Hoppity-hop
Hop the little kangaroo was bored. He looked around the village and pulled a face.
He looked up at his mother, Bounce, and pulled a face. Bounce laughed. “Hop,” she
said, “you are a kangaroo, not a horse. Why the long face?” Hop scowled. “I’m
bored,” he moaned. I’m so boooooooooooored.” At this the baby kangaroo, Peep,
looked out of Bounce’s pouch to see what the fuss was all about. Hop scowled at him
too. He was certainly bad-tempered today!
“Hm,” said Bounce, “let’s cheer you up.” She looked around the neat little kangaroo
village with its rows and rows of houses, and a thought struck her. “I know!” she
said. “Let’s play Sudoku!” “Sudoku?” echoed Hop sulkily. “Never heard of it.” “It’s a
game,” began Bounce, “you have these numbers that are arranged in…” “But I don’t
want to do math,” broke in Hop, stamping his feet. “You said it was a game.” “It is a
game,” said Bounce patiently. “Here, let me show you.”
She picked up a twig and drew a very large square on the ground, like this:
“How many squares do you see?” “One, of course,” answered Hop. Bounce then drew
two lines inside the square, like this:
“And now?” “Four,” said Hop, who was beginning to think this a very stupid game.
Bounce now drew four more lines inside the squares, like this:
“And now?” she asked. Hop squinted. There were many squares now. It was all too
large for him; he was, after all, a very young kangaroo. “Hm…twelve?” he asked
hopefully. Bounce shook her head no.
Hop then began to hop from square to square. Hoppity-hop he went, counting the
number of jumps he had to make to cover all the squares. This time he got it right.
“Sixteen,” he announced proudly. “Right you are,” said Bounce. This is a Sudoku
grid. And each of these, she added, pointing with her twig to the four squares in the
big square, is a mini-grid.
She also showed Hop the rows and columns in the Sudoku grid.
Bounce began to collect pebbles. Forty smooth round pebbles she found, and these
she placed gently in her pouch, beside a surprised Peep. She then walked up to the
grid and placed some of her pebbles in different squares, leaving a few squares
empty.
When she finished the grid looked like this:
“Now, Hop,” she said, “listen carefully. You must fill in the blank squares with
pebbles in such a way that every column, row and mini-grid contains 1, 2, 3 and 4
pebbles in different squares. Since there are only four squares in each row, column
and mini-grid, you cannot repeat a number. Do you understand?” Hop nodded,
already deep in thought. He did not understand completely, but he was no longer
bored. Bounce handed him the pebbles and bounced away with a smile.
Questions
How many rows does this grid contain?
How many columns does this grid contain?
How many mini-grids does this grid have?
“Stupid Sudoku”
Hop the little kangaroo looked sadly at the Sudoku grid his mother had set for him.
He was annoyed. Try as he might, he could not get the numbers to behave. The grid
remained incomplete, and he was beginning to sulk. “Stupid grid,” he said out loud.
“Stupid game.” He was very fond of the word stupid.
His mother, Bounce, who heard him, came bounding. “Having trouble?” she asked,
smiling. Hop looked up at her, his chin trembling. “Stupid Sudoku,” he said tearfully.
“Well, we’ll see about that,” she said. “Here, let me help you.” She swept away his
pebbles, so that the grid looked like this again:
“Let us begin,” she said, “with a column that already contains two numbers. Which
ones are those?” Hop did not need to look at the grid to answer. “Columns 2 and 3,”
he said, cheering up a bit. “Okay,” said Bounce. “Look at column 3.
The first square has 2 pebbles and the third square has 3, right?
Hop nodded. “Now, we need 1 pebble in one square and 4 pebbles in another, right?”
Hop nodded again. Bounce looked at him. “Can we place 4 pebbles in the second
square?” she asked. “Think hard.”
Hop thought hard. “I don’t see why not,” he said finally. Bounce patted him on the
head. “Because, little one, there is already a square with 4 pebbles in the same row,”
she said. Look.
Hop looked. “Ah,” he said. “Of course. So it must be 1.” He placed a pebble in the
second square of column 3.
