The History of Sudoku: Number Place, Japan and a New Zealander

Sudoku is American and it is from 1979. The name is Japanese and it is from 1984. The craze has an exact date, 12 November 2004, and the person who started it was a retired judge from New Zealand who had spent six years writing a computer program.

Three countries, three decades, and no single inventor. That is the honest answer to "who invented Sudoku", and everything below is the detail behind each of those three steps.

One thing to drop before we start. The puzzle is not ancient. It gets sold as an old Japanese pastime because the name is Japanese and a grid of boxes looks timeless on a page, but the modern puzzle is younger than the pocket calculator.

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The timeline on one screen

Two gaps in that table are the interesting part. Five years pass between an American magazine puzzle and a Japanese reprint. Twenty years pass between the Japanese reprint and the London column that made it global. Nothing about the puzzle changed in either gap. What changed was who could get hold of it.

Latin squares, and the piece the 1780s were missing

A Latin square is a grid where every symbol appears once in each row and once in each column. Leonhard Euler worked on them in the 1780s, and that work is a real ancestor of the puzzle you solve on the bus. Here is a small one.

1234
2341
3412
4123

Every row holds 1 to 4. So does every column. It is a flawless Latin square, and it is not a Sudoku, because the top left 2x2 block holds 1, 2, 2 and 3.

That block is the entire difference. Sudoku is a Latin square plus a third constraint: the grid is carved into boxes, and each box must also hold every digit once. Nine rows, nine columns and nine boxes is 27 regions, and it is the nine boxes that make the puzzle solvable by scanning rather than by algebra. Cross-hatching, the first technique anyone learns, is a move you can only make because the boxes exist.

Sudoku was not created in the 1780s by a Swiss mathematician. Euler gave the world the rows and the columns. The boxes arrived roughly two hundred years later, and that claim about him is the most repeated wrong fact in Sudoku writing.

May 1979: Number Place, in a magazine that named nobody

The modern puzzle appears in the May 1979 issue of Dell Pencil Puzzles & Word Games, under the title Number Place. Same 9x9 grid, same three constraints, same one answer. If you solved one today you would not notice anything unusual.

It is credited to Howard Garns, a retired architect from Indianapolis. Read that verb carefully, because it is doing real work.

Dell did not print an author line on its puzzles. Nobody signed Number Place. The attribution rests on a pattern instead: Garns's name appears in the contributor lists of the issues that carried the puzzle and is absent from the issues that did not. That is strong circumstantial evidence, and it is not a signature.

So the correct sentence is "credited to Howard Garns", not "invented by Howard Garns". Pages that state the second one are treating a list of names in a magazine as a byline. The difference matters here more than it usually would, because Garns died in 1989, fifteen years before The Times printed its first, and never knew what the puzzle became. He cannot confirm anything, and no document does it for him.

1984: Japan gave it a name and two rules the rules do not require

Nikoli, a Japanese puzzle publisher, ran it in 1984 under the title 数字は独身に 限る, roughly "the digits must be single". Maki Kaji, of Nikoli, cut that down to two characters, 数独, and that abbreviation is the label the rest of the world was eventually handed. What the word carries takes the two characters apart.

Japan gave the puzzle more than a name. Nikoli added two conventions that a solver still sees every day, and neither is part of the rules:

1. At most 32 givens.
2. Symmetric clue placement.

Symmetry is easier to see than to describe. Here is the clue pattern of a real puzzle with the digits stripped out, X for a given and a full stop for a cell you have to fill.

.X.X..X.X
...X..X..
..X....XX
..X.XXXXX
X.......X
XXXXX.X..
XX....X..
..X..X...
X.X..X.X.

Turn that upside down and it is unchanged. The X at r1c2 has a partner at r9c8, the one at r3c9 pairs with r7c1, and the middle row reads the same backwards. Rotate the page 180 degrees and every clue lands on another clue.

Now put the digits back.

