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diff --git a/project/templates/numberwall.xhtml b/project/templates/numberwall.xhtml new file mode 100644 index 0000000..0e343ed --- /dev/null +++ b/project/templates/numberwall.xhtml @@ -0,0 +1,1234 @@ + +{%- extends "base_math.xhtml" -%} + + + +{%- block title -%}Number Wall Visualiser{%- endblock -%} + + + +{%- block footer -%}{{ math_footer ("numberwall.xhtml") }}{%- endblock -%} + + + +{%- block style %} + <link href="/css/numberwall.css" rel="stylesheet" /> + <link href="/css/accordian.css" rel="stylesheet" /> +{% endblock -%} + + + +{%- block content %} +<h4>Number Wall Visualiser</h4> + +<p>Git repository: <a href="/cgi-bin/cgit.cgi/number-wall">Link</a><br /> +Mathologer video: +<a href="https://invidious.namazso.eu/watch?v=NO1_-qptr6c" class="external">Invidious</a> +<a href="https://www.youtube.com/watch?v=NO1_-qptr6c" class="external">Youtube</a></p> + +<h5>9/11/2022</h5> + + +<h5>Overview</h5> + +<p>On August 27, 2022, a youtube mathematician called the Mathologer published an interesting video +on the mathematical constructs known as number walls. In the description of that video he also +published a coding challenge of creating an implementation of the number wall algorithm.</p> + +<p>This is not an entry into that coding challenge. This is just a tribute.</p> + +<p>For various reasons this didn't end up finished in the time period required, and in any case it +is not an online implementation and so doesn't qualify. But this project still has some fun tools to +play around with for anyone with an interest in math. Amongst other things, elaborate pictures of +the fractal square patterns produced by large number walls can be made.</p> + +<div class="figure"> + <img src="/img/pagoda_wall_mod_5.png" + alt="Number wall of the Pagoda sequence modulo 5" + height="600" + width="600" /> + <div class="figcaption">Number wall of the <a href="https://oeis.org/A301849" class="external"> + Pagoda sequence</a> with cells equal to 0 mod 5 in orange and others in blue</div> +</div> + +<p>It is strongly recommended that you watch the Mathologer's video for an explanation of what +exactly a number wall is. If, for some reason, you don't want to do that or the video is +inaccessible, then the rules of how to construct a number wall are detailed in the appendix.</p> + + +<h5>Tools</h5> + +<p>There are three programs provided:</p> +<ul> + <li><i>wallgen</i>, a command line program that generates a number wall from a given + input sequence in comma separated value format.</li> + <li><i>wallsolve</i>, a command line program that uses a number wall and polynomials constructed + from a given input sequence to find a recurrence relation that generates that sequence, and then + calculates the next integer using that recurrence relation.</li> + <li><i>visualwall</i>, a graphical program that creates a visual representation of a number wall + with cells equal to zero modulo a given number highlighted in one color, and all other cells + highlighted in a different color.</li> +</ul> + +<p>An example of using <i>wallsolve</i> to calculate the next number in the sequence of integer +cubes:</p> +<div class="precontain"> +<pre> +> wallsolve -i 1,8,27,64,125,216,343,512,729,1000 +S(n) = 4S(n-1) - 6S(n-2) + 4S(n-3) - S(n-4) +Next: 1331 +> +</pre> +</div> + +<p>An example of <i>visualwall</i> has already been given in the overview in the form of the picture +of the Pagoda sequence number wall, and the output of <i>wallgen</i> is too verbose to show here. +</p> + + +<h5>Implementation Details</h5> + +<p>All calculations to produce the number walls in these programs are done using the simple +construction rules in the appendix. No determinants were used, as that method would be more +computationally intensive.