The New York Times Crosswords

This is about crossword puzzles and people who do them. We specialize in making molehills out of mountains. If you find this article interesting, there’s probably no hope for you either. Article in My Notes. (Cross-reference)

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Hard Sudoku

A “Very Hard” Sudoku that turned out to be pretty easy …

The San Francisco Chronicle’s Saturday Sudokus are rated “6 stars”, and are considered very hard. Many weeks, they won’t even “validate” when I load them into my Sudoku program, which supposedly indicates there is no unique solution. Some weeks, I don’t even bother trying these puzzles. My experience with this one was different.

Newpaper Sudoku

In solving Sudoku, the generally accepted approach is to begin by filling out the obvious entries that you can “eyeball”. Once you’ve exhausted the empty squares that can plainly hold only one number, you’re left with squares that could be any two or more different digits, but, usually could not be some other subset of digits without violating a rule.

To the left, it is easy to see which numbers are “given” in the puzzle, and which numbers I have penned in. For these, I use a non-erasable black pen because it is easy to see, and because there is very little chance of making a mistake at this stage. I’ve been playing the game for about half a year, but I was surprised at how many numbers I was able to fill in this way today.

Once you fill in a number, it forms a new clue for our puzzle, and you can build on that. The ‘2’ in the top right cell (3×3 grid) is obvious: a ‘2’ can only go in that cell’s middle row, and there is only one open space in that row.

The ‘2’ in the lower right is much less obvious and you can skip this if it gets difficult. Where can the ‘2’ in the middle right cell go? I don’t know, but it can’t go in the middle row because I already had a ‘2’ in that row. It can’t go in the left column because of the ‘2’ in the top right that we just added. So it could be in either of the empty squares in the middle column.

Looking at the bottom right 3×3 cell again, now we know the ‘2’ can’t go in the middle column or the left column, and it can’t go in the bottom row. We already had the ‘9’ position. The ‘2’ has to go in the upper right square. It takes a little practice before you can “see” that without penciling in the possibles with teeny little numbers.

The newspaper scan above represents my best ability to jumpstart the solution. Sharper or more experienced players will undoubtedly do better. The next step is to pencil in “possibles”: the possibles for the lower right square of the lower right cell are, by simple elimination, {7,8} at this point.

Unfair Puzzle warning!

(1) Pencil Tips filled in:

Filling in the pencil tips

However, my Pappocom PC puzzle solver doesn’t think this is a fair puzzle. What an ego boost! But not every program can be perfect. At least it accepted this puzzle. For this article, I used the program to fill in all my “pencil tips” at once. If your eyes are very good, you can do this on the newpaper or puzzle book, but my eyes aren’t that good, and I make too many mistakes.

I already solved this puzzle, but here I have reverted it and re-filled in my pencil tips with the “possibles”. You will get a few “freebies” like the ‘8’ in the middle 3×3 cell, since that middle square can’t contain any of the other numbers. [Why not ‘7’? See “(3) Tricks and Rules” below].

This means you can eliminate any other 8’s in that middle 3×3 cell, and in that middle row, and in that middle column. In the row where we can fill in the ‘8’, we now also know which square is the ‘1’ and which is the ‘7’, further eliminating 1’s and ‘7s in their associated 3×3 cells, rows, or columns:

(2) Partial Solution:

Partial Solution

Rationalizing the 2 and the 8 Filling in the ‘1’ and ‘8’ we found for the middle row (fifth row down), we’ve eliminated the associated 1’s and 8’s, since a number can only occur once in any row, column or 3×3 cell.

This reveals a definite solo ‘4’ and a definite ‘8’ in the middle and bottom right-hand cells, respectively.

Filling these in, and further eliminating possibles that are no longer possible, allows us to complete the middle right 3×3 cell. Given the single ‘4’, can you see which is the ‘2’ and which is the ‘8’?

We’re well on our way to solving this puzzle. There is not much more we can do with the right-hand 3×3 cells until we complete some of the other grids. In the upper left cell, there is an extra ‘4’ pencil tip that should not be there, in the middle square, since there is already a ‘4’ in that column. But fixing that error does not give us a new grip on the puzzle.

If you are intending to try to complete this on your own, I might next look for the ‘2’ in the fourth row. While you’re in that middle cell, look for the ‘7’.

When you complete a square, don’t ever forget to rid your pencil tips of eliminated possibles right away, or the patterns will be much harder to spot.

