Kenken: Sudoku With Math

Kenken is a Sudoku variant invented by Tetsuya Miyamoto as a way to teach math. Frankly, Sudoku bores me senseless, but I found the addition of math makes the puzzles much more interesting.

The game is played on grids ranging from 4x4 to 9x9, and the rules are as follows:

• Do not repeat a number in any row or column.
• The numbers in each heavily outlined set of squares, called cages, must combine (in any order) to produce the target number in the top corner of the cage using the mathematical operation indicated.
• Cages with just one box should be filled in with the target number in the top corner.
• A number can be repeated within a cage as long as it is not in the same row or column.

Short version: the numbers you place in the bold boxes need to yield the number in the corner by multiplication, division, addition, or subtraction. 

Here's a video with Will Shortz (former editor of Games Magazine) explaining how to play.You can find playable puzzles at Kenken.com.

PUZZLE: Tabling the Question

I haven't done a math puzzle in a while, so here's an old one to strain the average brain. (And by that I mean my brain: in the past, my readers have solved these so fast it's almost embarrassing.)

Mr. Jenkins comes home to find his son Todd reading a comic book rather that working on his quadratic equations. He sets about berating the boy, when Todd interrupts him with a deal. "If I can pose a problem you can't solve, you have to finish my homework, and I can to finish my comic." Mr. Jenkins, considering himself something of a math whiz, takes this bet in a heartbeat.

Todd pushes a large, perfectly round table into a corner, with its edge touching both walls. He places a spot and then turns to his father. "That spot," says Todd, "is exactly 9 inches from one wall and 8 inches from the other. Without measuring the table, tell me its diameter."

My Jenkins puzzled and puzzled 'til his puzzler was sore, and then wound up finishing Todd's homework while the boy returned the adventures of Uncle Scrooge. Would you have done any better?

PUZZLE: Cell Division

A biochemist is cultivating living cells. Each cell splits into two cells after one minute.

One minute later the two cells split to make four, then the four become eight, and so on. Every minute, the number of cells doubles.

Assume that it takes an hour for one cell to grow until a bottle is filled. If a chemist starts with two cells, how long will it take to fill the same bottle.

This version is from The Tokyo Puzzles, by Kobon Fujimura

PUZZLE: Cheap Labor

Huey, Dewey, Louie, Donald, and Gyro are each working on a single day this week guarding Uncle Scrooge's money bin.

Huey is working as many days before Louie as Dewey is after Gyro.

Donald is working two days before Gyro.

Louie is working on Wednesday.

When are the other four working?

App O' The Mornin': Theseus

Robert Abbott is one of the masters of logic mazes, which add layers of complexity to the standard linear point-to-point maze. (You can find his home on the web here.) He does a lot of work for us at Games Magazine, including some fine cover puzzles. We even made an interactive version of one of his cover puzzles, called Starry Night, which you can try online.

One of his most enduring designs is Theseus and the Minotaur, which has evolved over several iterations, including pen-and-pencil, computer, Java, and the game Mummy Maze (PopCap), which belatedly acknowledged its dept to Abbott’s original concept.

It now finds a welcome home in the App store with a very simple, clean visual style and control system. Developed by Jason Fieldman, the app offers 17 levels in the free Lite version and almost 90 in the full $4 version. The goal is to maneuver “Theseus” (a blue ball) to the exit by first trapping the “Minotaur” (a red ball with horns) in one of the niches on the map. Since the Minotaur is always closing on Theseus with two moves to Theseus’s one, the trick is to find ways to trap him with his own rules of movement, which favor horizontal motion over vertical.

Although the rules are easy to understand, the puzzles get larger and increasingly complex, and some a real mind-benders. There are many maze games in the App stores, including rolling ball games that use the device’s motion control to emulate tilting mazes, but Theseus is the most clever. It’s a classic design done with a solid implementation.

Weekend Post: Math and Me

I’m about as far from being a mathematician as one can get while still being able to add 20% to a restaurant check. Math was my bane all throughout school. I hated it and held it in contempt. I failed Algebra I … twice. That takes a special kind of stupid.

I know my subjects: literature, art, history, philosophy, and theology. But the math thing always bugged me. I didn’t like the fact that I’d let it defeat me, and suspected I had done so for bad reasons, such as arrogance and laziness.