</p> + +<p>Since the methods of constructing a number wall given by the Mathologer assume a sequence that is +infinite in both directions and any sequence used by these programs is finite, some compromises had +to be made. In particular, <i>visualwall</i> uses a longer sequence to generate its number wall and +then fits the largest square viewing area it can inside the resulting triangle of calculated values. +</p> + +<p>The polynomial library made for <i>wallsolve</i> assumes that coefficients will always be +integers. This applies even to the division implementation, with only integer division being used +and the remainder term becoming larger than it may otherwise be expected to be if rationals were +permitted. As a number wall calculated from an integer sequence (or a sequence of polynomials with +integer coefficients) will always have integer values, this isn't actually a problem. But care had +to be taken in number wall construction so that division operations were always left until last.</p> + +<p>The modulus values allowed by <i>visualwall</i> are not limited to prime numbers. If a composite +modulus is selected then the highlighted portions of cells equal to zero modulo that number may not +produce squares. The resulting pictures are, however, still interesting to examine.</p> + +<p>At present <i>visualwall</i> only has the option to display either the Pagoda sequence or a +sequence of random numbers, unless a comma separated value formatted sequence is supplied in an +external file. It would not be hard for anyone with a passing knowledge of OCaml to add in different +sequences, but at this stage there is no need.</p> + + +<h5>Appendix: How to Construct a Number Wall</h5> + +<p>A number wall is a grid of numbers that extends infinitely left, right, and down. It begins with +an infinite row of ones at the top, then below that a row containing the input sequence. All other +cells are computed according to the following rules.</p> + +<div class="accordian"> + <input type="checkbox" name="collapse" id="handle1" /> + <label for="handle1">Toggle Cross Rule</label> + <div class="hidden_content"> + +<table class="rule"> + <tr> + <th>Pattern</th> + <th>Relation</th> + <th>Notes</th> + </tr> + <tr> + <td> + <table class="wall"> + <tr> + <td> </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>a</mi> + </math> + </td> + <td> </td> + </tr> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>b</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>c</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>d</mi> + </math> + </td> + </tr> + <tr> + <td> </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>e</mi> + </math> + </td> + <td> </td> + </tr> + </table> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mi>a</mi> + <mo>⁢</mo> + <mi>e</mi> + <mo>+</mo> + <mi>b</mi> + <mo>⁢</mo> + <mi>d</mi> + <mo>=</mo> + <msup> + <mi>c</mi> + <mn>2</mn> + </msup> + </mrow> + </math> + <br /> + <br /> + or alternatively + <br /> + <br /> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mi>e</mi> + <mo>=</mo> + <mfrac> + <mrow> + <msup> + <mi>c</mi> + <mn>2</mn> + </msup> + <mo>-</mo> + <mi>b</mi> + <mo>⁢</mo> + <mi>d</mi> + </mrow> + <mi>a</mi> + </mfrac> + </mrow> + </math> + <br /> + <br /> + where + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mi>a</mi> + <mo>≠</mo> + <mn>0</mn> + </mrow> + </math> + </td> + <td> + This rule is used to calculate the vast majority of cells. + </td> + </tr> +</table> + + </div> +</div> + +<div class="accordian"> + <input type="checkbox" name="collapse" id="handle2" /> + <label for="handle2">Toggle Long Cross Rule</label> + <div class="hidden_content"> + +<table class="rule"> + <tr> + <th>Pattern</th> + <th>Relation</th> + <th>Notes</th> + </tr> + <tr> + <td> + <table class="wall"> + <tr> + <td> </td> + <td> </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>a</mi> + </math> + </td> + <td> </td> + <td> </td> + </tr> + <tr> + <td> </td> + <td> </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>b</mi> + </math> + </td> + <td> </td> + <td> </td> + </tr> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>c</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>d</mi> + </math> + </td> + <td> </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>e</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>f</mi> + </math> + </td> + </tr> + <tr> + <td> </td> + <td> </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>g</mi> + </math> + </td> + <td> </td> + <td> </td> + </tr> + <tr> + <td> </td> + <td> </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>h</mi> + </math> + </td> + <td> </td> + <td> </td> + </tr> + </table> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <msup> + <mi>b</mi> + <mn>2</mn> + </msup> + <mo>⁢</mo> + <mi>h</mi> + <mo>+</mo> + <mi>a</mi> + <mo>⁢</mo> + <msup> + <mi>g</mi> + <mn>2</mn> + </msup> + <mo>=</mo> + <msup> + <mi>d</mi> + <mn>2</mn> + </msup> + <mo>⁢</mo> + <mi>f</mi> + <mo>+</mo> + <mi>c</mi> + <mo>⁢</mo> + <msup> + <mi>e</mi> + <mn>2</mn> + </msup> + </mrow> + </math> + <br /> + <br /> + or alternatively + <br /> + <br /> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mi>h</mi> + <mo>=</mo> + <mfrac> + <mrow> + <msup> + <mi>d</mi> + <mn>2</mn> + </msup> + <mo>⁢</mo> + <mi>f</mi> + <mo>+</mo> + <mi>c</mi> + <mo>⁢</mo> + <msup> + <mi>e</mi> + <mn>2</mn> + </msup> + <mo>-</mo> + <mi>a</mi> + <mo>⁢</mo> + <msup> + <mi>g</mi> + <mn>2</mn> + </msup> + </mrow> + <msup> + <mi>b</mi> + <mn>2</mn> + </msup> + </mfrac> + </mrow> + </math> + <br /> + <br /> + where + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>b</mi> + <mo>≠</mo> + <mn>0</mn> + </math> + </td> + <td> + If cell + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>a</mi> + </math> + goes out of bounds of the wall, it is considered equal to zero. + </td> + </tr> +</table> + + </div> +</div> + +<div class="accordian"> + <input type="checkbox" name="collapse" id="handle3" /> + <label for="handle3">Toggle Horseshoe Rule</label> + <div class="hidden_content"> + +<table class="rule"> + <tr> + <th>Pattern</th> + <th>Relation</th> + <th>Notes</th> + </tr> + <tr> + <td> + <table class="wall"> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>a</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>b</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>c</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>d</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>e</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>f</mi> + </math> + </td> + </tr> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>g</mi> + </math> + </td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>h</mi> + </math> + </td> + </tr> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>i</mi> + </math> + </td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>j</mi> + </math> + </td> + </tr> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>k</mi> + </math> + </td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>l</mi> + </math> + </td> + </tr> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>m</mi> + </math> + </td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>n</mi> + </math> + </td> + </tr> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>o</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>p</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>q</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>r</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>s</mi> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>t</mi> + </math> + </td> + </tr> + </table> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mfrac> + <mrow> + <mi>W</mi> + <mo>⁢</mo> + <mi>X</mi> + </mrow> + <mrow> + <mi>Y</mi> + <mo>⁢</mo> + <mi>Z</mi> + </mrow> + </mfrac> + <mo>=</mo> + <mi>S</mi> + </mrow> + </math> + where + <br /> + <br /> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>W</mi> + </math> + is the common ratio of the arithmetic sequence + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mi>a</mi> + <mo>,</mo> + <mi>b</mi> + <mo>,</mo> + <mi>c</mi> + <mo>,</mo> + <mi>d</mi> + <mo>,</mo> + <mi>e</mi> + <mo>,</mo> + <mi>f</mi> + </mrow> + </math> + <br /> + <br /> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>X</mi> + </math> + is the common ratio of the arithmetic sequence + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mi>t</mi> + <mo>,</mo> + <mi>s</mi> + <mo>,</mo> + <mi>r</mi> + <mo>,</mo> + <mi>q</mi> + <mo>,</mo> + <mi>p</mi> + <mo>,</mo> + <mi>o</mi> + </mrow> + </math> + <br /> + <br /> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>Y</mi> + </math> + is the common ratio of the arithmetic sequence + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mi>a</mi> + <mo>,</mo> + <mi>g</mi> + <mo>,</mo> + <mi>i</mi> + <mo>,</mo> + <mi>k</mi> + <mo>,</mo> + <mi>m</mi> + <mo>,</mo> + <mi>o</mi> + </mrow> + </math> + <br /> + <br /> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>Z</mi> + </math> + is the common ratio of the arithmetic sequence + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mi>t</mi> + <mo>,</mo> + <mi>n</mi> + <mo>,</mo> + <mi>l</mi> + <mo>,</mo> + <mi>j</mi> + <mo>,</mo> + <mi>h</mi> + <mo>,</mo> + <mi>f</mi> + </mrow> + </math> + <br /> + <br /> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>S</mi> + </math> + is equal to + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mn>1</mn> + </math> + if the dimension of the zero square is even, or + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mn>-1</mn> + </math> + if the dimension is odd + </td> + <td> + All cells surrounding a square of zeros are guaranteed to be nonzero.<br /> + <br /> + This rule is applicable to squares of zeros of any finite size, but for the sake of + simplicity the previous rule should be used for isolated zeros.<br /> + <br /> + The cells that this rule should be targeted at calculating are those on the bottom row + below the square of zeros. Reformulating the math to solve for those cells is left as an + exercise to the reader. + </td> + </tr> +</table> + + </div> +</div> + +<div class="accordian"> + <input type="checkbox" name="collapse" id="handle4" /> + <label for="handle4">Toggle Broken Cross Rule</label> + <div class="hidden_content"> + +<table class="rule"> + <tr> + <th>Pattern</th> + <th>Relation</th> + <th>Notes</th> + </tr> + <tr> + <td> + <table class="wall"> + <tr> + <td> </td> + <td> </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>a</mi> + <mn>1</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>a</mi> + <mn>2</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>a</mi> + <mn>3</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>a</mi> + <mn>4</mn> + </msub> + </math> + </td> + <td> </td> + <td> </td> + </tr> + <tr> + <td> </td> + <td> </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>b</mi> + <mn>1</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>b</mi> + <mn>2</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>b</mi> + <mn>3</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>b</mi> + <mn>4</mn> + </msub> + </math> + </td> + <td> </td> + <td> </td> + </tr> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>c</mi> + <mn>1</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>d</mi> + <mn>1</mn> + </msub> + </math> + </td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>f</mi> + <mn>4</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>e</mi> + <mn>4</mn> + </msub> + </math> + </td> + </tr> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>c</mi> + <mn>2</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>d</mi> + <mn>2</mn> + </msub> + </math> + </td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>f</mi> + <mn>3</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>e</mi> + <mn>3</mn> + </msub> + </math> + </td> + </tr> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>c</mi> + <mn>3</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>d</mi> + <mn>3</mn> + </msub> + </math> + </td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>f</mi> + <mn>2</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>e</mi> + <mn>2</mn> + </msub> + </math> + </td> + </tr> + <tr> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>c</mi> + <mn>4</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>d</mi> + <mn>4</mn> + </msub> + </math> + </td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>f</mi> + <mn>1</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>e</mi> + <mn>1</mn> + </msub> + </math> + </td> + </tr> + <tr> + <td> </td> + <td> </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>h</mi> + <mn>4</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>h</mi> + <mn>3</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>h</mi> + <mn>2</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>h</mi> + <mn>1</mn> + </msub> + </math> + </td> + <td> </td> + <td> </td> + </tr> + <tr> + <td> </td> + <td> </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>g</mi> + <mn>4</