.5.3..1.8
...1..5..
..1....67
..2.74631
7.......2
64352.8..
46....2..
..7..3...
1.9..2.4.

That is 32 givens, exactly on Nikoli's ceiling, arranged in the pattern above, and it has one solution. Solve it if you like. Nothing in the rules of Sudoku demanded either property, and a grid breaking both would be just as valid a puzzle.

What the conventions actually buy is a look. A symmetric grid reads as made rather than generated, and the 32 cap keeps a puzzle from being a fill-in exercise. Neither is a difficulty setting. This is the point where most Sudoku history goes wrong: the clue count is not the difficulty, and a 30-given grid can be far harder than a 24-given one. What sets difficulty is the hardest technique the grid forces on you, and how often it forces it. What actually makes a puzzle hard works through the grading.

I built Sudoku Master. Its four grades sit either side of that Nikoli line: easy carries 36 givens, medium 31, hard 26 to 28, extreme 21 to 27. Only the easy set breaks the 32 ceiling, and that is deliberate, because a first puzzle should not double as a test of stamina. Now look at the last two grades. They overlap. A 27-given puzzle can be graded hard or graded extreme, so the count is not the thing telling them apart.

1997 to 2004: a New Zealander, and six years of programming

Wayne Gould, a retired judge from New Zealand living in Hong Kong, saw the puzzle in a Tokyo bookshop in 1997. He then spent six years writing a program to generate them, and took the output to The Times of London, which printed its first on 12 November 2004.

The trip to the bookshop gets told as the story. The six years are the story.

A daily paper needs a puzzle tomorrow, and the day after, and every day after that, each one graded, each one with exactly one answer, none of them repeats. Hand-built puzzles in the Nikoli tradition are craft objects and take a person time. A program does not get tired and does not run out. Gould did not bring newspapers a puzzle. He brought them a supply, and supply is what turned a magazine curiosity into a fixture that every paper with a spare column could carry.

That date is worth keeping because so much gets dated from it. Sudoku had existed for twenty-five years by then, in two countries, under two names, read by people who liked puzzle magazines. From that column onward it was in newspapers, then in books, then on phones.

What mathematics did once it noticed

By the time newspapers were printing one a day, people who count things had started counting.

The first number is about 6.67 sextillion. The second is the one worth thinking about: shuffle a grid's rows within a band, swap two digits, reflect it, rotate it, and you get a different arrangement of the same puzzle. Discount all of that and roughly five and a half billion grids remain. Still more than anyone will ever solve, and about a trillion times fewer than the first figure.

The 17 result was settled by exhaustion. McGuire, Tugemann and Civario had a program check every 16-clue configuration and found that none of them has a single answer. That is an accepted result and it is not a pencil-and-paper proof, so "settled by computer search" is the fair way to describe it. Seventeen-clue puzzles exist and get collected. Sixteen clues can be printed, but no arrangement of them pins down a single answer, which leaves you with a shape rather than a puzzle. Seventeen clues has the full result.

One more finding gets quoted badly, so here it is properly. Yato and Seta showed in 2003 that Sudoku generalised to n²xn² grids is NP-complete. That is a statement about arbitrarily large grids. It says nothing at all about the one in your newspaper, which is a fixed, finite 81 cells that a computer finishes in milliseconds. Anyone using it to argue that Sudoku is hard for machines has read the theorem backwards. How a solver actually works is the plain version.

So who invented Sudoku?

Four names come up. Here is what each of them did, and how solid the evidence for it is.

If you need one name for a pub quiz, it is Howard Garns, with the word "credited" attached. One date? 12 November 2004. And if somebody tells you a Japanese monk was doing these in the eighteenth century, they have merged three separate facts into one wrong one.

Myths worth dropping

"Sudoku is an ancient Japanese puzzle." Two of those words are wrong. It is not ancient, and the grid is not Japanese. The label is, and the label was coined in 1984.

"Euler came up with it in the 1780s." He worked on Latin squares, which are rows and columns only. The box constraint, the thing that makes the puzzle scannable, is not in that work.

