# Kakuro (cross sums) Tips and Tricks **URL:** \ **Category:** The Game Room\ **Created:** [August 20, 2014, 7:45pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031 "2014-08-20T19:45:14Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [August 20, 2014, 7:45pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/1 "2014-08-20T19:45:14Z")
My name is\*\* RTFirefly\*\*, and I’m an addict. In the past several months, I’ve gotten quite hooked on Kakuro. Right now, I’m early on in [Absolutely Nasty Kakuro, Level 4](http://www.amazon.com/Absolutely-Nasty%C2%AE-Kakuro-Level-Series/dp/1402799926/ref=sr_1_7?ie=UTF8&qid=1408559914&sr=8-7&keywords=kakuro). Out of the first 10 puzzles in the book, I’ve solved 4, screwed up with careless mistakes on 3, and have gotten totally stuck about halfway through with the remaining 3, which (if you’re familiar with this series) gives a pretty good definition of my current ability level. While there’s been little mention of Kakuro on the Dope, I know I can’t be the only one in this crowd who enjoys beating his head against a challenging Kakuro puzzle. I figure we can trade tips, discuss which puzzle books and apps are good, etc. One thing I do (using a fine-tip pen - pencils and me don’t get along) is if the possible numbers that can go in a square are much more restricted than the entire range from 1 to 9, I note the remaining possibilities in very small writing at the top of each square. For instance, in a 5-space row or column totaling to 34, you’ve got 9-8-7-6-4 in some order. I write that in each square, leaving plenty of room for that square’s ultimate solution. So when that row intersects with a 5-space column summing to 16 (6-4-3-2-1), the intersection square can only be a 6 or 4. If I’ve already written the 9-8-7-6-4 in that square, I scratch out the 9-8-7 part. This way, I’m not figuring out everything from scratch if I set down a puzzle for an hour or a day: my accumulated knowledge about the squares I haven’t solved is still all there. Which is good, because it can take me a day or three to do a puzzle at my current level. But that’s more about keeping track, rather than actually solving the puzzles. So on we go: For the 7- and 8- space rows/columns, complementation is key. The numbers from 1 to 9 sum to 45, so if you’ve got an 8-space row that sums to 39, you’re going to use all the numbers except 6. If you’ve got a 7-space row that sums to 41, you’ll use all the numbers except 1 and 3. If your 7-spacer sums to 32, you’re going to be missing two numbers summing to 13: either a 4 and 9, a 5 and 8, or a 6 and 7. If you already know that two of your spaces in that row are a 6 and an 8, you know the 4 and the 9 are the odd pair out. I have a mental widget I call ‘filling the zone.’ It works like this: say you’ve got two squares in a row that you know have to be an 8 or a 9. That’s two squares that can only be filled by those two numbers. You’ve just filled the 8-9 zone: no other squares in that row can be an 8 or a 9, because that would mean you’d have to use one number in that zone twice. And the idea is generalizable. Take the example of a 6-space row summing to 24. The only possibilities for the numbers in that row are: 9-5-4-3-2-1 8-6-4-3-2-1, and 7-6-5-3-2-1. The 7-8-9 zone is going to consist of a single square, and the 4-5-6 zone is going to consist of just two squares. If you have a square in that row that’s in a 2-space column summing to 16, it has to be a 7 or 9, and the rest of the squares in that row all have to be in the 1-6 range. (Plus you get to toss out the 8-6-4-3-2-1 possibility. You’ve got 5-3-2-1 all there for sure, and either a 4 or a 6.) That’s all I can think of for now. Your turn! ---
**Author:** ![Zakalwe](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zakalwe/32/270_2.png) [@Zakalwe](https://boards.straightdope.com/u/Zakalwe)\ **Post date:** [August 20, 2014, 8:13pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/2 "2014-08-20T20:13:02Z")
Googled it to see what the hell you were on about. I hate you, you heartless bastard. 🙂 ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [August 20, 2014, 8:29pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/3 "2014-08-20T20:29:53Z")
Always happy to inject a new timewaster into someone else’s life. 😃 Another thing: every step in solving a Kakuro puzzle is like a miniature theorem and proof. (_Very_ miniature; most of them just take seconds to do, and you barely notice one before moving to the next.) As a math geek, I rather get off on that. ---