Now, said Bounce, what happens to the fourth square of column 3? Hop looked at
the grid. Since 1, 2, and 3 were already in the column, he counted out 4 pebbles and
placed them in the remaining square.
“Wonderful,” said Bounce. “Now do the same for column 2.” Hop looked hard at the
grid. Two and 3 pebbles needed to be placed in the two empty squares in column 2.
Since the first square in the grid already contained 2 pebbles, Hop placed 3 pebbles
in it and 2 pebbles in the remaining square.
He then looked up at Bounce, who beamed. “Now we’re talking, she said. What’s our
next move?”
Hop noticed that the first mini-grid contained 3 pebbles in its second square and 4
pebbles in its fourth square. So, he thought, working it out in his head, we need 1
pebble in one square and 2 in another. Since the top row already contains 2 pebbles,
we must have 1 pebble in its first square. The remaining square in the mini-grid will
obviously need 2 pebbles.
Hop placed the pebbles in the squares. The grid now looked like this:
Hop looked up from the grid at his mother. She was smiling. “You’ve got the hang of
it, haven’t you?” she said. “Let’s see you wrap it up.”
Hop looked at the grid. The figured that there must be 4 pebbles in the remaining
square of the top row and 3 pebbles in the remaining square of the second row.
Hop was deep into the game now. He looked at the second and fourth mini-grids.
Both had two empty squares each. The second mini-grid needed 3 pebbles in one
square and 4 pebbles in another to be complete. Hop saw that there were already 3
pebbles in the third square of row 3. Thus, the first square of row 3 could only
contain 4 pebbles. The remaining empty square of mini-grid 2 would obviously
contain 3 pebbles.
It was now easy to fill the remaining two squares. Hop placed one pebble in square 4
of row 3 and two pebbles in square 4 of row 4. The grid was now complete.
Hop looked at it proudly and grinned at his mother. “Stupid Sudoku, is it?” she said,
laughing.
Exercise:
Try to solve this easy Sudoku.
Big Sudoku
Two days after she had taught him Sudoku, Bounce the Kangaroo spied her little son
Hop sitting glumly under a tree. All around him were 4X4 Sudoku grids.
She had set him five puzzles in the morning, and now, barely two hours later, he had
worked them all out.
She bounced up to him. “Hello, Hop,” she said. “Want another puzzle?” Hop scowled
his scowl at her. “What’s the matter?” she asked. Hop said nothing. Instead, he put
his paw into her pouch and poked the baby kangaroo, Peep, who was sleeping there.
“Wake up, Peep,” he said. “Wake up.” Peep opened a reluctant eye and closed it.
Bounce gently pushed Hop’s paw away. “Now, Hop, she said, you know you’re not
supposed to do that. Let me give you another puzzle.” “But they’re all so easy,”
burst out Hop. “I’m bored bored bored.” Bounce laughed. “Ah,” she said, “so that’s
it. You’ve outgrown the 4X4, have you? Hm…let’s play some big Sudoku, then. Come
with me.”
Hop sullenly trailed after Bounce and watched as she swept away the 4X4 puzzles.
Then she drew a very large square and several lines inside it. When she had finished,
the grid looked like this:
“This,” she said, “is a big Sudoku. How many squares do you think it has?” Hop
shrugged. “Okay,” said Bounce, “how many squares does each column and row
have? Surely you can tell me that.”
Hop squinted at the grid. Then he leapt on to it and hopped from square to square,
counting. Hoppity-hop he went, and at the end of it he knew the answer. “Nine
squares in each row and column,” he cried to Bounce. “Right,” said Bounce. “Now, do
you remember your multiplication tables?” Hop frowned. This was getting a bit too
educational. “Ye---s,” he answered, “a bit.”
“Well then,” said Bounce, “in your opinion, what is 9 times 9?” Hop knew the answer
to this one. “81,” he yelled happily. “Good,” said Bounce, pleased. Pointing to the
grid, she added: “This is a 9X9 grid. It has 81 squares instead of 16. It’s a very big
Sudoku. We’ll use numbers instead of pebbles in this one, okay?” Hop nodded. Using
her twig, Bounce began to write numbers inside the squares of the big Sudoku,
leaving some squares blank. Which she finished, the grid looked like this:
“The rules remain the same. But there are 9 rows and 9 columns,” she said, pointing
them out.
