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05 March 2009 @ 12:03 am
Where's The Beef?  
Difficulty: Harder 7x7

A lot of people in the puzzle community are currently having a negative reaction to KenKen because of the excessive hype from St. Martin's and the NYT. I understood KenKen appearing at the US Sudoku Championship as another logic puzzle to market there, but it felt much more forced when it was a large component of the extra program at the ACPT this year, a likely cause for some puzzle bloggers to state they had mean things to say about the puzzle. I can understand some of that disgust, but that is not why I dislike KenKen.

The one time I managed to attend the ACPT was when Wayne Gould was the honored guest and the focus of Friday night was sudoku, a "non-crossword" puzzle that was garnering a negative reaction from some in the word puzzle community as it loomed as another puzzle that might diminish the great crossword, when in reality it could not and will not. I could ignore the knee-jerk reaction then because I'd come to respect sudoku as a very elegant and addicting puzzle which deserved the success it received after Japanese constructors improved Number Place from a good idea without polish to a superb puzzle (yes, most domestic sources of sudoku are incredibly poor, but I digress). Hearing a similar reaction to KenKen at a similar time post-ACPT is not unexpected from a crowd where the feelings for KenKen will most likely range from, at best, apathy to disappointment or disgust.

My beef with KenKen starts elsewhere, specifically with never seeing a correct mathematical description of the puzzle from the company selling the puzzle around the world. It was created to teach elementary school Japanese students the basics of mathematics without feeling like arithmetic or times tables. This is a valid goal, and I am maybe most interested right now, if I were to choose a career in puzzles, in focusing on using puzzle designs to teach math and science concepts. The problem is that the math is never well described or used for two of the operations, namely subtraction and division.

From one of Will's books: "Cages with more than two squares will always involve addition or multiplication. Subtraction and division occur only in cases with two cells." Why must this be?

Wikipedia says this: "In the English-language KenKen books of Will Shortz, the issue of the non-associativity of division and subtraction is addressed by restricting clues based on either of those operations to cages of only two cells." I got a similar blog comment yesterday mentioning the problem with associativity at play here. This seems to be the company line and people are eating it up. However, there is nothing truly associative [ a + (b+c) = (a+b) + c] at play in KenKen. I'm sorry. There isn't. The cells aren't always naturally ordered, and parentheses can't really exist in a KenKen universe as is. People tend to solve in running tallies anyway so the first time they ID a thing, it becomes the "a" in their equation and whatever they find next is the "b". As long as a + b and b + a are the same, it doesn't matter what order people find/assign things.

The problem with KenKen and operations comes exclusively in the non-commutativity of subtraction and division. Here, the puzzle "cheats" with division and subtraction and never formally says it. a-b is not equal to b-a, but you choose a>b so a-b is positive and matches the value. Incorporating this identification of a maximum cell into the rules, the correct mathematical description of the puzzle should be this:

Fill in an n x n grid of cells such that the digits 1 to n appear once in each row and column. In the bold regions of size j and value v, the cells x1, x2, ..., xj with x1 = max(x1,x2,...,xj) can have a binary operation $ (from the set {+,-,/,x}) applied such that, evaluating from left to right, x1$x2$...$xj = v.

The puzzle is now defined without any inelegance, and in a subtle way where I don't need to separate out subtraction and division and certainly don't limit them to two-cell regions. Once the "largest element" is identified, to make - or / work for any sized region, the remaining $xi operations are commutative (and associative) regardless of the order of xi's for both subtraction and division as they've become the addition of negative numbers and the multiplication of divisors respectively. I'm tired of being told by the creators of KenKen that subtraction and division only go into two cells, and when asked for an explanation, being told associativity limits them. This is not the case.

What does the new rule definition mean puzzle-wise? As is, if I saw two cells with 4 and 2, I could figure out it was valid for 2- because 4>2 and 4-2 = 2. Well, if I have four cells with 5,1,2,1 and a value 1-, I can also figure out that is valid in the same way for 5-1-2-1 (or any other arrangement with 5 first). I don't have to write a puzzle where I say "Subtraction and division occur only in cases with two cells" when that makes no sense as a "necessary" constraint. If removing the identification of the operator is the be-all end-all of KenKen difficulty, I can now have a puzzle where an encountered logical step is never "well, its three cells, so it must be either + or x".

So, that's a real inelegance in KenKen that I've never ever ever ever been happy about where I've seen it presented. I don't have the original Japanese books so I can only comment on the Times (of London and New York) and how they've handled it, but the arbitrary two-cell limitation is everywhere I've looked. The rule on - or / is not specifically given at KenKen.com, the place where NexToy highlights their trademarked puzzle name, but I've never seen a three-cell handling of the operation in either place because I doubt the "Kenerator" can handle it.

[Djape.net and potentially others are ahead of the curve here, but that's not the "highly commercial" form of the puzzle. They also do not use the name KenKen for trademark reasons, so djape having the correct rule-set as I propose here does not fix the problem on the crowded bookshelves.]

So, while there is plenty of room to complain about the hype surrounding the puzzle and also room for some envy of not being able to saturate the market with good logic puzzle books because shelf space is being given to crummy logic puzzle books based on the Shortz brandname (do we really need KenKen books "by the seaside" and "for the coffee break" already?), the incredible inelegance of the math in a puzzle that is supposed to be teaching math has always bothered me. Coupled with the computer-generation that leads to bland, uninteresting puzzles, and you get in my opinion the exact opposite of what the "next sudoku" should look like.