"Sudoku means difficult." It does not. The word is Nikoli's shorthand for the rule about repeats, and the grade printed above a puzzle has nothing to do with it.

"Wayne Gould made it up." He found it, wrote the generator, and sold it to newspapers. Every one of those matters, and none of them is authorship.

"Fewer clues means a harder puzzle." No. A 24-given grid full of hidden singles is easier than a 30-given grid that needs an X-Wing. Clue count and difficulty are two different measurements that happen to correlate loosely.

"Symmetric clues are one of the rules." They are a Nikoli house style, adopted because it made the puzzles look hand-built. An asymmetric grid with 40 givens is a perfectly legal Sudoku.

"Nobody knows who made the first one." Overcorrecting the other way is also wrong. The Garns attribution is well supported. It just is not signed, so the right word is "credited".

Questions people ask

Is Sudoku Japanese?

The name is, the puzzle is not. It was published in the United States in 1979 as Number Place and reached Japan five years later, where Nikoli gave it the name that travelled and the conventions that shaped how it looks. Calling it a Japanese puzzle is half right, and the half that is right is the half everybody says out loud.

Who was Howard Garns?

A retired architect from Indianapolis, credited with the Number Place puzzles that ran in Dell Pencil Puzzles & Word Games from May 1979. Dell did not name its puzzle authors, so the attribution rests on his name appearing in the contributor lists of the issues that carried it and not in the others. He died in 1989, before the puzzle became famous anywhere.

Why is it called Sudoku and not Number Place?

Because the copy that reached the West came from Tokyo, not from Dell. Wayne Gould bought a book of them there in 1997, and the Japanese name came home in the same bag. Dell's English title never left the American magazine racks.

Who was Maki Kaji?

The person at Nikoli who named the puzzle in 1984. That one decision is why the English-speaking world uses a Japanese word for an American puzzle.

Did Leonhard Euler create Sudoku?

No. He studied Latin squares in the 1780s: one of each symbol per row, one per column. Sudoku bolts boxes on top, and the boxes are the thing you scan. Two centuries sit between the two.

When did Sudoku become popular?

In Japan, from 1984 onwards through Nikoli's magazines. In the English-speaking world, from 12 November 2004, when The Times of London ran its first. The global boom is dated from that column, not from the 1979 original, which circulated quietly in a puzzle magazine for a quarter of a century.

How many Sudoku puzzles are there?

There are 6,670,903,752,021,072,936,960 valid completed grids, and 5,472,730,538 of those are genuinely different once rotations, reflections and digit relabelling are discounted. Both figures count finished grids rather than puzzles: one completed grid yields an enormous number of puzzles, depending on which cells you keep as clues.

Are Sudoku puzzles made by computer now?

At newspaper scale, yes, and that is exactly what changed in 2004. Gould spent six years on a generator because a daily paper needs a fresh graded puzzle every single day. Nikoli's cap of 32 givens and its symmetric clue placement come out of a slower, hand-made tradition, and a program can be told to respect both.

Has the puzzle itself changed since 1979?

Not the classic one. Nine rows, nine columns, nine boxes, digits 1 to 9, one answer. What has grown up around it is the variant family, where the boxes get reshaped or the diagonals get added or arithmetic gets bolted on. Types of Sudoku goes through them.

Keep reading

• What the word Sudoku actually means, for the two characters and why "difficult" is not one of them
• Seventeen clues: the smallest Sudoku that can exist, for the 2012 search that settled the minimum
• How to play Sudoku, the same three rules the 1979 puzzle already had
• What actually makes a Sudoku easy, medium, hard or extreme, which is not the clue count
• Types of Sudoku, from classic to Samurai, for everything built on top of the 1979 grid
• The full Sudoku guide, the whole series in one page, if you want the map before the detail

Get it: Sudoku Master, free on iPhone and Android. The link sends you to whichever store your phone uses.