“And the mini-grids have 9 squares instead of 4.
That means that you must fill in the blank squares in such a way that every column,
row, and mini-grid contains the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 in different
squares.
Since there are only 9 squares in each row, column, and mini-grid, you cannot
repeat a number. Also, these are the horizontal blocks and these the vertical blocks.
The grids are marked 1 to 9 and the blocks are marked 1 to 3. Do you understand?”
Hop shrugged. “It’s the same game,” he complained, “only bigger. It will be as easy
to do as the others. I can solve it in a jiffy.” Bounce raised her eyebrows. “Is that
so?” she said archly. “We’ll see. Why don’t you give it a try?”
Exercises:
How many rows does big Sudoku contain?
How many squares are there in a big Sudoku?
How many mini-grids does a big grid have?
In a big Sudoku, how many squares does a mini-grid have?
Do you remember what you first looked for when solving a small 4X4 Sudoku?
Exercises:
Mark the horizontal blocks in this grid.
Mark the vertical blocks in this grid.
Scanning for triplets
Bounce the Kangaroo was frowning. She had been frowning all morning. Hop, her
little son, was a little scared. His gentle mother rarely frowned. The problem was
that, try as they might, they were unable to solve the big Sudoku that she had set
the previous day.
As time went by and morning turned to afternoon, Bounce’s frown deepened. She
tried one number, then the other, but every time she was stuck. Hop was concerned.
He had thought his mother could do anything!
Suddenly they heard a loud laugh. “Ooh ooh ahh ahh ahh ahh. Ooh ooh ahh ahh ahh
ahh.” Hop looked up. The laugh was coming from up a tree. He squinted at the tree
closest to the grid. Again the laugh sounded: “ooh ooh ahh ahh ahh ahh,” and Hop
saw that a rather large white bird was making the sound.
Puzzled, he looked at Bounce, who smiled. “It’s Syd, the Kookaburra,” she said. “Hi
Syd,” she called loudly. Syd did not reply but laughed again. “I see you’re getting
nowhere,” he said. “Ooh ooh ooh ahh ahh ahh ahh ahh ahh ahh.” Hop was annoyed.
“You might help us instead of laughing,” he said feelingly.
Syd flew over the grid. “Perhaps I will, my lad” he said, “I have a nice aerial view.”
Then he said to Bounce rather severely: “you’ve been guessing, haven’t you?”
Bounce looked shamefaced. “Never,” said Syd, “never never never guess. That’s the
first rule of Sudoku. Always use logic. Now, clear the game.”
When the game was cleared, the grid looked like this:
Now, said Syd, here’s how we start. Scan the grid for what are known as triplets. For
example, look at vertical block 2. Is any number in the block repeated?
Hop looked carefully.
He did not want Syd to laugh at him again. “1?” he asked finally. “Yes,” said Syd.
“One of the 1s is in mini-grid 2 and the other in mini-grid 5, right?”
Hop nodded. “And we need a 1 in mini-grid 8, right?” Hop nodded again. “In minigrid 8, then, where does 1 fit in?” asked Syd. Hop looked at the mini-grid. Since
columns 1 and 3 already had a 1 each, there was only one possible square for 1 in
mini-grid 8.
Hop drew 1 in the square and looked at Syd. “That’s right,” said Syd in a
businesslike voice. Some people call this scanning, because you scan the grid for
triplets. Now, is any number in any other block repeated?”
Hop scanned both horizontal and vertical blocks. He saw that several numbers in
several blocks were repeated. For example, the number 7 was repeated in vertical
block 1 in mini-grids 2 and 3. However, there were three squares, not one, that were
possible homes to the number 7 in mini-grid 1.
Hop explained this to Syd, who nodded. “In such cases,” he said, “you must make a
small mark of the number 7 in the corner of the square, like this. As your grid fills
up, these will solve themselves.”