As I'm showing this week, there is some depth available to the puzzle for theming and interesting logic. If you stop compromising 2 of the 4 basic operations, you can have a much more elegant and challenging puzzle. The example above, a union jack for all the beef-eaters out there dealing with my dislike of KenKen this week, has no 2-cell regions, but has all 4 operators used. Before now, you might have thought that impossible, at least if you were only getting your KenKen from the official provider of them.

(PS: For those that find it ironic that a person who could post a puzzle yesterday where he could neither correctly multiply 3^6*5^5 nor correctly orient a division symbol (/) dislikes KenKen because of problems with its mathematical formalism, my defense is that the people I know that care the most about mathematical things are probably the worst at arithmetic. In my many years at Caltech playing Scrabble, I never let the mathematicians tally word scores.)
Mike Selinkerselinker on March 5th, 2009 09:31 am (UTC)
Aw. You broke character.

Anyway, you've hit on the real problem with KenKen, which is the problem with sudoku and the growing problem with many crosswords: take away the human spark, and you're just left with arrangements of ink.

As a side note, I was talking to a non-puzzler businessman yesterday, and I mentioned there was a puzzle type called KenKen, and he said, "Oh yeah, I do those all the time. I love those!" So take that as a single data point that the marketing is working, possibly because it might be a really catchy puzzle type.

motrismotris on March 5th, 2009 04:55 pm (UTC)
The KenKen book I got at the US Sudoku tournament will hold the distinction of being the only book of puzzles I ever finished in less than an hour (45') and also the only book I ever completed during a single flight (a Philly-DC connection). That's 30 seconds a puzzle or thereabouts.

I'm not surprised by this. The puzzle was designed for 7 year olds and if you don't try to make them hard, most adults should be able to do them rather well. While there will be people who enjoy the sense of accomplishment of getting into the puzzle when they first start, the chance of them buying a second or third book will require better puzzles than I'm seeing. Or maybe I'm wrong and the world has forgotten how to do arithmetic and seeing tons of 3+ in two cells and remembering, oh, 1+2 = 3, is fun.
Caziquecazique on March 5th, 2009 05:50 pm (UTC)
I think you guys might be being a little harsh. It sort of sounds like you're saying that from your perspective, this is a bad puzzle, and therefore if anyone likes it they like it for the wrong reasons.

I find your analysis incredibly interesting and valid, but remember that our crowd isn't the only audience for these things and is FAR from the biggest. Will and others are in this as a business, and the things you talked about above simply don't matter to most people who are going to drive it as a business. This is the sort of thing that us crossworders forget sometimes when debating some fine point or looking at pangrams and word counts in puzzles - most people don't give a rat's ass whether their puzzle has 72 words or 74.

Your frustration is completely valid vis a vis your books of what I think of as "artisanal sudoku" vs. the shelves-ful of regular ones. But this is a free market. And while I'm no huge fan of Kenken, I can certainly see doing one - even a regular old boring computer-generated one - as a diversion. And that's what most people do puzzles as - diversions. The fact that something works well as a diversion for a lot of people does not, in and of itself, devalue it. Yes, figuring out a 2-cell 3+ region is not the biggest leap of mathematical or any other kind of logic, but all puzzles don't have to be a Sam Loyd visual puzzle, your sudoku with the states, a Patrick Berry themeless or the "Landslides" crossword to have some merit.
zundevilzundevil on March 5th, 2009 06:19 pm (UTC)
It is easy to get wrapped up in the hardcore puzzler world and somewhat neglect the concerns of the typical solver. Points taken. Two things:

1. This market isn't quite as free as one might like. Sometimes the Kenken push reeks downright of puzzle payola. But even aside from that -- why hasn't there been any real improvement to the puzzles? Why do they adhere to the chumpy rules for division and subtraction? Is there something inherently fun to them? I think the best non-LJ puzzles I'd ever done were parts of the first batch from the good London Times. It's been all downhill from there.

2. On the other hand -- it's not as if there's a bunch of other Kenken producers waiting to fill a possible void. So I guess these books are still the best ones being made? Hopefully this'll motivate someone with writing talent to go make it happen. Tho would the St. Martin's people be interested in publishing a book of handmade Kenken, from easy to very hard?

Well I know I would! There's one purchaser / puzzle evangelist on-board!
motrismotris on March 5th, 2009 06:24 pm (UTC)
I think the first London Times wave puzzles were from the hand-written phase of the books before the Kenerator. There has been some mixing since, but the loss of a human constructor who was laying out a clever path robbed the puzzle of a lot of its potential value. This is not to say a computer-generated KenKen can't have some value for some people. But I'd think you could increase your market by uniformly increasing the quality of the puzzle for more people. As you say, with KenKen trademarked, NexToy/St. Martin's controls the puzzle they put out there under the name and could do more with it, in my mind, to make more profits.
motrismotris on March 5th, 2009 06:19 pm (UTC)
You make many valid points.

There are many puzzles for sale out in the marketplace that I would not buy, but that I know people enjoy doing. The word search is a great example, with a solid market for people who want a relaxing diversion. The issue for me, in part, is that the NYT does not have a word search - it doesn't even have a sudoku - because it didn't think it could have those puzzles stand out from those of other papers. Since that denied a market for artistic, themed sudoku, it is a bit insulting to see this puzzle get pushed into the NYT in a similarly artless way.

