first commit

.dockerignore

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+# Byte-compiled / optimized / DLL files
+__pycache__/
+*.py[cod]
+*$py.class
+
+# C extensions
+*.so
+
+# Distribution / packaging
+.Python
+build/
+develop-eggs/
+dist/
+downloads/
+eggs/
+.eggs/
+lib/
+lib64/
+parts/
+sdist/
+var/
+wheels/
+share/python-wheels/
+*.egg-info/
+.installed.cfg
+*.egg
+MANIFEST
+
+# PyInstaller
+#  Usually these files are written by a python script from a template
+#  before PyInstaller builds the exe, so as to inject date/other infos into it.
+*.manifest
+*.spec
+
+# Installer logs
+pip-log.txt
+pip-delete-this-directory.txt
+
+# Unit test / coverage reports
+htmlcov/
+.tox/
+.nox/
+.coverage
+.coverage.*
+.cache
+nosetests.xml
+coverage.xml
+*.cover
+*.py,cover
+.hypothesis/
+.pytest_cache/
+cover/
+
+# Translations
+*.mo
+*.pot
+
+# Django stuff:
+*.log
+local_settings.py
+db.sqlite3
+db.sqlite3-journal
+
+# Flask stuff:
+instance/
+.webassets-cache
+
+# Scrapy stuff:
+.scrapy
+
+# Sphinx documentation
+docs/_build/
+
+# PyBuilder
+.pybuilder/
+target/
+
+# Jupyter Notebook
+.ipynb_checkpoints
+
+# IPython
+profile_default/
+ipython_config.py
+
+# pyenv
+#   For a library or package, you might want to ignore these files since the code is
+#   intended to run in multiple environments; otherwise, check them in:
+# .python-version
+
+# pipenv
+#   According to pypa/pipenv#598, it is recommended to include Pipfile.lock in version control.
+#   However, in case of collaboration, if having platform-specific dependencies or dependencies
+#   having no cross-platform support, pipenv may install dependencies that don't work, or not
+#   install all needed dependencies.
+#Pipfile.lock
+
+# poetry
+#   Similar to Pipfile.lock, it is generally recommended to include poetry.lock in version control.
+#   This is especially recommended for binary packages to ensure reproducibility, and is more
+#   commonly ignored for libraries.
+#   https://python-poetry.org/docs/basic-usage/#commit-your-poetrylock-file-to-version-control
+#poetry.lock
+
+# pdm
+#   Similar to Pipfile.lock, it is generally recommended to include pdm.lock in version control.
+#pdm.lock
+#   pdm stores project-wide configurations in .pdm.toml, but it is recommended to not include it
+#   in version control.
+#   https://pdm.fming.dev/#use-with-ide
+.pdm.toml
+
+# PEP 582; used by e.g. github.com/David-OConnor/pyflow and github.com/pdm-project/pdm
+__pypackages__/
+
+# Celery stuff
+celerybeat-schedule
+celerybeat.pid
+
+# SageMath parsed files
+*.sage.py
+
+# Environments
+.env
+.venv
+env/
+venv/
+ENV/
+env.bak/
+venv.bak/
+
+# Spyder project settings
+.spyderproject
+.spyproject
+
+# Rope project settings
+.ropeproject
+
+# mkdocs documentation
+/site
+
+# mypy
+.mypy_cache/
+.dmypy.json
+dmypy.json
+
+# Pyre type checker
+.pyre/
+
+# pytype static type analyzer
+.pytype/
+
+# Cython debug symbols
+cython_debug/
+
+# PyCharm
+#  JetBrains specific template is maintained in a separate JetBrains.gitignore that can
+#  be found at https://github.com/github/gitignore/blob/main/Global/JetBrains.gitignore
+#  and can be added to the global gitignore or merged into this file.  For a more nuclear
+#  option (not recommended) you can uncomment the following to ignore the entire idea folder.
+#.idea/