As Hop bent over to scan the grid again, Syd flapped his wings. “That’s enough
Sudoku for me today,” he said. “I have to be someplace at seven. I’ll be back
tomorrow, to see how you’re doing, eh?” “Thank you, Syd,” called both Hop and
Bounce together as Syd flapped away.
Exercise:
Find a home for 4 in mini-grid 4 by scanning for a triplet in horizontal block 2.
Eli-mi-na-tion
Tomorrow came, but brought no Syd with it. Hop the little kangaroo was sad. He
hopped over his Sudoku grid unhappily. He had used Syd’s “triplets” tip where he
could, and the grid now looked like this:
Now he was stuck again. Who would help him out this time?
“Hop! Ho-ooop!” Someone was calling him. Hop looked about him. Bounce was
bouncing towards him, shouting uncharacteristically. Running alongside her
was…wait a minute…it was Dingo!!! Hop gave a whoop of joy and bounded away to
meet him.
“Dingo!” he cried. “Oh Dingo Dingo. You must help me. I’m stuck.” Dingo the
Australian dog looked at Hop with a twinkle in his eye. “Hello, Hop,” he said,
laughing. “Slow down, will you? Begin at the beginning.” “It’s Sudoku,” cried Hop. “I
can’t solve it. Come and see.”
So the three walked together towards the grid, with Hop chattering at the top of his
voice to Dingo. When they reached the grid Dingo stopped and sniffed the air. Then
he looked long and hard at the grid. “Hm,” he said, “have you heard of the twins
rule, Hop?”
Hop hadn’t. He was all ears. “Look at mini-grid 3,” began Dingo. “Both squares 1 and
3 of row 3 must contain either 1 or 2, right?” Hop nodded. He had already worked
this out while scanning the grid.
“Well then, while we don’t know which number goes into which square, we do know
that both these squares must be occupied by either 1 or 2, right?” Hop thought
about this for a while. “Hm,” he said. “That means both the squares are occupied
and both numbers taken, right?” continued Dingo. Hop nodded.
“This is called the twins rule,” said Dingo. “It is very useful in ruling out squares and
numbers. Now, once we know that both these squares are taken by either 1 or 2, we
can solve square 1 of mini-grid 3. We shall do this by elimination, okay?”
“By eli…mi…what?” Hop did not know the word. “Eli-mi-na-tion,” repeated Dingo. “It
means coming to a single conclusion by ruling out all your other options.” Hop
frowned. Dingo’s words were swimming in front of his eyes. He looked at Bounce
helplessly. She laughed. “He simply means that you can test each number to see
which one fits in a square,” she explained, adding, “Dingo is fond of using many
words.” Dingo smiled sheepishly.
“Do go on,” said Hop impatiently. “To eliminate,” Dingo continued, “you must first
write down all the numbers from 1 to 9.” Hop drew the numbers 1 to 9 beside the
grid, like this Said Dingo: “now, look at square 1 of mini-grid 3. Since 3, 5, 9, and 4
are already in the grid, they cannot be in square 1. They are eliminated. Cross them
out, please.”
Hop crossed out the numbers, like this “The numbers 1 and 2 also cannot go into
square 1 of mini-grid 3, right?” asked Dingo. “Yes, because of the twins rule,” replied
Hop. “Good,” said Dingo. “Cross them out too.” Hop did so. “What numbers are we
left with?” asked Dingo$.
Hop answered: “6, 7, and 8.” “Right,” said Dingo. “There are 7s in both row 2 and
column 8.” “Of course,” cried Hop, suddenly understanding, “that means that 7 can
only fit in square 1.” Dingo looked very pleased. “Well, there you have it,” he said.
Exercises:
Use elimination to mark (with a pencil) all the possible solutions for all the empty
squares in column 5.
Scanning in both directions
Syd the Kookaburra and Dingo the Australian dog had done Hop’s Sudoku grid a
whole lot of good. The squares were finally filling up. The grid looked like this:
Hop was peering into the grid as he waited for Dingo to wake up. Dingo was sleeping
under a tree, having given Hop strict instructions not to disturb him. Hop fairly
bristled with impatience. To him it seemed that Dingo had been sleeping for hours.
