**These are probably the first plans of experiments. Its history dates back to the eighteenth century by the Swiss mathematician Euler and we meet him in the games of SUDOKU.**

## Introduction

These are probably the first plans of experiments. Its history dates back to the XVIII^{} century by the Swiss mathematician Euler and the problem of the ” *36 officers *“. We all know this type of plan: they are the basis of SUDOKU. Popularized since the middle of the year 1980 by the Japanese, the term SUDOKU is the contraction of the Japanese SUji wa DOKUshin ni Kaguri数字は独身に限る, which can be translated as *“digits can appear* only once”.

In statistics, this type of plan was used in the early twentieth century^{} by **Fisher** to study agricultural productivity issues.

## The principle

This is ultimately a **fractional experience plan**. They have the advantage of being the most ” *logical* “, and for this reason, statisticians appreciate it very much. They are based on the principle of permutation. The rule is as follows:

**In each row and column of a square, the rule requires that each element appear only once and once.**

From here we find the so-called Greco-Latin plans where for each element ” *Latin* ” We find another level possible. At the outset, Euler’s interest in logic was to study the possible combinations of N (Latin) letters to which he associated another (Greek) letter.

In the following representation:

- The vertical axis of the square represents a factor: for example the parcels of a field.
- The horizontal axis of the square is another factor: for example, the weather if we admit that it can have an effect on the growth of our corn and the fact that the weather is different depending on the terrain.
- The letters: they represent another factor. For example the type of fertilizer that will be used.
- For the Greco Latin square, the Latin letters represent the levels of another factor that may be the type of maize.

**Example of a Latin square plan**

**Example of a Greco-Latin plan**

#### EULER and the 36 officers

Euler produces various methods to construct plans of this type odd, even, multiple of 6, of 4… During this research, he noticed that some types of plans were not solvable. It illustrates this through the famous problem of the 36 officers:

*“36 officers of six different ranks and drawn from six different regiments, to be stowed in a square, so that on each horizontal and vertical line there were six officers with a different rank and regiment.”*

It will be that much later, in 1901, that the French Tarry demonstrated why no square Greco Latin of order 6 is possible.

## 1-Choice of factors

The first step is to choose the factors that we want to study and their respective levels. If we want to study three factors, we’ll choose a Latin square plan, if we want to study 4, we’ll take a Greco Latin plan.

Despite their name, we do not have to build ” *squares *“. Indeed, the notion of square dictates that each factor studied has a number of equal level. But in fact, we can quite have a number of different levels between factor.

## 2-Construction of a Latin or Greco-Latin square plan

For example, we want to study three factors. For simplicity, our three factors each have three levels. If we wanted to do all the experiments, we would have to carry out 27 experiments. The Latin square plan makes it possible to make only 9.

For this, we represent in the first time our experimental field in the following way:

Factor 2 may have levels I, II and III

Factor 3 can have levels A, B and C

### 2.1 – Choice of starting point

Our first step is to choose our first experience. There is not really a rule, but most generally we will choose an experience that we know is significant.

For our example, we will choose the position C, II, b

### 2-2-choice of the N + 1 point

The following experiments are chosen according to the initial rule:

**In each row and column of a square, the rule requires that each element appear only once and once.**

In concrete terms, by taking our example, this means that my next point cannot be position II for factor 2 and B for factor 3. So we will choose for example C, I, C

### 2.3 – Finishing the plan

Continuing like this, we realize that by respecting the rule, we get 9 maximum points, which will represent our 9 experiences. In our example, we’ll have:

- C, II, b
- C, I, C
- C, III, a
- b, I, b
- B, II, a
- B, III, C
- A, I, a
- A, II, C,
- A, III, b

By following this rule of construction, we will be able to build plans with experimental domains that are not cubic but parallelepipedic.

## 3 – Significance analysis

Once the experiments are done, we will calculate the influence of our factors and then the significance in:

**for a Latin square:**use a 3-factor Anova**for a Greco-Latin square:**use a 4-factor Anova