People can write elegant, artistic logic puzzles. Nikoli and other Japanese sources have been doing it for years. In this golden moment for number/logic puzzles in the US, I wish the powers that be would take advantage of the opportunity to showcase the possibility of logic puzzles, rather than just using their market power to cash out, and burn out, a fad.
Mike Selinkerselinker on March 5th, 2009 06:23 pm (UTC)
Zeek, I want you to start your own line of KenKenKen books. You could call it "Ken Can KenKenKen, Can You?" I'll bet Mr. Jennings would write the intro.


Edited at 2009-03-05 06:25 pm (UTC)
stigant on March 5th, 2009 09:29 pm (UTC)
While you are, of course, correct that people who find kenken fun are not (and certainly, as motris is demonstrating, it IS possible to create a decently challenging kenken puzzle), seeing a logic puzzle which I consider to be inferior gain popularity AGAIN (I've never really liked Sudoku either), over so many other, more elegant, puzzle types is understandably frustrating. I don't know exactly what it's like in Japan, but I do know that Nikoli has carved out a considerable market for more challenging and elegant puzzles, all or most of which are hand-created (and those which aren't are usually more interesting than computer generated kenken and sudoku). In the states, though, we have nothing BUT sudoku (and now kenken) because those puzzles have monopolized the market to the detriment of more interesting puzzles. It wouldn't be so bad if Sudoku and Kenken were creating a market for other puzzle types, but the gateway-drug syndrome has yet to materialize. In fact, they are suffocating the market for other types of puzzles.

I don't know a lot about crosswords, so forgive me if I'm off base here. But if someone wanted to publish a book of "artisinal" crossword puzzles among all the "low-brow" crosswords, I imagine that their "only" struggle would be making their book stand out from all the rest of the crossword books on the shelf. To be sure, that's not a trivial fight. But such a book would at least be accessible to the average crossword enthusiast since while the particular puzzles may be more interesting (harder vocabulary? better fill? whatever it is that separates an artisinal crossword from a hack crossword) the rules are largely the same.

But if someone wants to publish a book of artisinal logic puzzles (other than sudoku and kenken), they face an uphill battle just to get a publisher to listen to them, let along agree to publish them. THEN they have to fight the shelf battle. And even there, they're at more of a disadvantage than the artisan crossword constructor because people won't pick up their book by accident. Joe-schmoo sudoku enthusiast can't just pick up a book filled with other types of logic puzzle because he would have to learn new rules. Experience and annecdotal evidence suggest to me that typically, people who are comfortable with Sudoku are usually willing to learn some new sudoku theorem (technique) but VERY loath to learn a new set of axioms (and then the theorems) required to start an entirely new puzzle type.

I guess what I'm saying is that marketing a logic puzzle book other than Sudoku is a bit like marketing a Russian crossword book in an English-speaking market. If the majority of your potential market doesn't speak Russian, nay, doesn't even know the alphabet, what chance does such a book possibly have?
(Anonymous) on April 26th, 2009 01:16 am (UTC)
Alright, the puzzle is SUPPOSED to be simple.

In Japanese, the word 'Ken' means simple, modest, or thrifty. Therefore, KenKen can be these words combined.
'Simple Simple,' 'Simple Modest,' 'Simple Thrifty,' ect...

It's named to be simple, alright? Testuya Miyamoto designed it to be simple, but that doesn't mean it can't be made a little harder... right?

Of course... there is the other use of the word to mean sword... but I highly doubt that's what it stands for here...

motrismotris on April 26th, 2009 06:35 am (UTC)
Re: KenKen
My understanding is that 'Ken' in Japanese means wisdom, and certainly as applied here. I have never heard it connected to the word simple.
Craig K.canadianpuzzler on March 5th, 2009 12:50 pm (UTC)
ShortzKen will probably be at least a modest success, because it is a puzzle that has little chance of making people feel stupid, marketed by a puzzle promoter with a track record of not making people feel stupid.

Yes, ShortzKen puzzles will likely have a pronounced trend toward the bland. To some degree, however, that's because of the necessity of having a low risk of making people feel stupid. (The rest of that is probably attributable to the use of computer-generated puzzles which have not been strongly tweaked to produce non-bland puzzles.
Mike Selinkerselinker on March 5th, 2009 03:47 pm (UTC)
Are you using that as shorthand, or are they really called "ShortzKen"?

Craig K.canadianpuzzler on March 6th, 2009 12:43 am (UTC)

I've also decided to use ShortzDoku as shorthand for the Shortz-branded sudoku puzzles as well.
(Deleted comment)
(Anonymous) on March 5th, 2009 08:38 pm (UTC)
Sigh I though coming home in the evenings I could well and truly take off my mathematical hat.

Ok - the "problem" with associativity is that it doesn't fix a unique answer - it gives lots of answers. So for a set of values x_1 , ... , x_n you cannot say that it uniquely satisfies a cage valuation v for - or / ; you can do so with + and X.

So with your example:

(((5 - 1) -2)-1) = (((5 - 2 ) -1)-1) = (((5 - 1) -1)-2) = 1 which is all gravy, but:

5 -(1-2) -1 = 5
5 -2 -(1-1) = 3
5 -(2-1) -1 = 3

However I agree with you that in principle this isn't a problem. You just have to consider that given an n-cage with valuation v, allowing the possibilities the operation is - or / you have to be VERY careful about ruling out possibilities for what the x_i could be.

For example, given a 4-cage with valuation 3-, it might be tempting to rule out 5,2,1,1 thinking this could only possibly be 1; but you would be wrong to do so, as you can arrange them with the operation to get 3 out. Essentially, the publishers ruling out - or / in any of the n-cages (for n > 2) boils down to a restriction of convenience.