(3) Sudoku Tricks and Rules

There are a number of logical rules one can memorize and practice to help speed solutions. I am learning them now, and at some point will publish my own notes. But they are just rehashes of compilations I have already found elsewhere. Visit for an excellent set of tips, and through this site, or Google, you can also find other excellent sites with tutorials.

I do not think I found it necessary to use many of these tips to solve this puzzle. There are reports of young kids who solve these puzzles without tricks or tips just by looking at the possibles and eliminating impossibles, one square at a time. There is also an excellent article in the current Scientific American, The Science Behind Sudoku, which gives the clearest expression of the general principles that I’ve seen yet.

The only example I will discuss here is PAIRS:

Naked Pairs:
If two cells in a group contain an identical pair of candidates and only those two candidates, then no other cells in that group could be those values. [Solving Sudoku, Angus Johnson]

(The term “group” is always used to refer to any row, column or 3×3 cell of 9 squares.)

In our Sudoku partial solution above, we have a pair of {4,5} possibles in the middle 3×3 cell. We don’t know which square is the ‘4’ and which the ‘5’, but we do know that no other square in the cell can be a ‘4’ or a ‘5’. The pairs rule means we can eliminate the ‘4’ in the {2,4,7} square, so this square is really only a {4,7} — it can really only contain a ‘4’ or ‘7’.

But, this is the only square with a ‘7’ in the middle cell, so, the pairs rule doesn’t help tell us (this time) which is the ‘7’. We already knew that, since no other square in that cell (or in that row) was a possible ‘7’.

In reviewing this article, I found a PAIR which was used to solving the puzzle. Look at figure (1) again where we have penciled in a lonely ‘8’ in the center 3×3 cell:

We know that this square can’t be a 2,3,9 or 6 because its row already contains these values. It can’t be a 1, 4 or 5 because its column already contains those values. That leaves a ‘7’ or ‘8’, so why can’t it be the ‘7’?

This column contains a pair of {6,7} possibles. By filling in our pencil tips for the column, we have learned that one of those squares must be the ‘6’, and the other, the ‘7’, though we don’t know which is which yet. Since one must be the ‘6’ and one must be the ‘7’, we do know no other squares in the column can be a ‘6’ or a ‘7’, as the Pairs rule formalizes in logic. So, by elimination, we know this square can’t be a ‘7’. It has to be the ‘8’.

(4) Solved Sudoku:

I finished this one in the PC puzzle application 15 minutes 29 seconds, which is pretty nearly a personal record for me for any difficulty level. That doesn’t include “pencil time” for “obvious” entries on the newspaper itself. There is no exact order required for a solution. I don’t always fill in all the pencil tips at once, but only for the cells, rows or columns on which I am actively working. This helps reduce clutter. When I can go no further any other way, I continue filling in pencil tips until the whole puzzle is filled.

I don’t know why this puzzle was rated “very hard”. There must be a mathematical algorithm or reason for the difficulty ranking. Whatever the reason, it is a great confidence-builder. Any puzzle which does have a unique solution can be solved by using the simplest logic and rules.

It is possible to use the PC Sudoku program to “cheat” by guessing; this particular program will flag an erroneous entry in red. But I try not to do that, and didn’t have to guess to complete this solution.

I loaded the original newspaper puzzle into the Pappocom Sudoku program. The program displays the original puzzle clues in blue letters. My solved entries are in dark gray.

Finished Puzzle!

Alex Forbes ©May 27, 2006

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Sudoku Puzzles

We were late getting into this worldwide favorite. We’ve done crossword puzzles for decades. Recently we paid attention to a numbers “crossword”. The first one we tried had a difficulty rating of five stars (hard). It took several hours and we had to cheat a couple of times, but we solved it.

On easy puzzles (one star), after we learned the rules we could pencil in the solution. A few erasures, half an hour, and we’re done.

On difficult puzzles, we’re still printing out blank forms with plenty of room to pencil in our “pencil mark” guesses. The newspaper forms are too small, and they don’t like erasers very well. We’re getting better, but a five-star puzzle still takes us several false starts and several hours.

Because these puzzles can be so challenging, we did a web search to see if anyone had posted tips for solving. It turns out – wouldn’t you guess – there are several dedicated sites. At you can also order a program that calculates solutions, helps you solve without cheating, and (if you prefer pencil and paper) print out today’s puzzle.

Here’s an easy puzzle that gratifies the ego. This one came from today’s San Francisco Chronicle. Be sure and check out the tips on the linked sudoku website. I posted the solution below.