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>g</mi> + <mn>3</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>g</mi> + <mn>2</mn> + </msub> + </math> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>g</mi> + <mn>1</mn> + </msub> + </math> + </td> + <td> </td> + <td> </td> + </tr> + </table> + </td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mi>Y</mi> + <mo>⁢</mo> + <mfrac> + <msub> + <mi>a</mi> + <mi>n</mi> + </msub> + <msub> + <mi>b</mi> + <mi>n</mi> + </msub> + </mfrac> + <mo>+</mo> + <mi>s</mi> + <mo>⁢</mo> + <mi>W</mi> + <mo>⁢</mo> + <mfrac> + <msub> + <mi>c</mi> + <mi>n</mi> + </msub> + <msub> + <mi>d</mi> + <mi>n</mi> + </msub> + </mfrac> + <mo>=</mo> + <mi>Z</mi> + <mo>⁢</mo> + <mfrac> + <msub> + <mi>g</mi> + <mi>n</mi> + </msub> + <msub> + <mi>h</mi> + <mi>n</mi> + </msub> + </mfrac> + <mo>+</mo> + <mi>s</mi> + <mo>⁢</mo> + <mi>X</mi> + <mo>⁢</mo> + <mfrac> + <msub> + <mi>e</mi> + <mi>n</mi> + </msub> + <msub> + <mi>f</mi> + <mi>n</mi> + </msub> + </mfrac> + </mrow> + </math> + <br /> + <br /> + where + <br /> + <br /> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mi>W</mi> + <mo>,</mo> + <mi>X</mi> + <mo>,</mo> + <mi>Y</mi> + <mo>,</mo> + <mi>Z</mi> + </mrow> + </math> + are the common ratios of the arithmetic sequences along the sides of the square of zeros + as in the Horseshoe Rule<br /> + <br /> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>s</mi> + </math> + is equal to + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mn>-1</mn> + </math> + if + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>n</mi> + </math> + is odd and + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mn>1</mn> + </math> + if + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mi>n</mi> + </math> + is even + </td> + <td> + As with the Horseshoe Rule, this rule is applicable to squares of zeros of any finite + size.<br /> + <br /> + The cells that this rule should be targeted at are those along the bottom denoted as + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>g</mi> + <mi>n</mi> + </msub> + </math> + but as with the previous rule, reformulating the math to solve for those cells is left + as an exercise to the reader.<br /> + <br /> + As with the Long Cross Rule, if any of the cells + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <msub> + <mi>a</mi> + <mi>n</mi> + </msub> + </math> + go out of bounds of the wall they are considered to be equal to zero. This can occur if + there are non-isolated zeros in the input sequence. + </td> + </tr> +</table> + + </div> +</div> + +<p>An example of a number wall:</p> + +<table class="wall"> + <tr> + <td>⋯</td> + <td>1</td> + <td>1</td> + <td>1</td> + <td>1</td> + <td>1</td> + <td>1</td> + <td>1</td> + <td>⋯</td> + </tr> + <tr> + <td>⋯</td> + <td>64</td> + <td>125</td> + <td>216</td> + <td>343</td> + <td>512</td> + <td>729</td> + <td>1000</td> + <td>⋯</td> + </tr> + <tr> + <td>⋯</td> + <td>721</td> + <td>1801</td> + <td>3781</td> + <td>7057</td> + <td>12097</td> + <td>19441</td> + <td>29701</td> + <td>⋯</td> + </tr> + <tr> + <td>⋯</td> + <td>2016</td> + <td>4140</td> + <td>7344</td> + <td>11844</td> + <td>17856</td> + <td>25596</td> + <td>35280</td> + <td>⋯</td> + </tr> + <tr> + <td>⋯</td> + <td>1296</td> + <td>1296</td> + <td>1296</td> + <td>1296</td> + <td>1296</td> + <td>1296</td> + <td>1296</td> + <td>⋯</td> + </tr> + <tr> + <td>⋯</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>0</td> + <td>⋯</td> + </tr> + <tr> + <td>⋰</td> + <td>⋮</td> + <td>⋮</td> + <td>⋮</td> + <td>⋮</td> + <td>⋮</td> + <td>⋮</td> + <td>⋮</td> + <td>⋱</td> + </tr> +</table> + +<p>The above excerpt was taken from the number wall generated by the sequence of integer cubes. In +this case, the wall essentially ends in a row of all zeros as all subsequent rows below that will +also be zeros. Note however that this does not always happen.</p> + +<p>A property of number walls is that zeros will always be arranged in squares. A singular isolated +zero is considered to be a 1x1 square and a wall ending in rows of zeros as above is considered to +be the start of a square of zeros of infinite size. This property can be used to supplement the +calculation rules.</p> + +{% endblock -%} + + |