For example, imagine a 5-cage with the valuation 1-. There would be a LOT of possibilities for that, perhaps so many so that actually 1- isn't much more of a clue than leaving the box without any label or valuation and simple assuming latin square rules. On the other hand, other values would be more interesting - for example a 5-cage with the valuation 16- [so 6 - (1-6) - (1-6) =16] ...

Large cell with division are more restrictive but I'm not going to give that more thought for now

Finally, all of this discussion reminds me of the key to a little teaser Simon shared with me at the WSC in Prague: Using only the 4 arithmetic binary operations, make 24 using precisely the set {1,3,4,6} - so 4x6 isn't a solution because 1 and 3 remain unused.

motrismotris on March 5th, 2009 08:59 pm (UTC)
My problem with the associativity argument is that it does not apply to how the operations are being performed in this puzzle. Before we bring in magic parentheses that don't make sense, we have to answer how you get - and / in two cells anyway. You get this by saying there is a largest number, and you subtract/divide from it. This solves the commutation problem for two cells. It also says you always use a largest x1 as the starting point. Now, every other non-largest number is actually a (-xi) in the grid and you do it like addition. 5 + (-1 + -2) + -1 = 5 + -1 + (-2 + -1) = 1 and so on. It is associative and commutative. Don't say you'd be confused and do 6 - (1-6) - (1-6) = 16 when (1-6), from the two cell definition, isn't a valid subtraction in KenKen to begin with. KenKen is afraid of negative numbers.

Identifying such a max cell can be an interesting step and clues including it must be at least (v+j-1) in my definition for subtraction, and at least v in my definition for division. If you had a 3- in 4 cells and identified a 1 and 2, you now have a 6- left in 2 cells, with a big number still unidentified. I know this because 3 > 1 and 2. Its not that difficult, but is fully consistent with how the rule should operate.

The {1,3,4,6} is a classic puzzle so I unfortunately know the answer already, and Wei-Hwa at some point, in college I believe, made a generator to find other good ones. He started his Puzzle Challenges with {3,3,8,8} which is another challenging option even with the duplicated numbers.
motrismotris on March 5th, 2009 09:19 pm (UTC)
To follow up, you basically need a deterministic definition of how to subtract and divide. I've given such a definition. For any v, any j, and any x1 to xj, I only ever return one value. The KenKen rules without such a clean rule are tremendously inelegant and lead to this "argument" we are having about two of my favorite operations.
(Anonymous) on April 22nd, 2009 05:53 pm (UTC)
Returning one value
From my understanding the subtraction and addition rules return only one value just as well as the addition and multiplication rules do. Take a two square cage of a larger puzzle, say 10x10, and that for the cage you have 12x, then the cage could be 3 and 4 or 2 and 6. I think that maybe what you mean to say is that you can right an equation for the square and a*b=12, while in the case of subtraction and division, say 2- was given for the cage, you wouldn't know if you had a-b=2 or b-a=2, though in both cases you don't have enough information to know which of the two numbers goes in which spot in the cage without relying on the row and column rules. I'm pretty sure that the rules of subtraction and division are well defined though and that they were accidentally ignored by some other posters by using parentheses that were actually changing what operation was being performed on some of the numbers.
(Anonymous) on March 5th, 2009 11:00 pm (UTC)
If you want -,/ to act as closely as is possible to +,x then that's what you've achieved. The motive for doing so is that actually + and x share similar properties already and so I guess there's a vague "homogeneous" feel to the puzzle.

It's another argument to say that puzzles without this strong sense of homogeneity are somehow inelegant though. It's true that the inhomogeneity here somewhat reduces the amount of information a clue can give you - but I don't think that's necessarily terminal. Perhaps something can be made of different operations having wildly different behaviour? I haven't looked into that so couldn't say for sure either way.

For example, placing the maximum constraint on commutativity solves the problem of having intermediate parentheses which don't make sense - but going back to my silly set of examples, both (5-2)-(2-1) and 5 -(2-1)-1 both satisfy that constraint. And in your example what's to stop the assumption that the remaining two satisfy 4- rather than 6- [so 4 - (2-1) = 3]?

Like I say, your interpretation provides a nice sense of homogeneity - but - that doesn't mean more exotic interpretation of the rules shouldnt be looked into. (Lack of) associativity and commutativity arguments basically give sets {x_1,...,x_n} multiple valuations under a certain operation, rather than just 1. This has the effect of having less deducible information associated with that operation - but by no means leaves the puzzle undefined, as these publishers claim.

motrismotris on March 5th, 2009 11:25 pm (UTC)
An elegant puzzle design should have a simple set of rules that can be uniformly applied. A puzzle in mathematics that uses 4 operations, but in different ways, is certainly inelegant to me and is not a separate argument. I also do not think a "basic" puzzle in arithmetic should allow for multiple interpretations of how to perform its operations so my deterministic subtraction far outweighs (5-2)-(2-1) and 5-(2-1)-1 which might exist in crazy variant space, but not KenKen-space.

Sudoku has a simplicity to the statement of its rules. Fill in the cells with digits such that in each row, column, and 3x3 region the digits 1 through 9 appears once each.

Consider the bizarro world where Ukodus was the big puzzle with this ruleset: "Fill in the cells with digits such that in each row and column the digits 1 through 9 appear once each. In the 3x3 regions the sum of the nine digits will be 45." Somehow, ukodus (while having a broader puzzle space than sudoku I guess) sounds like a much less elegant puzzle to me. It adds in math where the rest of the puzzle doesn't have math. It almost describes 1-9 appearing in those 3x3 regions, but not cleanly. A very, very, very minor tweak to the rules of ukodus fixes the puzzle to have a more elegant rule and a better puzzle.

