신종 숫자퍼즐 스도쿠

1. 스도쿠란?

''숫자들이 겹치지 말아야 한다''는 뜻의 일본어 조어인 스도쿠는 가로와 세로 9칸씩 총 81칸의 정사각형의 가로.세로줄에 1∼9의 숫자를 겹치지 않도록 적어 넣는 게임으로 이미 숫자로 채워진 일부 칸을 제외한 나머지 빈 칸을 채워야 한다.

단, 큰 사각형 안에 있는 가로.세로 3줄로 이뤄진 작은 사각형 9개 각각 내부에서도 1∼9는 겹치지 않아야 한다.

이 게임은 가로.세로줄의 개수를 늘려 난이도를 조정하거나 숫자 대신 알파벳을 사용할 수도 있다.

스도쿠는 스위스의 수학자 레온하르트 오일러가 고안한 ''마술 사각형''이라는 퍼즐게임에서 유래했는데 80년대 일본의 한 출판사가 뉴욕에서 이를 발견한 뒤 일본에 스도쿠라는 이름으로 들여왔으며 이후 다른 나라들로 퍼지기 시작했다.

스도쿠는 일본에서 한달에 60만권 이상의 게임 책자와 해설서가 팔리고 있으며 영국으로 건너온 뒤에는 거의 모든 주요 신문들이 이 게임을 지면에 싣고 있다.

더 타임스는 휴대폰을 통해 퍼즐게임을 공급하고 스도쿠 게임 책자 12만부를 팔았으며, 인디펜던트는 스도쿠 선수권대회 개최도 계획하고 있고 타블로이드판 미러지는 스도쿠 퍼즐이 알츠하이머병의 진행 속도를 늦출 수 있다고까지 주장한다.

홍콩의 판사 출신이자 스도쿠 게임을 영국에 도입한 웨인 굴드는 은퇴 후 스도쿠 컴퓨터 프로그램을 개발해 100만달러의 수입을 올렸는데 그가 만든 스도쿠 웹사이트는 하루 1만명이 방문하고 있다.

굴드는 이 게임이 수학게임이 아니라 일종의 논리게임이라며 오랫동안 퍼즐을 풀다보면 ''약한 고리''가 드러나게 되고 사각형과 숫자 사이의 분명한 조합이 이뤄진다고 말했다.

2. 게임의 방법 및 규칙

이 게임은 가로.세로 9칸씩 총 81칸으로 이뤄진 정사각형의 가로.세로줄에 1~9의 숫자를 겹치지 않게 적어 넣는 퍼즐방식이다. 다만 가로.세로 3줄로 이뤄진 작은 사각형 안에서도 1~9가 겹치지 않게 들어가야 한다. 이코노미스트는 "게임 규칙이 워낙 단순해 누구나 쉽게 도전할 수 있지만 풀기가 만만치 않은 지능형 게임이란 게 최대 매력"이라고 소개했다.

스도쿠는 다양한 방법으로 난이도를 달리할 수도 있다. 가로.세로줄의 개수를 늘리거나 숫자 대신 알파벳을 사용하기도 한다. 또 아이들이 즐길 수 있도록 숫자를 사용하지 않고 가로.세로 네 줄 정도의 단순한 형태에 색색의 도형을 이용한 것도 있다.

" Fill in the grid so that every row,
every column, and every 3x3 box
contains the digits 1 through 9. "

Roll your mouse on and off the grid.
You may have to wait for the page to load fully.

Here are the rules of the puzzle —

The digits to be entered are 1, 2, 3, 4, 5, 6, 7, 8, 9.

This is a row, 9 cells wide. A filled-in row must have one of each digit. That means that each digit appears only once in the row. There are 9 rows in the grid, and the same applies to each of them.

This is a column, 9 cells tall. A filled-in column must have one of each digit. That means that each digit appears only once in the column. There are 9 columns in the grid, and the same applies to each of them.

This is a box, containing 9 cells in a 3x3 layout. A filled-in box must have one of each digit. That means that each digit appears only once in the box. There are 9 boxes in the grid, and the same applies to each of them.

You can't change the digits already provided in the grid.
You have to work around them.

Every puzzle has just one correct solution.

The Sudoku program has an option which tells you if you've put a number in the wrong place. You might like to leave that option switched On while you're getting used to things.

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There is really only one rule: Fill in the grid so that every row, every column, and every 3 x 3 box contains the digits 1 through 9.
	This means that —

How to solve

There's no right or wrong place to start – but we do have to start somewhere, so let's look at the three boxes at the top of the puzzle.

There's a 1 in the middle box, and a 1 in the box on the right – but the box on the left still needs a 1. At first glance you might think the 1 could go into any of the four empty cells. However, the 1 cannot go in the top row of the box, because the top row of the grid already has a 1. A grid-row has room for only one of each number. Nor can the 1 go in the second row of the box because the second row of the grid already has a 1. There's only one place left for the 1 to go. Roll your mouse to see.

Look for similar patterns elsewhere. You'll see something similar with the 2s in the stack of boxes down the middle. The top box doesn't have its 2 yet. There is a 2 in each of the center and bottom boxes, so we can rule out five of the seven empty cells in the top box. This is not quite the same situation as before, because there are two places the 2 could go. Some people just guess where the 2 should go, but it is better to use logic. It's best to have a reason why a particular number must go in a particular place.

