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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.)
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.