Somehow, I feel I'm in a bizarro world with KenKen and am just waiting for those NekNek puzzles to come out that understand subtraction and division.
(Anonymous) on March 5th, 2009 11:57 pm (UTC)
Haha there was a reason I guess I mentioned commutativity and associativity being no-no's for division and subtraction yesterday - this has opened up a big can of worms. Talking about elegance sort of blows the can up - although i'm certainly not saying homogeneity isn't elegant - and for what it's worth i believe what you've done offers a small improvement to the original formulation.

However, a lot of what i have said has been pure exploratory hypothesis. I still remain sceptical as to the value of kenken as a "proper" logic puzzle to enjoy, rather than some sort of brain training (or i guess arithmetic teaching) exercise. I don't think that even the possible emergence of any sort of nekneks will change this for me.

(Anonymous) on April 22nd, 2009 06:07 pm (UTC)
Simplifying and solvability
While that would be simplifying, I don't think that it is needed. That is, you don't need to have the largest number be first in subtraction. I've been working on a project on KenKen for class and haven't had any trouble working just under the minimal rules at http://www.kenken.com/. While they do not present cages larger than 2 for subtraction and division, they are not outlawed and work fine under the same rules. They also do not use negative numbers or fractions in their cage requirements though it is perfectly valid under the rules and I have been considering them for my work. Actually you can turn the kenken board into a system of equations, but for subtraction and division there are multiple possibilities for what equation to use. If the puzzle was well made though, that is it has a unique solution, then only one of the multiple equations for a subtraction or division cage will result in a solvable system with the other equations.
(Anonymous) on April 22nd, 2009 05:40 pm (UTC)
I'm pretty sure that you're making rather liberal use of parentheses. From my understanding of the rules the order of the subtraction is not important, but by adding parentheses the way you are you are actually adding numbers...I mean just distribute the minus sign (negative ones that is), which means that you're actually adding some of the numbers and breaking the rules. I thought about this a bit because I'm studying KenKen for a class and I'm pretty sure that the number of different possible answers is then the same as the number of different elements in the cage because only changing the starting number will change the difference. Everything that I just said about subtraction holds for addition as well. This still doesn't tell you enough to solve the cage though, but I feel that the idea is that you should have to also use information from the rules regarding no repetition in rows and columns. Applying these rules to 2x2 squares and consideration rotation redundancy of cage rotations I think that there are exactly 14 such puzzles with unique solutions, which is counting the two trivial cases where each square is given. This calculation also depends on the number of 2x2 Latin squares, which is trivial for the 2x2, but is very significant for larger games.
(Deleted comment)
motrismotris on March 6th, 2009 05:48 am (UTC)
Is that observation a jrivet original? Its both cute and apt, given my recent problems.
(Deleted comment)
stigant on March 6th, 2009 01:14 pm (UTC)
>> Limiting the operations to + and x would probably be more elegant, but less interesting.

I don't know about that. I really enjoy Killer Sudoku which only has half of those options. There's plenty of interesting deductions and techniques to be had from just addition.
Henrytahnan on March 6th, 2009 04:08 pm (UTC)
I wonder whether what you're saying about commutativity and so forth is true in a pure-mathematics sense but not in an applied-mathematics sense. (Or, really, pure-arithmetic and applied-arithmetic.)

That is, what you're suggesting one should do in all four cases is "pick the largest number and then {add/subtract/multiply/divide} the rest of the numbers {to/from/by} it". That's uniform; but I don't get the sense it's what people do. Faced with a bunch of numbers, I can "add them" or "multiply them" but I can't "divide them" or "subtract them". That's almost a fact about word usage as opposed to math—if I'm working on something, I can call out to someone nearby, "Hey, could you add 5, 8, and 16 for me?", but not "Hey, could you subtract 5, 8, and 16 for me?", and similarly for multiplying and dividing.

So while as a puzzler, I have no trouble getting my mind around the variant rule you're suggesting, I can entirely understand a nonpuzzler who look at a three-square "3/" wholly differently than they look at a three-square "10+". (If you'd like to test this empirically, albeit with a small data set, my wife kind of likes KenKen even though she's not the kind of puzzle solver we are; perhaps I'll give her today's, though perhaps a mediumer or easierer puzzle would be preferable. Preferabler.)

Actually, larger regions are another corollary of this. Certainly as a pedagogical tool, "2278125*" is probably more than an elementary school student wants to deal with; it's almost certainly more than my wife wants to deal with. Even "50+" is a lot to keep track of casually, even though the math is faster.

So, right. In some ways, it's what Cazique was saying.
motrismotris on March 6th, 2009 04:30 pm (UTC)
For the casual solver, I'd just hand over the 3/3/3*3 puzzle and leave it at that until my GAMES KenKen PuzzleCraft is out. Its the only one I sent on to Willz, and he liked it (despite the initial feeling of intimidation when looking at the 3000x).

