gives that square a 5, making the far left square of the 35-across a 1. (To be fair with math and hindsight we could have figured out before now that a 3 couldn't be used to make a 10 in this situation anyway, heh, but I don't think that would have quite solved the puzzle on its own yet) Finishing up the 1 gives us a 3 underneath it for the 4-down, 7 adjacent for the 10-across, 9 above for the 20-down, and 6 adjacent for the 15-across. This all leaves us with only a 4, 8 for the top two squares of the 37-down, making the leftmost square of the top-left 10-across a 5 and the square below that for the 14-across a 6.

Edited by Fonze

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@Fonze, have you given Epic Sudoku from Kristanix, which I mentioned earlier, a try yet? The Killer Sudoku mode is basically sudoku with sums like kakuro.

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I only wound up trying a short couple, but I didn't really get into it. Killer sudoku is certainly an interesting twist on sudoku, but I've done literally thousands of sudoku puzzles at this point and have grown tired of that form of puzzle; it just doesn't seem to offer the same level of logic I want in a puzzle, being more tedious in its design of meticulously and repetitively going through, number by number, columns/rows/boxes. That's not to downplay sudoku though; I've had a lot of fun with it over time and a love for it got me into kakuro.

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