About ten years ago, I decided to do something about it. I bought a couple of “teach yourself” books and began with straight mathematics and “pre-algebra” and then did a full course of algebra self-study. It was hard going, but eventually I began to listen to the numbers like I could listen to words or music or rhetoric. I realized that they were a language and I just needed to let the numbers speak.

And, finally, I understood it. I’m still no mathematician, but I get it now. I can follow problems and see the special beauty in numbers.

My faith had a lot to do with helping me clear that hurdle. I study a particular stream of Catholic philosophy known as Thomism, which is a rigorously logical approach to all the questions of Creation. St. Thomas Aquinas was my first big personal “discovery” since college, when I first dug deeply into Plato. He opened up a doorway to a vast storehouse of logic. It was not merely a belief system, but a way of approaching any question in an orderly way, giving equal time to contrary arguments, assuming nothing, and testing everything.

Logic is at the core of Creation, just as it is at the core of all good gaming. Games and puzzles are logic made manifest. They are concrete. They can be cracked open and understood. I was always good at pattern puzzles, conundrums, riddles, and lateral thinking. But with St. Thomas in one hand and a new respect for math in the other, I started tackling the kind of puzzles and problems I used to avoid.

I’m still not all that great at them, but I like to think I understand them and appreciate them better than I used to. I also think they are terribly important, particularly in the education and parenting of children. We live in a world where extravagant emotion always trumps logic and plain old common sense. We could do with a little more logic and a little less hysteria. We’ve spent several generations nurturing our inner child, when we should have been nurturing our inner Mr. Spock.

Games and puzzles are almost always intertwined with mathematics, and all of them taken together sharpen the mind. They help us understand ourselves and our world. They form a language with its own poetry. We have to make sure the next generation knows how to listen to that poetry.

Thursday Puzzle SOLUTION: Startling Subtraction

STARTLING
STARLING
STARING
STRING
STING
SING
SIN
IN
I

See the comments on the original post for some alternative solutions. 

About Puzzles

I had fun preparing the puzzles for the past two weeks. A lot of them are classics, since even the master puzzlers didn't invent most of their own puzzles: they just reformulated them. Some I did from memory, and some were drawn from my collection of puzzle and math books. I only botched one answer, which my astute readers quickly caught.

This was never meant to be a regular daily feature, but I will continue to post puzzles, maybe a couple of times a week. As I said in a previous post, I'm about to disappear into the thickets of a massive special supplement for Games Magazine, but will continue to post an App O' The Mornin' and whatever else I can until I complete it sometime in the next two weeks.

I also plan to post something about the future of the site. It's about to be one month old, and I've been steadily populating it with posts. It will continue, and begin to take shape as I explore this whole blogging thing.

Thanks to the regular puzzle solvers, especially Eye of the Frog, who answered every one correctly, which puts him one-up on me. I think I posted today's "1105" puzzle just to see if I could stump him. I hope to heck I can get it right.

Friday Puzzle (Bonus): A Riddle

And now for a riddle:

What can go up a chimney down, but can't come down a chimney up?

Friday Puzzle (Hard): 1105

Let's finish off two weeks of puzzles with some challenging math and a more gentle riddle.  First, the math:

The sum of the squares of two consecutive numbers is 1105.

What are the numbers?

Thursday Puzzle: Startling Subtraction

Take a look at the 9-letter word below:


S T A R T L I N G


It is possible to remove a single letter at a time, forming a new word each time, until you have formed a total of 9 words (including the starting word). The letters cannot be rearranged. Each time one letter is removed, a new word must be formed.


What are the words?


[Adapted from Martin Gardner, though I don't think he originated it]

Wednesday Puzzle SOLUTION: The Bookworm

UPDATED:  I've often used this puzzle to stump people, and this time I fell into my own trap. I tried to create even numbers for easy mental calculations, but I kept changing the numbers of volumes. In typing up the solution, I got sloppy and forgot that the central trick of the puzzle is that there are TWO FEWER BOOKS  eaten than would appear to be the case. In my original solution, I only calculated one.  Thanks to readers Eye of the Frog and Ethan C. for catching the error. This post has been revised to correct the mistake. (I plead exhaustion: aside from the puzzles that are directly quoted, I wrote the last two weeks worth of material from memory in a single sitting. No, Martin Gardner would not have bought that excuse either, but it's my story and I'm sticking to it.)

