.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/combinatorics/designs/plot_01_latin_squares.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_combinatorics_designs_plot_01_latin_squares.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_combinatorics_designs_plot_01_latin_squares.py:


Euler's officers: orthogonal Latin squares
================================================

Builds a pair of orthogonal Latin squares of order 5 and superimposes
them so that every (rank, regiment) pair appears exactly once, the
arrangement Euler's 36-officers problem asks for. It then checks by
brute force that no such pair exists for order 2.

.. GENERATED FROM PYTHON SOURCE LINES 12-19

.. code-block:: Python

    from itertools import permutations

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.combinatorics import are_orthogonal, is_latin_square, orthogonal_latin_square_pair








.. GENERATED FROM PYTHON SOURCE LINES 20-22

An orthogonal pair of order 5
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-37

.. code-block:: Python


    ranks, regiments = orthogonal_latin_square_pair(5)
    print("ranks:\n", ranks)
    print("regiments:\n", regiments)
    print(f"orthogonal: {are_orthogonal(ranks, regiments)}")

    fig, axes = plt.subplots(1, 2, figsize=(8, 4))
    for ax, square, title in zip(axes, (ranks, regiments), ("rank", "regiment")):
        ax.imshow(square, cmap="tab10")
        for (i, j), v in np.ndenumerate(square):
            ax.text(j, i, str(v), ha="center", va="center", color="w")
        ax.set_title(title)
        ax.axis("off")
    fig.suptitle("Every (rank, regiment) pair appears exactly once")




.. image-sg:: /api/gallery/combinatorics/designs/images/sphx_glr_plot_01_latin_squares_001.png
   :alt: Every (rank, regiment) pair appears exactly once, rank, regiment
   :srcset: /api/gallery/combinatorics/designs/images/sphx_glr_plot_01_latin_squares_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    ranks:
     [[0 1 2 3 4]
     [1 2 3 4 0]
     [2 3 4 0 1]
     [3 4 0 1 2]
     [4 0 1 2 3]]
    regiments:
     [[0 1 2 3 4]
     [2 3 4 0 1]
     [4 0 1 2 3]
     [1 2 3 4 0]
     [3 4 0 1 2]]
    orthogonal: True

    Text(0.5, 0.98, 'Every (rank, regiment) pair appears exactly once')



.. GENERATED FROM PYTHON SOURCE LINES 38-40

No orthogonal pair of order 2
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 40-45

.. code-block:: Python


    squares = [np.array(rows) for rows in permutations(permutations(range(2)), 2)]
    latin = [s for s in squares if is_latin_square(s)]
    found = any(are_orthogonal(a, b) for a in latin for b in latin)
    print(f"order 2: {len(latin)} Latin squares, orthogonal pair exists: {found}")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    order 2: 2 Latin squares, orthogonal pair exists: False





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.028 seconds)


.. _sphx_glr_download_api_gallery_combinatorics_designs_plot_01_latin_squares.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_latin_squares.ipynb <plot_01_latin_squares.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_latin_squares.py <plot_01_latin_squares.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_latin_squares.zip <plot_01_latin_squares.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_