**Author:** ![Randy\_Seltzer](https://avatars.discourse-cdn.com/v4/letter/r/ee7513/32.png) [@Randy\_Seltzer](https://boards.straightdope.com/u/Randy_Seltzer)\ **Post date:** [August 20, 2014, 9:08pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/4 "2014-08-20T21:08:30Z")
I can’t do them, personally, but my SO is absolutely crazy about them. A couple of years ago, I got her the [Mensa Kakuro](http://www.amazon.com/Mensa%C2%AE-Kakuro-Conceptis-Puzzles/dp/1402739370#) book which she blew through pretty quickly. She said it was a little bit easy. So I picked her up the “Absolutely Nasty Series” numbers two and four… > [@RTFirefly](#): > > Right now, I’m early on in [Absolutely Nasty Kakuro, Level 4](http://www.amazon.com/Absolutely-Nasty%C2%AE-Kakuro-Level-Series/dp/1402799926/ref=sr_1_7?ie=UTF8&qid=1408559914&sr=8-7&keywords=kakuro). …she found “level two” to be about the same as the Mensa book, but level four to be much more her speed. She does them in bed before we go to sleep. To me, it’s amazing. It’s like watching Bob Ross. I can lie there and watch over her shoulder and she’ll just start putting numbers in squares faster than I can add them. It’s like watching a blank canvas turn into a landscape with a bunch of happy little trees: I could certainly never hope to do any such thing, but it’s absolutely breathtaking to behold. I can’t speak to her strategy, but her technique seems to be that she narrows a square down to two or three numbers and writes all of them in. Once she figures out which is correct, she circles it and scribbles out the others. If she immediately knows the correct number for a square, she just writes that in and circles it immediately. So a completed puzzle has a circled number in each square. Occasionally, I’ll interject with “How did you know that was a three?” and the explanation goes something like this: “See this square over here?” [indicating some distant square that seems totally unrelated to the one I inquired about] “Yeah” “Well it has to be either 8 or 9, because it’s 17-in-2.” “Okay I get that, but what does that have to do with…” “So that means that this one over here can’t be 8, so it has to be either 7 or 1.” “…um…” “And since it is either 7 or 1, that means it can’t be 4. Which means this one over HERE has to be either 4 or 9.” “…” “Which is why THAT one has to be three.” ---
**Author:** ![Biotop](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/biotop/32/321_2.png) [@Biotop](https://boards.straightdope.com/u/Biotop)\ **Post date:** [August 20, 2014, 9:50pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/5 "2014-08-20T21:50:40Z")
I’m pretty much obsessed with kakuro. I have completed the first three in the _Absolutely Nasty_ Series and am saving Book 4 for vacation next month. I also recommend the “_Second Degree_” Brown and Black Belt series from Conceptis — and the “_Three Dimensions_” book currently available from Amazon – which has smallish but actually quite challenging puzzles. “_Kakuro for Dummies_”, which you would think would be easy, is not. Try it if you can find it. The collections from DJape are also dynamite. Dell’s _Math and Logic_ series is available somewhat irregularly but has several 4 star level puzzles in each issue. As to strategy, it has been helping me a lot to add up across and down sections of a segment of the puzzle to determine the needed value for one or two squares who’s sum is otherwise unknown. Another helpful technique I find in the larger more difficult puzzles is to remember that, according to rules, each kakuro puzzle has a unique solution. So say I have a two-square block that adds up to 15, and I try “8-7” and “7-8” and the result shows that both results would work AND would not lead to anything different for the rest of the puzzle, than I can discard both possibilities because the final solution would not be unique were either correct. Therefore the doublet must be “9-6” or “6-9.” Did that make sense? ---
**Author:** ![Zakalwe](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zakalwe/32/270_2.png) [@Zakalwe](https://boards.straightdope.com/u/Zakalwe)\ **Post date:** [August 21, 2014, 4:07am UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/6 "2014-08-21T04:07:57Z")
Interestingly the very first puzzle I did had a block of four (17s down side-by-side with each also being part of a 4-number row (24 and 23)) where the 9s and 8s were interchangeable. So technically, that puzzle had two valid solutions. ---
