How many 4x4 latin squares are there

Partitions of Integers 4. Recurrence Relations 5. Catalan Numbers 4 Systems of Distinct Representatives 1. Existence of SDRs 2. Partial SDRs 3. Latin Squares 4. Introduction to Graph Theory 5. Matchings 5 Graph Theory 1. The Basics 2. Euler Circuits and Walks 3. Hamilton Cycles and Paths 4. Bipartite Graphs 5. Trees 6. Optimal Spanning Trees 7. Connectivity 8. Graph Coloring 9. The Chromatic Polynomial Coloring Planar Graphs Groups of Symmetries 2. Use a computer program such as LatinSquare.

These web pages show an example analysis of a 4-by-4 Latin square for a treatment factor A with blocking factors B and C. This is followed by further examples of the following common forms of replicated Latin square:. For Model-2 analyses, the replication increases q and n , and hence power to detect the treatment effect s.

For Model-1 analyses, replication that allows estimation of block by treatment interactions will reduce q and power unless the interaction effects are small enough to permit post hoc pooling of error terms. Click here to download a computer program LatinSquare. The best control for sequence effects is to vary the order of presentation of the conditions. In a random order of presentation, the order of presentation of conditions is randomly arranged and participants are randomly assigned to sequences.

In a counterbalanced order of presentation the order is systematically arranged and participants are randomly assigned to the conditions. To determine how many different orders of presentation sequences of conditions are needed for complete counterbalancing, we calculate X! X factorial , in which X is the number of conditions.

A factorial is calculated by multiplying the number of conditions by all integers smaller than the number. In a study with two conditions, there are only two orders of presentation X! Obviously, for more than three or four conditions, complete counterbalancing is not feasible.

Partial counterbalancing offers the best solution. To create a partially counterbalanced order we can randomly select some of the possible orders of presentation, and randomly assign participants to these orders. Alternatively, we could use a Latin square design, a more formalized partial counterbalancing procedure. Latin squares are named after an ancient Roman puzzle that required arranging letters or numbers so that each occurs only once in each row and once in each column.

Use of Latin Square assumes that, for each number of conditions used in studies, there is a theoretical "population" of all possible Latin Square arrangements. The task for the researcher, then, is to select one or more of those possible squares with as little bias as possible — hopefully in such manner that every theoretical square has the same probability of being selected as any other one.

Thus there are two related objectives in calculating a Latin Square for a study: 1 to select a subset from all possible orders of the conditions of the study and 2 to select the subset randomly, as an unbiased choice from all possible orders. Let us suppose that we have a within-subjects study with four conditions, A, B, C, and D. Each participant is to undergo each of the four conditions. Complete counterbalancing would require 24 experimental conditions 4!

By creating a Latin Square we can select an unbiased subset of the 24 conditions, and run our study with good control over sequence effects. The square is laid out in rows and columns, the number of which equals the number of levels or factors. A four-factor study will have four columns and four rows.

A relatively easy way of creating a Latin Square for this four-factor study would proceed as follows:. Begin by setting out the first row and first column of a standard square, that is, one that has its first row and first column arranged in "the standard order," 1,2,3,4, or A, B, C, D. To generate the second row, place the first letter A of the previous row in the last position of the second row. To generate the last two rows, continue that process.

You now have a Standard Latin Square that meets the criteria that each condition must appear exactly once in each row and column--that is, each condition appears once in each position. This Latin Square can now be used in your study as a fairly unbiased arrangement of the conditions of the study.

One or more of your participants can be randomly assigned to the rows, and that will determine the sequence of conditions that each participant undergoes.

However, this square is not completely a random choice from among all of the possible 4 X 4 squares, because we had decided to set the first row and column in the standard sequence i.