Scripts hip to be square: rubik's cubes and sudoku

Magic squares may seem esoteric, but their cultural impact is evident whenever you open the newspaper or walk into a toy shop. The two most popular puzzles of recent years – Sudoku and the Rubik's Cube – are both consequences of a centuries-long preoccupation with them.

In the 18th century, Leonhard Euler, the greatest mathematician of his day, was devising ways to create magic squares. In order to do this he started looking at another type of square that could be used as a kind of template for producing magic squares.

Euler's new concept was a square in which every number, or symbol, would appear once and only once in each row and column. While these squares had been known about since at least a few centuries before, Euler was the first mathematician to analyse them systematically and he coined their name "Latin square".

He also invented the sister concept in which two Latin squares are superimposed on each other, and such that each cell in the grid is unique. This he called a "Graeco-Latin Square".

In 1782, Euler set the "36 officers problem", a frivolous puzzle that led to much deep academic work and discoveries. Can you make a 6x6 Graeco-Latin made up of six regiments of six officers each of different ranks so that no rank and regiment is repeated in any row or column? Only in 1901 was it proved that this was impossible.

Unlike magic squares, Latin and Graeco-Latin squares have found many uses and applications in non-mathematical settings, for example in cryptography and biological experiments, experimental design.

The best-known occurrence of Latin squares now, however, is in newspapers and puzzle books. Sudoku is a puzzle to complete a partially completed 9x9 Latin square that contains the digits one to nine in each column and row, with the added specification that the 3x3 sub-squares also contain the numbers from one to nine.

The previous puzzle craze to Sudoku was the Rubik's Cube, whose history can also be traced back to the magic square. In the mid-19th century in upstate New York, Noyes Palmer Chapman, an amateur puzzle enthusiast, made a physical model of a magic square such that the numbers from 1 to 16 were on small wooden squares that could be fit in a 4x4 box. He realised that if he left out one of the squares, it was possible to slide the other 15 squares around. This became known as the "15 Puzzle", which was an international fad in 1880 – and is the original sliding block puzzle, versions of which you can still find in toyshops.

In the 1970s Hungarian designer Ernö Rubik was trying to reinvent the 15 Puzzle in three dimensions when he came up with the idea of the Rubik's Cube.

From the magic square to Sudoku we seem to have always liked our puzzle crazes to come in squares – although this is a matter not for mathematicians, but for psychologists.

Mini-dictionary Recreational Mathematics

an artist	художник
to fascinate	захоплювати, викликати захват
mesmerizing pattern	візерунок, що заворожує
a cell of a grid	елемент сітки
a row / a column	рядок / стовпчик
consecutive natural numbers	послідовні натуральні числа
magic square's order	порядок магічного квадрату
magic constant	магічна стала, константа
adjacent entries	зміст суміжних граф
statesman	державний діяч
matrix / matrices	матриця, форма
to trace back	прослідковувати
protective charm	захисний амулет
to equate with planets	ототожнювати з планетами
millennium / millenia	тисячоліття
to encapsulate the harmonies of the universe	утримувати в собі гармонію всесвіту
a tortoise	черепаха
flat	плаский
square	квадрат, квадратний
rhombus	ромб
an angle	кут
triangle (acute / equilateral / isosceles / obtuse )	трикутник (гострокутний / рівносторонній / рівнобедрений / тупокутний)
an array	сукупність, масив, сітка
face / edge / side	грань / ребро / сторона
quadrilateral	чотирикутник
regular / irregular polygon	правильний / неправильний многокутник
parallelogram	паралелограм
rectangle	прямокутник
crescent-shaped	серпоподібний
trapezium	трапеція
cube / cubic	куб / кубічний
cone / conical	конус / конусоподібний, конічний
plane / arc	площина / арка, дуга
area / volume	площа / об’єм