.gitignore

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+# Byte-compiled / optimized / DLL files
+__pycache__/
+*.py[cod]
+*$py.class
+
+.DS_Store
+# C extensions
+*.so
+
+# Distribution / packaging
+.Python
+build/
+develop-eggs/
+dist/
+downloads/
+eggs/
+.eggs/
+lib/
+lib64/
+parts/
+sdist/
+var/
+wheels/
+share/python-wheels/
+*.egg-info/
+.installed.cfg
+*.egg
+MANIFEST
+
+# PyInstaller
+#  Usually these files are written by a python script from a template
+#  before PyInstaller builds the exe, so as to inject date/other infos into it.
+*.manifest
+*.spec
+
+# Installer logs
+pip-log.txt
+pip-delete-this-directory.txt
+
+# Unit test / coverage reports
+htmlcov/
+.tox/
+.nox/
+.coverage
+.coverage.*
+.cache
+nosetests.xml
+coverage.xml
+*.cover
+*.py,cover
+.hypothesis/
+.pytest_cache/
+cover/
+
+# Translations
+*.mo
+*.pot
+
+# Django stuff:
+*.log
+local_settings.py
+db.sqlite3
+db.sqlite3-journal
+
+# Flask stuff:
+instance/
+.webassets-cache
+
+# Scrapy stuff:
+.scrapy
+
+# Sphinx documentation
+docs/_build/
+
+# PyBuilder
+.pybuilder/
+target/
+
+# Jupyter Notebook
+.ipynb_checkpoints
+
+# IPython
+profile_default/
+ipython_config.py
+
+# pyenv
+#   For a library or package, you might want to ignore these files since the code is
+#   intended to run in multiple environments; otherwise, check them in:
+# .python-version
+
+# pipenv
+#   According to pypa/pipenv#598, it is recommended to include Pipfile.lock in version control.
+#   However, in case of collaboration, if having platform-specific dependencies or dependencies
+#   having no cross-platform support, pipenv may install dependencies that don't work, or not
+#   install all needed dependencies.
+#Pipfile.lock
+
+# poetry
+#   Similar to Pipfile.lock, it is generally recommended to include poetry.lock in version control.
+#   This is especially recommended for binary packages to ensure reproducibility, and is more
+#   commonly ignored for libraries.
+#   https://python-poetry.org/docs/basic-usage/#commit-your-poetrylock-file-to-version-control
+#poetry.lock
+
+# pdm
+#   Similar to Pipfile.lock, it is generally recommended to include pdm.lock in version control.
+#pdm.lock
+#   pdm stores project-wide configurations in .pdm.toml, but it is recommended to not include it
+#   in version control.
+#   https://pdm.fming.dev/#use-with-ide
+.pdm.toml
+
+# PEP 582; used by e.g. github.com/David-OConnor/pyflow and github.com/pdm-project/pdm
+__pypackages__/
+
+# Celery stuff
+celerybeat-schedule
+celerybeat.pid
+
+# SageMath parsed files
+*.sage.py
+
+# Environments
+.env
+.venv
+env/
+venv/
+ENV/
+env.bak/
+venv.bak/
+
+# Spyder project settings
+.spyderproject
+.spyproject
+
+# Rope project settings
+.ropeproject
+
+# mkdocs documentation
+/site
+
+# mypy
+.mypy_cache/
+.dmypy.json
+dmypy.json
+
+# Pyre type checker
+.pyre/
+
+# pytype static type analyzer
+.pytype/
+
+# Cython debug symbols
+cython_debug/
+
+# PyCharm
+#  JetBrains specific template is maintained in a separate JetBrains.gitignore that can
+#  be found at https://github.com/github/gitignore/blob/main/Global/JetBrains.gitignore
+#  and can be added to the global gitignore or merged into this file.  For a more nuclear
+#  option (not recommended) you can uncomment the following to ignore the entire idea folder.
+#.idea/