I suppose what I'm trying to answer for myself this week by exploring KK construction is what is the value of KK as a puzzle for people at my level of solving, where I want to see either more difficult, or more interesting, applications of the mathematics. You are right that there is no great way to get all 4 operations into any puzzle that works for how all people do math in their head, but to extend it as a puzzle for WPC-type competitions and such, I'd imagine my adjustment is the one to go with.
motrismotris on March 6th, 2009 04:35 pm (UTC)
And indeed, your pure versus applied-arithmetic thinking is maybe the right way to frame the problem. Still, if you just rephrased your expression as "Hey, could you subtract 5 and 8 from 16 for me?", I think that's a sentence most people could deal with. An uncommon request, but possible. If I heard "could you subtract 5 and 8" I'd say -3 and I bet I'm in a minority as most would say 3.
Jammermeeko713 on March 6th, 2009 08:05 pm (UTC)
hey. just wanted to let you know that i did union jack. and it was awesome. thanks.
devjoedevjoe on March 6th, 2009 08:52 pm (UTC)
Detailed solution (a significantly harder puzzle than the 3/3/3*3 and great divide puzzles)

Start with the 8 single-cell regions, and the center of the 50+ which must be a 6 since 50 is the sum of the digits 1 to 7 twice each, minus the number in the center that only appears once.

Now we have a bunch of regions that are only partly solvable:
252x is the product of 7,6,6,1 or 7,6,3,2.
4- is 7-2-1 (with the 7 in the left cell since the column for the other cells already has a 7) or 6-1-1 (with the 6 at the bend).
2/ is 6/3/1 (3 in top cell) or 4/2/1 (4 in right cell) or 2/1/1 (2 at bend).
1/ is 6/3/2 (3 in top cell) or 4/2/2 or X/X/1 (unique number at corner). But the corner is limited to 1, 2, and 6 by the previously placed digits, and any like pair is similarly limited, so only 6/3/2, 6/6/1, and 2/2/1 are possible.
2- is 6-2-1-1 or 7-3-1-1 or 7-2-2-1, in each case with a pair of like digits on a diagonal.

None of these possibilities for the 2/ or 1/ allow a 5 to appear, so the 5 must go in the center of the 5th column. Likewise, the only place for a 4 in the 5th row is in the middle.

Now in row 6, there must be a 4 in one of the 4-cell regions. But the 2- cannot contain a 4, so the 22+ must contain a 4. In row 2, there must be a 4 in one of the 4-cell regions, but it cannot go in the 252x, so it must go in the 18+. The same logic applies to the 5s in columns 2 and 6; the 18+ and 22+ must both contain a 5. The 5s and 4s already placed prevent either of these regions from having two 4s or two 5s. So the 18+ can only be 3,4,5,6 or 2,4,5,7. The 22+ can only be 4,5,6,7; the 5 and 6 can be placed, while the 4 and 7 go into the upper cells in some order.

Since both of the bottom two rows now contain a 7, the 2- cannot contain one; it must be 6,2,1,1 with the 6 in the upper row. Also, since there is a 6 in the last column, the 252x cannot contain two 6s; it must be 7,6,3,2 and the 7 and 2 can be placed, which forces the 4 and 7 in the 22+. And now there are two 7s in the top two rows, so the 18+ cannot contain one; it must be 3,4,5,6, and the 4 and 6 can be placed. In turn, this forces the 3 and 6 in the 252x, the 3 and 5 in the 18+, all digits in the 2-, and all the digits at the outer ends of the arms of the 50+.

Now the 4- (which must be 7,2,1) can be filled, and then the 1/ (must be 6,3,2) and the last remaining cell in row 6. Elimination can now be used to fill the remaining cells. The clues for the remaining 3-cell groups check (including the 15+, which was not needed at all for the solution).

motrismotris on March 6th, 2009 09:13 pm (UTC)
Thanks for the detailed solution. Mine actually let you get the four 3-cell regions before the four 4-cell regions so its nice to see there is a bit of flexibility in the path. Its a bit ironic you mention the 15+ is not needed at all, when it is the second step in my path. After you place a 5 in R4C5, you'll see R2C3 needs a 5 as well as there is no other space for it in that column. The remaining digits must be 64 or 73 but the row eliminates the 73 and the column forces an ordering of the 64. This then helps you work both down and across to get the rest of the triples.
(Anonymous) on March 7th, 2009 12:26 pm (UTC)
on automatic generation
After reading the NY Times article on KenKen I tried to write a
decent "Kenerator". I didn't know about the rule that says
subtraction and division cannot be in cages with more than two
cells. The only restriction I used was that with subtraction the
result cannot be negative, and with division it cannot be a
fraction. The numbers in the cells should produce the result when
put in a certain order (so yes, 5, 2, 1, 1 can produce 1 with
subtraction, of course also using parentheses is out of the

Up to yesterday my way of generating easier and harder puzzles was
too simplistic: the easy version would have some cells of size 1,
and the hard version could not have cells of size 1. But what really
influences the difficulty is the number of possible solutions for
each cage. Especially cages with the subtraction operator usually
have lots. The challenge here is, given a KenKen puzzle, how to
compute a difficulty rating? I've come up with a heuristic solution
based on computing the number of possible solutions of each cage,
which seems to ensure that the easy puzzles really are easy, and
vice versa.

A next step is to try and make the puzzles more "human", perhaps by
introducing fixed interesting shapes from a library of such shapes.
However, with large cages the computation of a difficulty rating
becomes problematic.

For those willing to try some computer-generated ones, they're at:

The puzzles before March 7 have the less reliable difficulty rating,
the ones starting from March 7 should be better in that respect.
Any ideas on how to compute a good difficulty rating are welcome :-)


motrismotris on March 7th, 2009 02:37 pm (UTC)
Re: on automatic generation
Thanks for your comments on how you are approaching puzzle generation. Even with sudoku, difficulty rating can be a real challenge, but the main method is to use a formalized set of logical steps with a scoring function for the kind of steps used. Most graders are fairly consistent now, but the relative difficulties are often messed up primarily based on how naked singles, which are often hard to find, are considered compared to hidden singles, what most people use to solve.

