I've created my solution in JavaScript. Initially I considered doing a Python and JavaScript hybrid with ajax calls to a Django backend but this seemed like overkill to me. The final solution is implemented as a jQuery plugin (code/jquery.tictac.js).

To view the solution, open code/game.html in your browser.

Using the tictac plugin is easy. It's a basic jQuery plugin that doesn't allow for any customizations in its current form. Create an HTML page with a table marked up like so:

<table id="board" class="board">
    <tr>
        <td id="one" class="even"></td>
        <td id="two"></td>
        <td id="three" class="even"></td>
    </tr>
    <tr>
        <td id="four"></td>
        <td id="five" class="even"></td>
        <td id="six"></td>
    </tr>
    <tr>
        <td id="seven" class="even"></td>
        <td id="eight"></td>
        <td id="nine" class="even"></td>
    </tr>
</table>

Then add the following JavaScript to your HTML page before the </body> tag:

<script src="http://code.jquery.com/jquery-1.5.1.js" type="text/javascript" charset="utf-8"></script>
<script src="path/to/jquery.tictac.js" type="text/javascript" charset="utf-8"></script>
<script type="text/javascript" charset="utf-8">
    $(function() {
        $("#board").tictac();
    });
</script>

The HTML page in code/game.html has a full working example with three boards on it.

Tests were built using QUnit. To run them, simply open code/tests/qunit.html in your browser.