Dockerfile

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+FROM python:latest
+
+WORKDIR /app
+COPY ./requirements.txt /app/
+EXPOSE 8000
+
+RUN pip install -r requirements.txt
+
+COPY . /app
+
+CMD ["gunicorn", "app:app", "-w", "3", "-b", "0.0.0.0:8000"]

app.py

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+from flask import Flask, render_template, request
+import solver
+
+app = Flask(__name__)
+
+
+@app.route("/", methods=["GET", "POST"])
+def index():
+    if request.method == "GET":
+        return render_template("page.html")
+    else:
+        try:
+            M = int(request.form["M"])
+            N = int(request.form["N"])
+        except ValueError:
+            return render_template("page.html", invalid=True)
+        if not (1 <= M <= 40 and 1 <= N <= 40):
+            return render_template("page.html", invalid=True)
+
+        g = solver.construct(M, N)
+        if g is None:
+            return render_template("page.html", nosol=True, M=M, N=N)
+        else:
+            image, works = g.plot()
+            return render_template("page.html", image=image, works=works, M=M, N=N)
+
+
+if __name__ == "__main__":
+    app.run(
+        host="0.0.0.0",
+        port=8000
+    )

requirements.txt

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+blinker==1.7.0
+click==8.1.7
+contourpy==1.2.0
+cycler==0.12.1
+Flask==3.0.0
+fonttools==4.45.1
+gunicorn==21.2.0
+itsdangerous==2.1.2
+Jinja2==3.1.2
+kiwisolver==1.4.5
+MarkupSafe==2.1.3
+matplotlib==3.8.2
+numpy==1.26.2
+packaging==23.2
+Pillow==10.1.0
+pyparsing==3.1.1
+python-dateutil==2.8.2
+six==1.16.0
+Werkzeug==3.0.1