It's time to look in the other direction – at the rows of the grid, not just the columns. The third grid-row from the top already has its 2, so we can rule out one extra cell from the top box. Now there's only one place left for the 2 to go.

In fact, you might like to lightly pencil a small ‘2’ into each of the possible cells. In the Sudoku program these small numbers are called pencilmarks and they can be added and removed very easily.

Pencilmarks are useful reminders, but be sparing with them. Using too many can cloud the issues you are trying to clarify.

The techniques on this page will get you launched into solving, but to finish a puzzle you may need to discover other techniques as well. Maybe you would prefer to make those discoveries on your own. It's certainly possible – you don't need to read this stuff. So, if this page has sparked enough ideas for you to start solving right now, good luck and enjoy!

For those who would like to read more, take a look at 2. Making progress, 3. Reaching out, and 4. Raising numbers. They were designed to be read in that order, but it's not essential.

Using the techniques described in 1. Getting started, there's only one place left for the 4 in the bottom-right box. Roll your mouse to see. Now the box has only three empty cells. The missing numbers are 1, 6, 9.

When there are only two or three empty cells left in a row, a column or a box, it's worth checking to see if we can fill one of the cells – or perhaps all of them.

Looking at the bottom-right box, you can see that the 1 is easy to place. We don't even need help from the intersecting rows.

Roll your mouse on and off the diagrams.

We are still looking at the bottom-right box. The 9 is also easy to place. We don't need any help from the intersecting columns.

Notice that the order of entering the numbers was important: 4, 1, 9, 6. If we had tried the 9 first, for example, we would have had no luck.

The box did not put up much resistance, but things are not always that convenient. Look at the top-left box, for example. It has only three empty cells but we can't finish the box, just yet. Even though a puzzle has an easy corner, it's no guarantee the puzzle as a whole is easy.

2. Making progress

Now we are down to just one empty cell. Whenever any row, column or box has only one empty cell, it's just a matter of checking to see which number is missing.

Let's try and complete the 6s in the band of boxes across the middle.

The center box already has its 6.

Let's start with the lefthand box. No luck in actually filling in a number. There are two places a 6 could go. We will mark them with pencilmarks.

Roll your mouse on and off the diagrams.

Next, the righthand box. No luck there, either. There are two places a 6 could go. We will mark them with pencilmarks.

But don't move away yet!
You know more than you think you know.

These interplays and interactions are endless, in number and variety. You will discover them for yourself as you go along, and that's where the most enjoyment lies. Every puzzle will be different.

It's possible to place a 5 in the top-left box, using just the information we have now.
Roll your mouse for a sneak peek.

How can that be? Surely there isn't enough information! The puzzle has only one 5.

Actually, it's quite common for a puzzle to have only one of a particular digit – or even none!
In such cases, you have to make something out of nothing.

We don't really need the bottom-right 5 at all. We can add the 5 to the top-left box, without it.

There's not enough information to fix the position of the 4 in the top-left box.

There are two places a 4 could go. We can mark them with pencilmarks.

We could use pencilmarks to show the two places a 7 can go.

In real life, we might not bother actually writing the pencilmarks in – but here, it makes things clear.

Here are the combined pencilmarks. They share the same two cells. One of those cells must be the real 4, and the other must be the real 7. We don't know exactly where the 4 goes or exactly where the 7 goes, but we do know where the pair goes! There is only one empty cell left, and that must be for the only missing number – the 5.

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Well, it's over to you, now. There are still lots of moves in plain view, if you look in the right places! Using just the numbers we have so far, you should be able to add —

If you try to finish the puzzle but can't, don't lose heart. The puzzle was graded Medium by our Sudoku program which made it up. You could try an Easy puzzle next time, or even a Very Easy. If you found the puzzle too easy, let Sudoku make you some Hard ones!

Roll your mouse on and off the diagrams.

the 8 in the bottom-left box the 1 in the bottom-middle box the 9 in the top-right box the 2 in the bottom-left box the 8 in the middle-right box It doesn't matter which you do first. In Sudoku, you take your successes where you can.

If we looked at the 7, there would still be no luck.

4. Raising numbers

Start by noting that the three numbers missing from the top-left box are 4, 5, 7.

The only 4 that can influence the outcome is the 4 in the second column from the left.

3. Reaching out

Let's look again at the pencilmarks we made. You don't know exactly where the 6 goes in the lefthand box, but you do know that it will go in the bottom row of the box – because that's where both pencilmarks are. Whichever one is the real 6, it will not only be the 6 for the box, but for that entire row of the grid as well. Of course we can't have two 6s in one row, so the pencilmark we made in the bottom row of the righthand box must be wrong. That leaves us with only one pencilmark in the righthand box. We can fill in a 6 after all! There's interplay between rows, columns and boxes. There's also interplay between the snippets of information you have, so you have to be aware of what's happening in the grid as a whole – not just the rows, columns and boxes.

Let's play a typical puzzle. This one happens to be a Medium puzzle.

Roll your mouse on and off the diagrams.

1. Getting started

Roll your mouse on and off the diagrams.

Sometimes you can build on success. Let's see if that 2 leads us to success in the top-right box. After considering the rows, we can rule out only four of the seven empty cells. Even after considering the columns, we can rule out only five cells, still leaving two places for the 2 to go. Never mind, we learned something about where the 2 can go in the top-right box.

What next?