Complete the grid so that every row. column and 3-by-3 box contains every digit, 1 to 9.

date: 12/12/2005
difficulty rating: easy
Easy Sudoku
Easy Sudoku

Alex Forbes ©December 12, 2005

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Puzzle – Had Had

This puzzle also seems like it should be easy, but it’s not. It’s obnoxious, stoopid, annoying and it won’t go away until you solve it. Why? Because you know you can. HINT: Can you think of a sentence with “had had” in it?

Thanks to: Dave Norton

Punctuate the sentence below correctly, giving it a credible meaning without rearranging the words.

dave where alex had had had had had had had had had had had a better effect on the reader

Once you get the hang of it, there’s more than one credible way to punctuate the sentence. It helps to show your work.

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Puzzle – Trash Talk

This puzzle seems like it should be easy, but it’s not. It’s obnoxious, stoopid, annoying and it won’t go away until you solve it. Why? Because you know you can. HINT: Put your Crossword Puzzle thinking cap on. We’ve found two unique solutions so far.

Thanks to: Dave Norton

Restate the sentence below in four words, each word starting with the same four letters, all in the same order.

“Deny disenfranchised dumpster divers.”

Question: What would be a non-giveaway example of the format you are looking for?

Answer: chat chatty chatters chatterly

Find one, two or more UNIQUE solutions, without borrowing from the same word root for more than one solution. The solutions we found are credible and do not use any root word from the original.

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Puzzle – Garden of Eden

OK, so you couldn’t solve the Zebra puzzle. Here’s one submitted by Stacy on December 15, 2002. We sense there’s a solution, but our truth table approach isn’t working — so far. See how you do. Challenge yourself!

February 17, 2003 – we found the solution. Send us yours and we’ll tell you whether that was the result we arrived at. How to publish solutions without giving away the answer — that is the question! (There’s a HINT somewhere in the YaBB Forum).

2/20/2003 Someone named ’24fan’ sent us an email with a 560K total size. It was titled “Garden of Eden” and may have contained a solution. It also contained the W32.Klez.H@mm worm virus. Norton Antivirus intercepted it, so we never got to see its contents. If this is you, we recommend disinfecting your machine! Going forward, in this age of proliferating viruses and cyberterrorism, we ask readers to please write us before attempting to send us email attachments.

2/26/2003 Our first reader solution, from N.W., and a good one too!

Everything was perfect in the Garden of Eden, until one day somebody ate an apple from the tree of knowledge. The immediate result was that you could no longer trust people to tell the truth. In fact, when all the residents were questioned about the events, only one of them answered all the questions truthfully. All the others told some lies, though no two of them told the same number of lies.

1. The snake ate the apple.
2. The snake was in the garden.
3. Eve has not weeded the garden.
4. Abel failed to do his chores.

5. Adam ate the apple.
6. We are not all equally truthful.
7. I was out weeding the garden.
8. The snake lies.

9. Abel ate the apple.
10. Abel doesn’t always tell the truth.
11. Mother has always preferred Abel.
12. The snake never lies.

13. Cain ate the apple.
14. The snake can’t see over the hedge.
15. I have done my chores.
16. The snake ate the apple.

17. I was not in the garden.
18. Eve ate the apple.
19. Cain is Eve’s favorite son.
20. I can’t see over the hedge.

Who ate the apple?

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Puzzle – Zebra

Mailed to us in 1998, this puzzle has resulted in several correct responses every year since. In 2002, two people mailed in the correct answer. Others wanted to know, “OK, so who owned the zebra?”

Try solving this puzzle… It’s definitely solvable! There are 25 variables and 24 are given values. You are to solve the 25th. The trick is HOW? If you look at the problem mathematically, no sweat. If you get lost in the English, you are dead meat. You will know you are right by checking the answer with all the conditions. This test was given by The German Institute of Logical Thinking in Berlin, 1981. And 98% FAILED.

There are five houses.
Each house has its own unique color.
All house owners are of different nationalities.
They all have different pets.
They all drink different drinks.
They all smoke different cigarettes.
The English man lives in the Red House.
The Swede has a Dog.
The Dane drinks Tea.
The Green house is on the left side of the white house.
In the Green house they drink coffee.
The man who smokes Pall Mall has birds.
In the yellow house they smoke Dunhill.
In the middle house they drink Milk.
The Norwegian lives in the first house.
The man who smokes Blend, lives in the house next to the house with
In the house next to the house with the horse, they smoke Dunhill.
The man who smokes Blue Master drinks beer.
The German smokes Prince.
The Norwegian lives next to the Blue House. They drink water in the
house that lays next to the House where they smoke Blend.
Who owns the Zebra?

Good luck !

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