Latin Square

The easiest construction is a cyclic shift.

1. Write 1 2 3 4 along the first row.
2. Shift left by one for the next row: 2 3 4 1.
3. Continue for all four rows.
4. Check every row and every column.

Now construct a 5×5 without the shifting rule.

1. Fill cells one at a time, obeying both conditions.
2. When you reach a cell with no legal symbol, back up and change an earlier choice.
3. Record how many times you had to backtrack.

This is where the structure earns its place.

1. Mark out a 4×4 grid of plots or positions.
2. Assign four treatments using a Latin square.
3. Note that each treatment appears once per row and once per column.

Euler's question, and it is worth attempting before reading the answer.

1. Build two 4×4 squares, one with numbers and one with letters.
2. Overlay them so each cell holds a number-letter pair.
3. Try to arrange it so all 16 pairs are different — a Graeco-Latin square.
4. Now attempt the same for 6×6.

Euler named them Latin squares because he used Latin letters for the symbols and Greek letters for the second square in a pair — hence Graeco-Latin. His 1782 paper framed the 6×6 case as the 36 officers problem: arrange six regiments of six ranks in a square so that each row and column contains one officer of each rank and one from each regiment. It cannot be done, and Gaston Tarry proved it by exhaustive enumeration in 1901.

The refutation is a good story about conjecture. Euler's guess was reasonable, held for 177 years, and was false. Bose, Shrikhande and Parker — nicknamed Euler's spoilers — constructed the 10×10 case in 1959-60, and it made the cover of Scientific American. A long-standing conjecture by a great mathematician is still only a conjecture.

Where the structure is used now: R. A. Fisher brought Latin squares into experimental design at Rothamsted in the 1920s and 30s, and they remain standard in agriculture, clinical crossover trials and any sensory panel where each taster must try each product in a different order. In software, Latin squares generate balanced test schedules; in telecommunications they underpin certain error-correcting codes.

The condition of use is worth stating: a Latin square controls exactly two sources of variation and assumes they do not interact with the treatment. Where they do interact, or where there are three nuisance factors, you need a different design — Graeco-Latin for three, or a full factorial if you can afford the plots.