“He promised me another tip,” Hop grumbled to himself, “and now all he does is
sleep.”
Finally Dingo stirred. Hop ran across to him. “You’re awake, you’re awake,” he cried.
Dingo paid him no attention. He rested his head on his paws and closed his eyes
again. Hop hopped in irritation. “DINGO!!!” He cried. Dingo opened his eyes. “Let me
sleep, Hop,” he said in a dignified voice. But Hop would have none of it. He made
such a fuss that Dingo had to give up his nap. “Alright, young man, alright,” he said,
“let’s go.”
The two of them stood over the grid, deep in thought. Finally Dingo said, “listen
carefully. In vertical block 1, since there is a 7 each in both columns 2 and 3 and
mini-grids 4 and 7, the 7 in column 1 must be in mini-grid 1, right?
“Yes,” said Hop thoughtfully, “but it can occupy any of the three squares in the first
column in mini-grid 1.” “That’s right,” said Dingo. “But look at horizontal block 1.
Where are the 7s?” “Ah!” said Hop. “In rows 1 and 2. That means that 7 cannot be in
either square 1 or 4 of mini-grid 1. It HAS to be in square 7.”
“Good, Hop. You are really in top form, aren’t you?” said Dingo. “This technique is
called scanning in both directions. It is very useful once you’ve solved a few basic
squares. For example…Hop, Hop, are you listening?” But Hop wasn’t listening. He
was already scanning in both directions for more opportunities to use this tip. Dingo
smiled, tiptoed back to his tree, stretched, yawned, and fell asleep.
Exercises:
Solve the number 3 in mini-grid 9 by scanning in both directions.
Solve the number 4 in mini-grid 1 by scanning in both directions.
Common Sense
Hop the little kangaroo was stuck. He had used all the techniques that Syd the
Kookaburra and Dingo the dog had taught him, but his Sudoku grid remained
unsolved. His squares were filled with possible solutions, and the grid looked like
this:
But Hop could not figure out how to proceed. And what’s more, Dingo had suddenly
left for a long journey, telling Hop that if he only mixed the techniques with a bit of
common sense, he would be able to solve the grid in no time! In no time, thought
Hop. Humph.
Hop’s mother, Bounce, now bounced up to him. “Stuck again, are we?” she asked
sympathetically. Hop did not reply. “Hm,” said Bounce, bending over the grid. “What
did Dingo say before he left?” “He asked me to use common sense, complained Hop.
What does he mean, common sense?” “Common sense,” said Bounce, “simply means
good sense. Let me show you. What are the numbers possible in square 8 of minigrid 2?”
Hop looked at his grid, where he had marked the possible solutions for some of the
squares using elimination. He answered: “5, 6, and 9.” “Now,” said Bounce, “in minigrid 5, squares 2 and 8 have the same possible solutions, right?” “Right,” said Hop,
“5 and 9.” “What does that tell us?” asked Bounce. Hop pondered.
“If we apply the twins rule, he said slowly, and then 5 and 9 will occupy either
square 2 or 8 in mini-grid 5.” “Yes,” said Bounce, “very good. That means both these
squares are occupied and both numbers taken, right?” Hop nodded. “Notice that both
these squares, as well as square 8 in mini-grid 2, belong to column 5,” said Bounce.
“Now, since we know that 5 and 9 will definitely occupy either square 2 or square 8
in mini-grid 5, neither 5 nor 9 can occupy square 8 in mini-grid 2, because they are
all in the same column.”
Hop blinked. It took him a few minutes of staring hard at the grid to figure out what
Bounce was saying. Finally, he smiled delightedly. “Yes, of course,” he said. “That
means the only possible number that can fit square 8 of mini-grid 2 is 6!”
Bounce beamed. “Well done, son,” she said. “Now, whenever you find such a
combination in a single row or column, you can quite safely solve a number, right?
Now, for example… But Hop was already bent over the grid intently.
Exercise:
Use common sense to solve square 9 in row 7.
The solution: