The family of number-grid puzzles

Latin Square, Inequality Grid, Skyline Grid, Mini Number Place, Black & White Grid. They look different, but they all stand on one foundation: fill a square grid so that nothing repeats in any row or column. Change the kind of clue, and you get a new puzzle.

Latin Square (the site’s “Latin Square”)

A Latin square puts n kinds of symbols into an n × n grid so that each symbol appears exactly once in every row and every column. The name is often explained by Euler’s use of Latin letters (a, b, c…) as the symbols.

For 4 × 4, there are 576 complete squares in all. This site’s program counted every one of them to check. There is 1 square of size 1 × 1, 2 of size 2 × 2, and 12 of size 3 × 3, and then the count jumps to 576 at 4 × 4.

Euler did not invent the Latin square. Similar arrangements are said to have been used before him as a tool for building magic squares. Euler is the one who studied them systematically.

The 36 officers problem, and Euler’s wrong guess

Take 36 officers, one for each combination of 6 regiments and 6 ranks, and place them in a 6 × 6 grid so that every row and every column contains each regiment once and each rank once. In other words: lay two Latin squares on top of each other so that every pairing is different. Euler could not solve it. He guessed that such a pair cannot exist whenever the side of the grid leaves remainder 2 when divided by 4 (2, 6, 10, …).

At the start of the 20th century, the Frenchman Gaston Tarry showed that 6 × 6 is impossible by checking every case. Sources give the year as 1900 or 1901. But the rest of the guess was wrong. In 1959 and 1960, Bose, Shrikhande and Parker built examples for sides 22 and 10 and showed that such pairs exist for every size except 2 and 6. Parker’s discovery used a computer. So the answer to the 36 officers problem (impossible) was right, and the family of solvable cases was bigger than Euler thought.

Inequality Grid (the site’s “Inequality Grid”)

This adds clues of the form “smaller than the neighbour” to a Latin square. It is said to have been devised in Japan around 2001. It goes by other names too. Every puzzle on this site has exactly one solution and can be solved by narrowing candidates from the signs and from the rows and columns. The hints give a reason in words, such as “smaller than the largest candidate of the bigger cell.”

Skyline Grid (the site’s “Skyline Grid”)

Instead of digits you place building heights, and the numbers outside the grid say how many buildings can be seen from that side. A low building behind a tall one is hidden. This is also a Latin-square relative. It is said to have spread through Japanese puzzle magazines, but who devised it and when is not clear.

Four buildings can be lined up in 24 ways. Counted by how many you can see, a clue of 1 fits 6 orders, a clue of 2 fits 11, a clue of 3 fits 6, and a clue of 4 fits just 1 (24 in all). This site’s program checked those numbers. A clue of 4 means the buildings rise one after another; a clue of 1 means the nearest is the tallest. The bigger the number, the more tightly the order is fixed.

Mini Number Place (the site’s “Mini Number Place”)

A beginner’s version of the 9 × 9 Number Place, shrunk to 6 × 6 with 2 × 3 boxes. A digit may not repeat in a row, a column or a box. The history of Number Place itself is in the article “The story of Number Place.” Who first made the 6 × 6 version, and when, has not been traced.

Black & White Grid (the site’s “Black & White Grid”)

This one uses two colours instead of digits. The rules on this site: no three of the same colour in a row or column; each row and column has equal numbers of black and white; no two rows or two columns are identical. It is a relative of the others, with tricks of its own. For example, when two cells of the same colour sit side by side, the cells on both ends must be the opposite colour. Several stories exist about its origin, and who came first is unclear. It also has many other names.

All of them are “a Latin square plus a clue”

• No clue: the Latin square. You reason with rows and columns alone.
• Inequality signs: neighbouring sizes cut down the candidates.
• Visible counts: the outside numbers show which heights cannot go where.
• Boxes: on top of rows and columns, nothing repeats inside a box.
• Two colours: you work with the no-three-in-a-row rule and the equal-count rule.

In every one of these puzzles, hints come in the order idea, then which cell to look at, then the answer. When stuck, try just the idea hint first.