**Author:** ![Biotop](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/biotop/32/321_2.png) [@Biotop](https://boards.straightdope.com/u/Biotop)\ **Post date:** [August 21, 2014, 5:37am UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/7 "2014-08-21T05:37:10Z")
> [@Zakalwe](#): > > Interestingly the very first puzzle I did had a block of four (17s down side-by-side with each also being part of a 4-number row (24 and 23)) where the 9s and 8s were interchangeable. So technically, that puzzle had two valid solutions. Multiple solution kakuro is bad kakuro in my opinion. Much of the beauty of the puzzle is in the uniqueness of the solution. Kakuro is such a great pure logic experience because with one solution and no matter how dense the puzzle is, you can almost always get there without having to take a sneak peak at the back of the book. ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [August 21, 2014, 1:03pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/8 "2014-08-21T13:03:50Z")
> [@Randy\_Seltzer](#): > > I can’t do them, personally, but my SO is absolutely crazy about them. A couple of years ago, I got her the [Mensa Kakuro](http://www.amazon.com/Mensa%C2%AE-Kakuro-Conceptis-Puzzles/dp/1402739370#) book which she blew through pretty quickly. She said it was a little bit easy. So I picked her up the “Absolutely Nasty Series” numbers two and four… > > …she found “level two” to be about the same as the Mensa book, but level four to be much more her speed. She does them in bed before we go to sleep. Thanks for that tip. Since level 2 of the ‘Nasty’ series is well in my rearview mirror, I won’t bother with the Mensa book. > [@](#): > > I can’t speak to her strategy, but her technique seems to be that she narrows a square down to two or three numbers and writes all of them in. Once she figures out which is correct, she circles it and scribbles out the others. If she immediately knows the correct number for a square, she just writes that in and circles it immediately. So a completed puzzle has a circled number in each square. Sounds a lot like what I do, though I’m inclined to write down the possibilities even when I’ve only eliminated 3-4 of them. > [@](#): > > Occasionally, I’ll interject with “How did you know that was a three?” and the explanation goes something like this: > > “See this square over here?” [indicating some distant square that seems totally unrelated to the one I inquired about] > > “Yeah” > > “Well it has to be either 8 or 9, because it’s 17-in-2.” > > “Okay I get that, but what does that have to do with…” > > “So that means that this one over here can’t be 8, so it has to be either 7 or 1.” > > “…um…” > > “And since it is either 7 or 1, that means it can’t be 4. Which means this one over HERE has to be either 4 or 9.” > > “…” > > “Which is why THAT one has to be three.” Yeah, that’s what it’s like, alright. 😃 ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [August 21, 2014, 1:12pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/9 "2014-08-21T13:12:15Z")
> [@Biotop](#): > > I’m pretty much obsessed with kakuro. I have completed the first three in the _Absolutely Nasty_ Series and am saving Book 4 for vacation next month. I also recommend the “_Second Degree_” Brown and Black Belt series from Conceptis — and the “_Three Dimensions_” book currently available from Amazon – which has smallish but actually quite challenging puzzles. “_Kakuro for Dummies_”, which you would think would be easy, is not. Try it if you can find it. The collections from DJape are also dynamite. Dell’s _Math and Logic_ series is available somewhat irregularly but has several 4 star level puzzles in each issue. Is is just me, or was there a weird disconnect between the first and second Black Belt books? After doing ‘Nasty’ level 3, the first Black Belt book was _easy_ - seemed easier than the second-level Brown Belt book that I’d recently finished, actually. But the couple of puzzles I’ve tried in the second-level Black Belt book, I can’t even get started on. A much bigger jump than between any other two books in the series, IMHO. I figure I’ll save the second Black Belt book for after I finish Nasty, level 4. > [@](#): > > As to strategy, it has been helping me a lot to add up across and down sections of a segment of the puzzle to determine the needed value for one or two squares who’s sum is otherwise unknown. Could you give a for-instance? (Feel free to cite a particular puzzle from a particular book; sounds like we have a lot of the same kakuro books!) > [@](#): > > Another helpful technique I find in the larger more difficult puzzles is to remember that, according to rules, each kakuro puzzle has a unique solution. So say I have a two-square block that adds up to 15, and I try “8-7” and “7-8” and the result shows that both results would work AND would not lead to anything different for the rest of the puzzle, than I can discard both possibilities because the final solution would not be unique were either correct. Therefore the doublet must be “9-6” or “6-9.” Did that make sense? Actually, it did. I’ll have to add that to my mental toolchest! ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [August 21, 2014, 1:18pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/10 "2014-08-21T13:18:52Z")