With a KenKen, I'd imagine the kinds of steps that need evaluation are the number of possibilities for the cages as well as how often you need to find a single (this is the only place a 5 can go) in a row/column as well as if/when you have pairs/triples of numbers that interact to eliminate choices. A cage that can be 13/31 or 14/41 places an imaginary 1 in the row that interacts with other groups. It also excludes 2 and 5 to n from those cells too. Deciding how these steps should be scored, however, is non-trivial but should improve your grading.

Also glad to read that you stumbled into the best definition of subtraction and division (in my opinion) without being told you could only use two cages for these operations.
garethmoore on March 9th, 2009 02:51 pm (UTC)
Re: on automatic generation
In terms of difficulty rating of puzzles, one thing I've always assumed (but never explicitly tested because it seemed like a lot of effort for little reward!) is that it's not just the step-by-step difficulty of the easiest path through a puzzle (however you choose to define that) that should determine its overall difficulty rating, but rather the number of simultaneous useful deductions that can be made at each stage - and that in turn needs to be normalised by the size of the search space. So in other words, "how likely am I to find a [not 'the'] next move?" (given the chance of spotting one and the complexity of the reasoning required)

This must be relevant, at least at some level, because it stands to reason that a Sudoku that can be solved using (say) 50 hidden singles where there is only one possible placement at each stage must be harder than a similar puzzle where there are several possible placements at most stages.

The relevance of the size of the search space must also increase proportionally to the difficulty of the logic used, since the relative 'difficulty' of spotting a naked single is very high in a mostly empty puzzle, but much lower when the puzzle has very few squares left to fill.

To assess this using a computer solver is not trivial, however, since you'd need to decide what counts as a different path (eliminating two pencilmarks in different orders but essentially at the same time probably doesn't count, for example) and also search for and merge paths when they reach the same state. Plus you might even need some psychological weighting in there - for example, do the majority of casual solvers start with the digit '1'? And that's before you even start considering how to score and weight the myriad of possibilties and combinations.

So, counting the number of givens it is then... :)
motrismotris on March 9th, 2009 04:39 pm (UTC)
Re: on automatic generation
The breadth of a solving path is truly a very important consideration when it comes to "timing" how long a person will take to solve a sudoku. I'm not sure a lot of computer raters use this consideration at all when they evaluate difficulty, however, which is why they can't evaluate "time" of solve well (that, and different people solve the puzzle in different ways)
(Anonymous) on April 22nd, 2009 06:13 pm (UTC)
Re: on automatic generation
I don't think that it is valid to exclude negative number and fractions. While I have not seen anyone else actually use them they are not excluded in the rules and I have not had any trouble considering them in my work, though I understand the additional complexities involved in including them in your Kenerator.
Bram Cohenbramcohen on March 17th, 2009 02:19 am (UTC)
I'm finding these a lot less difficult than your and other peoples's ratings indicate. But they also take a lot longer than sudokus, and are generally more interesting and novel. It does feel like there's a lot more thinking and a lot less staring at it and waiting for something to pop out. Maybe I'll find them a bit routine after a while though. The extended division and subtraction definitely adds a lot to it.
motrismotris on March 17th, 2009 02:41 am (UTC)
I feel I learned somewhat well how to rate sudoku puzzles based on both my solving experiences with them, but also on watching other people solve them, isolating the different basic techniques people try, and seeing the commonalities in where solvers struggle. The most common error in computer ranking is with overestimation of naked single detection, for example, mostly where digits are not dense in any of the row/column/region but there is certainly just one possibility left for a cell.

With KenKen, I'm pretty much guessing how well other solvers can handle searching the possibility matrix of different values/operations, and I am likely underestimating that skill in general in my audience. What I have tried to do up to now though is to always leave a trail of breadcrumbs in each puzzle - good break-in points is what I guess I mean - that will lead to an answer. Getting a fair, but hard to spot, break-in point takes some doing and I am working at making better "fair" but very difficult puzzles.
hoskeebo on May 16th, 2009 09:27 pm (UTC)
Very impressed
Wow, this is a great post, and you've gotten some great comments. I've only been a passing puzzler, until I came up against Kenken. It's always bothered me, though, that it's been generally treated as a "diversion," and the math has not been exploited to much good.

I'm not a mathematician, but I love math, and had been disappointed that I couldn't find a discussion like this one, until now. This is wonderful food for thought. Thanks. I can't wait to to twist my brain with your Union Jack.

I've made some free videos about how to start solving a basic 4x4 Kenken, as well as an advanced 9x9 KenKen in which the operation signs are not given.

You can check them out at:

davidlevylondon on June 20th, 2009 11:26 pm (UTC)
Comments on KenKen from David Levy

First may I introduce myself. I am the person in the KenKen team who is responsible for providing our puzzles to newspapers, web sites and book publishers, etc.

There have been several comments about KenKen on your site during recent months, for which we are grateful. Not only are we willing to learn new ideas for improving our puzzles, we actively want to learn and to improve. So many thanks to those of you who have made constructive comments. The purpose of this response is partly to explain why we do certain things our way, and partly to encourage this forum to continue to make suggestions. I shall respond by picking out a few quotations from the various postings and elaborating on them.