58 inches

Remember, if a set of books is in order on a shelf, the first volume will be the furthest book on the left. This would place the first page of that book to the right of that volume. The front cover of each book (except for the last) touches the back cover of the next book in line. The back cover of the first volumes touches nothing.

Therefore, a worm starting on the first page of the first volume will not need to eat through the first volume. It only needs to eat through the cover.

Since it's only going to the last page of the last volume, it doesn't need to eat through the front cover or text of volume 26.

24 volumes at 2 inches each = 48 inches

50 covers (front and back for volumes 2 to 25, plus the front cover of volume 1 and the back cover of volume 26) at 1/5th of an inch each = 10 inches

Wednesday Puzzle: The Bookworm

If you've read many puzzle books, you've probably come across this one in some form. Everyone takes a stab at their own version sooner or later, so here's mine:

A 26-volume encyclopedia set is placed, in order, on a single shelf. Each volume is 2" thick. Each cover is 1/5" thick.

If a book worm starts on the first page of the first volume, how far will he have to travel to reach the last page of the last volume.

Franklin on Games

Games lubricate the body and the mind.
  Benjamin Franklin

Tuesday Puzzle SOLUTION: Matchstick Fieldgoal ... Intercepted

Push the horizontal matchstick over, and then move one upright to reform the goalpost.

Tuesday Puzzle: Matchstick Fieldgoal ... Intercepted

The opposing team made the mistake of building their goalposts out of matchsticks. (They're really tiny football players).

A ball (the match head) is about to pass through the uprights.

See if you can make this fieldgoal an incomplete by only moving two matchsticks. You must still have an intact fieldgoal at the end. The ball must be completely outside of the goalposts.

SOLUTION

Monday Puzzle SOLUTION: A Lady Never Tells...

Mrs. Perkins is 45. Her husband is 54.

UPDATE: Reader "Eye of the Frog" posted an excellent solution to this in the comments of the original puzzle.

Monday Puzzle: A Lady Never Tells Her Age

One day at the hairdresser, a beautician asked stately Mrs. Perkins how old she was. Mrs. P. glared at the woman over her glasses, and said "If you reverse the figures of my husband's age, you shall have mind."

This didn't satisfy the young woman at all, who said, "I don't know how old your husband is."

"I am younger than he. The difference between our ages is one-eleventh of their sum."

Mrs. P. thought this would confuse the hairdresser so much that she'd drop the subject, but the young woman did some quick calculations, smiled, and said, "I wouldn't worry about it, dear. You look very good for a woman of your age."

How old is Mrs Perkins?

Friday Puzzle SOLUTION: The Stopped Clock

Mr. Leen winds his clock before he leaves, and notes the time. Although the time is incorrect, the clock will now track his time away from the house.

When he arrives at Mr. Been's, he notes the correct time.

When he leaves Mr. Been's he notes the time once more. Thus, he knows how long he has been there, and the precise time upon his departure.

When he checks his own clock, all Mr. Leen has to do is subtract the time he spent at Mr. Been's. This gives him his total walking time. If he halves this, he then has the total time it takes to make the trip one-way. He adds this time to the time he noted on Mr. Been's clock upon leaving, and he gets the current time.

For example: He sets his own (incorrect) clock to 12:00, walks to Mr. Been's, and notes that the accurate time is 10:30. He spends 2 hours there, and upon leaving notes that the time is 12:30.

When he arrives home, his own clock reads 3:00. Three hours have passed since he left the house.

Two hours were spent at Mr. Been's, which means that 1 hour was spent walking round trip. Thus, one leg of the trip takes 30 minutes. Since he left Mr. Been's at 12:30, he sets his clock for 1:00.

Friday Puzzle: The Stopped Clock

Mr. Leen is the most punctual man in town. Every day, he takes the same walk, and has such a regular pace that he passes the same landmarks at exactly the same time.

One day, he returns home to find that his servant has forgotten to wind the clock. Since his watch is broken, he is unable to set the correct time.

However, he has a solution. His friend Mr. Been has recently moved to town. He decides to walk to Mr. Been's, pay his first call at his friend's new lodging, and see what time it is.

After spending a hearty afternoon of fellowship, he returns to set his clock. Although he had never made the trip to visit Mr. Been before, and thus has no idea how long the walk takes, he is able to set his own clock correctly.

How does he do it?