> [@Biotop](#): > > Multiple solution kakuro is bad kakuro in my opinion. Much of the beauty of the puzzle is in the uniqueness of the solution. Kakuro is such a great pure logic experience because with one solution and no matter how dense the puzzle is, you can almost always get there without having to take a sneak peak at the back of the book. I agree totally with this. ---
**Author:** ![Claptree](https://avatars.discourse-cdn.com/v4/letter/c/58f4c7/32.png) [@Claptree](https://boards.straightdope.com/u/Claptree)\ **Post date:** [August 21, 2014, 6:24pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/11 "2014-08-21T18:24:21Z")
I play a lot of kakuros, though I’ve not bothered with any books yet. [Kakuroconquest.com](http://Kakuroconquest.com) gives me my fix, even though they could do with a couple higher difficulty levels. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 21, 2014, 10:57pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/12 "2014-08-21T22:57:04Z")
> [@](#): > > Could you give a for-instance? (Feel free to cite a particular puzzle from a particular book; sounds like we have a lot of the same kakuro books!) Let’s see if I can draw a diagram: ```auto XX X???X X??X XXXX ``` The Xs are black spaces (where the sums are written), and the ?s are blanks to be filled in. If you add up the vertical clues, you can get the total value of the four blue spaces. If you then add up the horizontal clues, you can get the total value of all five ?s. Subtract the one from the other, and you have the value of the green space, even if you can’t get the rest of the blues just yet. Note that, any time that you can do this with a single “green space”, the puzzle is effectively broken into two smaller, unrelated puzzles: Nothing you do in that corner after that will influence the rest of the puzzle, or vice-versa. ---
**Author:** ![Biotop](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/biotop/32/321_2.png) [@Biotop](https://boards.straightdope.com/u/Biotop)\ **Post date:** [August 21, 2014, 11:12pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/13 "2014-08-21T23:12:27Z")
> [@Chronos](#): > > Let’s see if I can draw a diagram: > > ```auto > > XX > X???X > X??X > XXXX > > ``` > > The Xs are black spaces (where the sums are written), and the ?s are blanks to be filled in. If you add up the vertical clues, you can get the total value of the four blue spaces. If you then add up the horizontal clues, you can get the total value of all five ?s. Subtract the one from the other, and you have the value of the green space, even if you can’t get the rest of the blues just yet. > > Note that, any time that you can do this with a single “green space”, the puzzle is effectively broken into two smaller, unrelated puzzles: Nothing you do in that corner after that will influence the rest of the puzzle, or vice-versa. And you can use the same technique to sometimes find the value of two or more spaces. When you find the missing value for three spaces is say, 7, then you know it is the old 4-2-1 combination in some form. This technique doesn’t work right off the bat in most of the “Nasty” series, but as you begin to solve opportunities often arise. ---
**Author:** ![wolfman](https://avatars.discourse-cdn.com/v4/letter/w/a8b319/32.png) [@wolfman](https://boards.straightdope.com/u/wolfman)\ **Post date:** [August 21, 2014, 11:12pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/14 "2014-08-21T23:12:35Z")
Whenever I see number matrix puzzles, I start trying to solve it in my head. Then I start noting possible similarities to matrices solved with linear programming. So I start thinking if there is anyway to apply linear programming techniques to derive the answer. Then I remember using linear programming sucks Satan’s sweaty scrotum, and I hate the fucking stuff. ---
**Author:** ![Biotop](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/biotop/32/321_2.png) [@Biotop](https://boards.straightdope.com/u/Biotop)\ **Post date:** [August 21, 2014, 11:27pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/15 "2014-08-21T23:27:06Z")
Sometimes a solver can use this method even outside a closed area provided there is an opportunity to close off an area by putting in possibilities in a single or double square, counting the rows and columns and determining how these possibilities work out in the other part of the puzzle. In the \*\*Chronos \*\*example, replace the second X in the bottom row with a question mark. Now if we know that new question mark may be ,for example a 9, 8, or 6, we can now compute the value of the yellow question mark down to three possible digits, one or two of which may prove impossible in the context of the puzzle… ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [August 22, 2014, 10:10am UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/16 "2014-08-22T10:10:14Z")