There has been a criticism that we have never published “… a correct mathematical description of the puzzle”. This is true, but we are not attempting at the moment to cater for mathematicians who are expert puzzle solvers. For every one of you who fall into this category there are literally thousands of people who enjoy solving puzzles without even thinking about the mathematics involved. While KenKen is still in its relative youth we must focus on our core of enthusiasts.

There has also been a criticism of the two-cell limitation where we have subtraction and division symbols. This is quite deliberate, because we are following the KenKen rules of the puzzle’s inventor, Tetsuya Miyamoto. Adding the “completeness” of multi-squared cages that use subtraction and division is certainly something that can be considered as a possible future variation on KenKen, but we have not yet seen any significant level of demand for such puzzles. As the number of KenKen enthusiasts grows we will be better able to assess whether that variation is one that will meet with enthusiasm from a significant proportion of our core solvers. This response should not be taken to mean that we are not open to new ideas for variations on the original KenKen theme. In fact we originated the KenDoku variation fairly early in the life of KenKen, and two books of KenDoku puzzles have already been published (same rules as KenKen but with the additional block restrictions of SuDoku). Additionally, we are currently programming the Kenerator for another (and more difficult) KenKen variation and plan to launch that when we feel it is appropriate.

There has been some comment in your forum on the disadvantage of not having a human expert to tweak the puzzles as they come from the Kenerator. I agree that hand tweaked puzzles might offer a little more to those of you, the one-in-a-thousand solvers, who are able to tell the difference. For this reason, when the puzzles were being generated in Japan and published in small numbers, Mr Miyamoto himself would tweak the puzzles before they were published. But the demand for KenKen has long passed the stage where that would be a practical proposition. For example, one day this week I had to deliver 750 puzzles to just one of our customers and a few hundred more to others. With numbers like these you can understand that hand tweaking is no longer an option because of time and cost reasons, unless we could find sufficient volunteers willing to donate their own time!
There are of course other ways in which we could make the puzzles more challenging for the serious puzzle addicts, but as the difficulty level of puzzles increases so the number of enthusiasts able to solve the puzzles diminishes, so we have to be careful about when we introduce puzzles at higher levels of difficulty. We are looking at this aspect of KenKen right now, examining the statistics for the different levels of puzzles on our web site and others, and as soon as we feel that the time is right we will offer the equivalent of SuDoku’s fiendish and super-fiendish puzzles.

I would like to add an appeal to the above comments. Please continue to make suggestions, either here or by sending email to us at customercare@kenken.com or from our web site www.kenken.com, so that we will have even more ideas to consider for future variations on KenKen. Your ideas can help us to expand the world of KenKen and at the same time to improve the logical thinking and math skills of enthusiasts of all ages.
motrismotris on June 20th, 2009 11:44 pm (UTC)
Re: KenKen
Thanks for your many comments and indeed I also encourage further discussion on this puzzle type as time passes. Sudoku had ~25 years of polish before it became an international sensation so it is undoubtedly going to be the case that this puzzle type has some room to grow.
Mike Selinkerselinker on June 21st, 2009 07:30 am (UTC)
Re: KenKen
David, let me introduce myself as well. I'm Mike Selinker, a fairly well known American puzzle designer, and Thomas's co-author on Games magazine's PuzzleCraft series. I find some of your comments interesting because I think they show the same level of problematic logic that has bedeviled the crossword in recent years. While I don't expect KenKen to follow the same path of a century-old puzzle type, I think some lessons can be learned. I'm going to give you some unsolicited advice, which you can either take or leave as you please. It's just my opinion, but I think it's an informed one.

The American crossword went through three phases. The first was a long period of handcrafted but largely (by today's standards) uninteresting puzzles. Then Will Weng came to the New York Times and Stanley Newman to Newsday and Will Shortz and Mike Shenk to Games, and over a 20-year period, handcrafted puzzles were interesting. I'd call this the golden age of crosswords. But then some enterprising puzzlemakers like Eric Albert crafted lists of words, and computer programs were written that became much faster and much more capable of filling a wide-open grid than a human. As the number of crosswords ballooned, crosswords returned to their largely uninteresting state except in a number of places where handcrafted quality is still appreciated. Those places, like the New York Times for example, are large enough that crosswords still have a cachet of being interesting, despite the average crossword being rock-dull.

By what you've said, KenKen rapidly moved through all three of those phases. What Thomas is trying to show you is that while there are quite a few solvers who are content with average KenKen, many of those solvers are the same ones that want better than average crosswords. Specifically, the New York Times caters to those people. So if you're going to remain a relevant puzzle type with those who like variety in their puzzles, you might consider listening to what Thomas has to say (and asking Thomas's help in implementing it). Because otherwise, those very sharp readers of the New York Times will likely fall away from your puzzle type, and demand that Will seek something they continue to find interesting.

Whether you meant it this way or not, your answer to Thomas reads like someone who is content with what he has. You certainly can be, given how popular your puzzle has become. But Thomas is showing you a way to both keep your mass-market audience and improve your puzzle type's standing among the hundreds of thousands of people who prefer a richer puzzle experience. It's good advice, and Thomas is the best person that I know to dispense it.

Just my (1x2) cents.

davidlevylondon on June 21st, 2009 08:49 pm (UTC)
Re: KenKen
Thanks very much for this input Mike. I understand what you say and it all sounds sensible.

We certainly want to learn and improve. After we have launched the next variation of KenKen would be a good time for us to take further input and discuss what should be the next stage.

Mike Selinkerselinker on June 21st, 2009 08:59 pm (UTC)
Re: KenKen
Cool. I'm a big fan of puzzles getting the best airing they can, regardless of who comes up with them. Thanks for listening.