solver.py

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+#!/usr/bin/env python3
+
+import io
+import base64
+
+from matplotlib import pyplot as plt
+import numpy
+from matplotlib.backends.backend_agg import FigureCanvasAgg as FigureCanvas
+from matplotlib.figure import Figure
+
+
+class Grid:
+    def __init__(self, M, N):
+        self.N = N
+        self.M = M
+        self.grid = [[0 for _ in range(M+N-1)] for _ in range(M+N-1)]
+
+    def check_bounds(self, x, y):
+        return 1 <= x <= self.M+self.N-1 and\
+            1 <= y <= self.M+self.N-1 and\
+            self.M+1 <= x+y <= 2*self.M + self.N - 1
+
+    def get(self, x, y):
+        if not self.check_bounds(x, y):
+            return 0
+        return self.grid[x-1][y-1]
+
+    def select(self, x, y):
+        assert self.check_bounds(x, y)
+        self.grid[x-1][y-1] = 1
+
+    def grid_parity(self):
+        for x in range(1, self.M+self.N):
+            nums = [self.get(x, y) for y in range(1, self.N + self.M)]
+            if sum(nums) % 2 != 1:
+                print(f"Found a contradiction at x={x}! {nums}")
+                return False
+        for y in range(1, self.M+self.N):
+            nums = [self.get(x, y) for x in range(1, self.N + self.M)]
+            if sum(nums) % 2 != 1:
+                print(f"Found a contradiction at y={y}! {nums}")
+                return False
+        for s in range(self.M+1, 2*self.M+self.N):
+            nums = [self.get(x, s-x) for x in range(1, s)]
+            if sum(nums) % 2 != 1:
+                print(f"Found a contradiction at x+y={s}! {nums}")
+                return False
+        return True
+
+    def reflect(self):
+        res = [[0 for _ in range(len(self.grid[i]))]
+               for i in range(len(self.grid))]
+        for x in range(self.M+self.N-1):
+            # reflect across x + y = m + n - 2
+            for y in range(self.M+self.N-1):
+                # sm = 2 * (self.M+self.N-2) - (x+y)
+                # diff = x - y
+                res[self.M+self.N-2-x][self.M+self.N-2-y] = self.grid[x][y]
+        self.grid = res
+        tmp = self.M
+        self.M = self.N
+        self.N = tmp
+
+    def plot(self) -> tuple[str, bool]:
+        fig = Figure()
+        ax = fig.add_subplot(1, 1, 1)
+        ax.set_xticks(numpy.arange(1, self.M+self.N, 1))
+        ax.set_yticks(numpy.arange(1, self.M+self.N, 1))
+        ax.set_aspect("equal")
+
+        ax.set_xbound(0, self.M+self.N)
+        ax.set_ybound(0, self.M+self.N)
+        ax.autoscale(enable=False)
+        x = []
+        y = []
+        for i in range(1, self.M+self.N):
+            for j in range(1, self.M+self.N):
+                if self.get(i, j):
+                    x.append(i)
+                    y.append(j)
+        ax.scatter(x, y, color='b')
+        # plt.title(f"Construction for M={self.M}, N={self.N}")
+        ax.grid()
+
+        ax.plot([1, self.M], [self.M, 1], color='r')
+        ax.plot([self.M, self.N+self.M-1],
+                [self.N+self.M-1, self.M], color='r')
+        ax.plot([1, 1, self.M], [self.M, self.M +
+                                 self.N-1, self.M+self.N-1], color='r')
+        ax.plot([self.M, self.M+self.N-1, self.M+self.N-1],
+                [1, 1, self.M], color='r')
+        pngImage = io.BytesIO()
+        FigureCanvas(fig).print_png(pngImage)
+        pngImageB64String = "data:image/png;base64,"
+        pngImageB64String += base64.b64encode(
+            pngImage.getvalue()).decode('utf8')
+        return pngImageB64String, self.grid_parity()
+
+
+def one(M, N) -> Grid:
+    assert (N-M) % 4 == 0
+    assert M == 1 or N == 1
+    reflect = M == 1
+    if reflect:
+        tmp = M
+        M = N
+        N = tmp
+    g = Grid(M, N)
+    for k in range(1, M+1):
+        g.select(k, M+1-k)
+
+    for k in range(1, M//2+1):
+        g.select((M+1)//2, (M+1)//2+k)
+        g.select(M, (M+1)//2+k)
+
+    if reflect:
+        g.reflect()
+    return g
+
+
+def cong_mod4(M, N) -> Grid:
+    assert (N-M) % 4 == 0 and N != 1 and M != 1
+    reflect = M > N
+    if reflect:
+        tmp = M
+        M = N
+        N = tmp
+    g = Grid(M, N)
+
+    if M % 2 == 1:
+        g.select(M, M)
+
+    # Main Axis points
+    for k in range(1, N//2+1):
+        g.select(M, M+2*k-1)
+    for k in range(1, (M-1)//2+1):
+        g.select(M, M-2*k)
+
+    for k in range(1, (N-1)//2+1):
+        g.select(M+2*k, M)
+    for k in range(1, M//2+1):
+        g.select(M-2*k+1, M)
+
+    # Points on the diagonal
+    for k in range(1, M//2 + 1):
+        g.select(M+2*k-1, M-2*k+1)  # going down
+    for k in range(1, (M-1)//2+1):
+        g.select(M-2*k, M+2*k)  # going up
+
+    for k in range(1, (N-M)//2+1):
+        g.select(
+            M+2*(M//2) - 1 + 2*k,
+            M-2*(M//2) + 2,
+        )
+        g.select(
+            M - 2*((M-1)//2) + 1,
+            M + 2*((M-1)//2) + 2*k,
+        )
+    if reflect:
+        g.reflect()
+    return g
+
+
+def mod_0_1(M, N):
+    assert (M+N) % 4 == 1 and M % 4 in [0, 1]
+    reflect = (M % 4) == 1
+    if reflect:
+        tmp = M
+        M = N
+        N = tmp
+    g = Grid(M, N)
+
+    g.select(M, M)
+    # Main Axis Points
+    for k in range(1, N//2+1):
+        g.select(M, M+2*k-1)
+    for k in range(1, (M-1)//2+1):
+        g.select(M, M-2*k)
+
+    for k in range(1, (N-1)//2+1):
+        g.select(M+2*k, M)
+    for k in range(1, M//2+1):
+        g.select(M-2*k+1, M)
+    # Step 3
+    for k in range(0, M//2):
+        g.select(M-2*k, 1+2*k)
+
+    # Tail
+    for k in range(1, (N-1)//2+1):
+        g.select(2, M+2*k)
+        g.select(M+2*k-1, 3)
+
+    if reflect:
+        g.reflect()
+    return g
+
+
+def mod_2_3(M, N):
+    assert (M+N) % 4 == 1 and M % 4 in [2, 3]
+    reflect = (M % 4) == 2
+    if reflect:
+        tmp = M
+        M = N
+        N = tmp
+    g = Grid(M, N)
+
+    for k in range(1, M+1):
+        g.select(k, M+1-k)
+
+    for k in range((M+1)//2+1, M+1):
+        g.select((M+1)//2, k)
+        g.select(M+1, k)
+    g.select((M+1)//2, M+1)
+
+    for k in range(1, (N-2)//2 + 1):
+        g.select(M+1, M+2*k)
+        g.select(M+2*k, M)
+
+        g.select(1, M+1+2*k)
+        g.select(M+1+2*k, 1)
+
+    if reflect:
+        g.reflect()
+    return g
+
+
+def construct(M, N) -> 'Grid | None':
+    if (M-N) % 4 == 0:
+        if M != 1 and N != 1:
+            return cong_mod4(M, N)
+        else:
+            return one(M, N)
+    elif (M+N) % 4 == 1:
+        if (M % 4) in [0, 1]:
+            return mod_0_1(M, N)
+        else:
+            return mod_2_3(M, N)
+    else:
+        assert ((N+M-1)*(N-M)) % 4 == 2
+        return None