Cool! This is a _definite_ add to the toolchest. As a matter of fact, in the puzzle I’m working on right now, I think this may be what I need to get unstuck. I’ll let you know how it works! ---
**Author:** ![CT\_Damsel](https://avatars.discourse-cdn.com/v4/letter/c/dfb087/32.png) [@CT\_Damsel](https://boards.straightdope.com/u/CT_Damsel)\ **Post date:** [August 22, 2014, 1:59pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/17 "2014-08-22T13:59:14Z")
My name is CTDamsel and I am also a Kakuro addict! I see Kakuro grids in my sleep! I do at least three every morning. I could not find many books in my local store. I purchased many online; but solved them so quickly my purse could not keep up with my need. :eek: I found a web site called KrazyDad, started downloading and printing them out. (I make a donation once a year.) Then when I got an ipad I still used the puzzles from that site but now do them on the ipad in pdf. I do have a couple of ipad apps for Kakuro; but I still prefer filling them in by hand. One thing I do off the top of my head is quickly scan a puzzle and fill in the obvious (17 w/ 2 spaces 8/9, 24 w/3 spaces 789.) When I find an 8 space I note the missing digit near there with an arrow pointing. In the example of an 41 w/8 spaces the missing digit must be 4. Or 40 w/7 is missing either the 1-4 or the 2-3. ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [August 22, 2014, 3:10pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/18 "2014-08-22T15:10:28Z")
> [@RTFirefly](#): > > Cool! This is a _definite_ add to the toolchest. As a matter of fact, in the puzzle I’m working on right now, I think this may be what I need to get unstuck. I’ll let you know how it works! And the answer is, it worked wonderfully - thanks for the tip, guys! Used it twice this morning in the puzzle I’d been stuck on last night - the first time helped me fill in several new squares, and then petered out (but that’s normal: thinking you’ve finally gotten that big breakthrough, and quickly finding out that no, it was just a little breakthrough before getting stuck again, is a routine experience in kakuro), but the second use of this method gave me enough new info to go the rest of the way. It involves a lot of care in the setting up so you’re sure of what’s in and what’s out of your horizontal and vertical counts, but it’s definitely worth the effort when you’re otherwise stuck. Absolutely a great tool to have for those tough jobs. 🙂 ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [August 22, 2014, 4:49pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/19 "2014-08-22T16:49:58Z")
> [@CT\_Damsel](#): > > My name is CTDamsel and I am also a Kakuro addict! > I see Kakuro grids in my sleep! I do at least three every morning. I could not find many books in my local store. I purchased many online; but solved them so quickly my purse could not keep up with my need. :eek: > I found a web site called KrazyDad, started downloading and printing them out. (I make a donation once a year.) Then when I got an ipad I still used the puzzles from that site but now do them on the ipad in pdf. I do have a couple of ipad apps for Kakuro; but I still prefer filling them in by hand. > > One thing I do off the top of my head is quickly scan a puzzle and fill in the obvious > (17 w/ 2 spaces 8/9, 24 w/3 spaces 789.) > > When I find an 8 space I note the missing digit near there with an arrow pointing. In the example of an 41 w/8 spaces the missing digit must be 4. Or 40 w/7 is missing either the 1-4 or the 2-3. Welcome to the thread! I’ll have to check out that website. And I’m with you on printing them out: until there’s an app that lets me indicate which numbers are and aren’t possibilities for each square, I really need to do them by hand. Narrowing down the possibilities is most of the game, really. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 22, 2014, 8:07pm UTC](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031/20 "2014-08-22T20:07:08Z")
The very first hit if you Google “Kakuro puzzles” is an app that lets you enter the possible numbers. [Next page](https://boards.straightdope.com/t/kakuro-cross-sums-tips-and-tricks/696031.md?page=2)