Test the Pattern
Numbers in puzzles are often just symbols. Sudoku could use nine letters or nine colours and work identically. Where arithmetic does appear, the lesson stays the same: a pattern is a hypothesis, and hypotheses need testing.
Sudoku: a short, true history
The puzzle we call Sudoku first appeared in 1979 as "Number Place" in Dell Pencil Puzzles and Word Games in the United States. It is generally credited to Howard Garns, a retired architect. In 1984 the Japanese publisher Nikoli introduced it to Japan, and its president Maki Kaji is credited with the name, a shortening of a phrase meaning "the digits must remain single".
Its worldwide boom came after Wayne Gould, a retired judge from New Zealand, wrote software to generate puzzles and The Times in London began printing them in November 2004. Within months, Sudoku was in newspapers across the world.
The appeal is clarity. The rules fit in one sentence, a good puzzle has exactly one solution, and every correct placement can be justified by the clues already on the page.
The 17-clue result was established by Gary McGuire, Bastian Tugemann and Gilles Civario. The grid total of 6,670,903,752,021,072,936,960 was calculated by Bertram Felgenhauer and Frazer Jarvis.
Two techniques that carry you far
Pick a digit and a box. Ask: where in this box can the digit go? If rows and columns already containing that digit rule out every cell but one, the digit goes there. Justification: "the 6 has nowhere else to live in this box."
Pick a cell. List every digit already seen in its row, column and box. If eight digits are ruled out, the ninth is forced. Justification: "this cell can only be 3."
If two cells in a row can only hold 2 and 7, those two digits are spoken for. Remove 2 and 7 from every other cell in that row. You have not placed anything yet, but you have made the grid simpler.
KenKen and Kakuro: when arithmetic joins in
KenKen was created in 2004 by the Japanese maths teacher Tetsuya Miyamoto as a way to help students think rather than memorise. Rows and columns hold each digit once, like Sudoku, and outlined "cages" show a target number and an operation. A two-cell cage marked "3−" in a 4×4 grid must be 1 and 4.
Kakuro, sometimes called Cross Sums, looks like a crossword filled with digits 1 to 9. Each run must add up to its clue with no repeats. The trick is learning a few unique combinations: a two-cell run summing to 3 can only be 1 and 2; a two-cell run summing to 17 can only be 8 and 9.
- Entry: 4×4 KenKen
- Steady: 6×6
- Stretch: large Kakuro
The sequence that fools almost everyone
Place points on a circle and join every pair with straight lines, keeping the points arranged so no three lines meet at one place inside. Count the regions. With 1, 2, 3, 4 and 5 points you get 1, 2, 4, 8 and 16 regions. Doubling, surely.
Exercise: how many regions with 6 points?
Hint
Do not predict. Draw it carefully and count, or trust the pattern only after a test.
Solution and lesson
Six points give 31 regions, not 32. This is known as Moser's circle problem. The pattern fitted five terms perfectly and still failed. The lesson for every number puzzle: a rule earns trust when you understand why it works, not merely that it has worked so far.
The Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, is a happier example. Each term is the sum of the previous two, and you can say exactly why every new term follows. That "why" is what separates a rule from a coincidence.
For a deeper, self-paced challenge that rewards this way of thinking, Project Euler offers hundreds of mathematical problems, best approached with pencil first and code second.
Where number reasoning goes wrong
Stopping at the first fit
Two rules can match the same opening terms. Look for a second check before committing.
Pencil marks left stale
After each placement, update candidates in that row, column and box. Out-of-date notes are the commonest source of Sudoku errors.
Doing sums in your head
In KenKen and Kakuro, list combinations on paper. Small arithmetic slips look like logic failures.