templates/page.html

@@ -0,0 +1,70 @@
+<!DOCTYPE html>
+<html lang="en">
+  <head>
+    <title>USAMTS 2/5/35 Solution</title>
+    <meta charset="UTF-8">
+    <meta name="viewport" content="width=device-width, initial-scale=1">
+    <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/@picocss/pico@1/css/pico.min.css">
+    <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.9/dist/katex.min.css" integrity="sha384-n8MVd4RsNIU0tAv4ct0nTaAbDJwPJzDEaqSD1odI+WdtXRGWt2kTvGFasHpSy3SV" crossorigin="anonymous">
+    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.9/dist/katex.min.js" integrity="sha384-XjKyOOlGwcjNTAIQHIpgOno0Hl1YQqzUOEleOLALmuqehneUG+vnGctmUb0ZY0l8" crossorigin="anonymous"></script>
+    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.9/dist/contrib/auto-render.min.js" integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05" crossorigin="anonymous"
+      onload="renderMathInElement(document.body);"></script>
+  </head>
+  <body>
+    <main class="container">
+    <strong><h3>Problem</h3></strong>
+    <p>
+      Let \(m\) and \(n\) be positive integers. Let \(S\) be the set of all points
+      \((x, y)\) with integer coordinates such that \(1 \leq x, y \leq m + n − 1\)
+      and \(m + 1 \leq x + y \leq 2m + n − 1\). Let L be the set of the \(3m + 3n −
+      3\) lines parallel to one of \(x = 0\), \(y = 0\), or \(x + y = 0\) and passing
+      through at least one point in \(S\). For which pairs \((m, n)\) does there
+      exist a subset \(T\) of \(S\) such that every line in \(L\) intersects an odd
+      number of elements of \(T\)?
+    </p>
+    <strong><h3>Proof</h3></strong>
+    <p>
+      The answer is just all \((m,n)\) such that \( (m+n-1)(m-n) \) is divisible
+      by 4. The sum of all the x and y coordinates should clearly be even, but
+      the sum of all \(x+y \mod 2\) is \( \frac{(m+n-1)(3m+n)}{2} \). For a
+      construction to even be viable, this quantity clearly needs to be even.
+    </p>
+    <strong><h3>Constructor</h3></strong>
+    {% if invalid %}
+      <p> Invalid Input! </p>
+    {% endif %}
+    {% if nosol %}
+      <p> No Valid Solution is Possible! </p>
+    {% endif %}
+    {% if image %}
+      <img src="{{ image }}" alt="Solution Image" />
+    {% endif %}
+    {% if works == False %}
+      <p> Claimed Construction Fails!!!! </p>
+    {% endif %}
+    <form method="POST" action="{{url_for ('index')}}">
+      <table role="grid">
+        <tbody>
+          <tr>
+            <td>
+              <label for="M">Value for M:</label>
+            </td>
+            <td>
+              <input type="number" name="M" min="1" max="40" value="{{M}}" required />
+            </td>
+          </tr>
+          <tr>
+            <td>
+              <label for="N">Value for N:</label>
+            </td>
+            <td>
+              <input type="number" name="N" min="1" max="40" value="{{N}}" required />
+            </td>
+          </tr>
+        </tbody>
+      </table>
+      <input type="submit" value="Construct">
+    </form>
+  </main>
+  </body>
+</html>