中国剩余定理的几何可视化演示
2020-06-19 15:49
本范例通过几何图形直观演示中国剩余定理的核心概念。利用多边形、正方形和序列等工具,将抽象的整数同余方程组转化为可视化的几何结构。通过调整参数,用户可观察互质整数条件下的最小公倍数及余数分布,帮助深入理解数论中同余理论的基本原理及其几何意义。
GeoGebra 画板及绘图内容加载中,请稍等……
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.588.0" app="classic" platform="w" id="23814eef-1b65-4061-83cd-4469c7984c76" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1200" height="800" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.4083333333333333" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="true" location="1,3" size="463" window="50,50,500,500" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1" size="763" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="244" window="100,100,600,400" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="437" window="683,169,1139,902" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
<view id="1" visible="true" inframe="false" stylebar="true" location="" size="798" window="100,100,600,400" />
</views>
<toolbar show="false" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6 | 100004 100003 100002 100001" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="2"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1198" height="798"/>
<coordSystem xZero="269.0644195917808" yZero="618.1024449477751" scale="51.570974753915046" yscale="51.57097475391508"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="0" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="64" g="64" b="64"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="0" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="false" positiveAxis="true" selectionAllowed="false"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="false" positiveAxis="true" selectionAllowed="false"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="m">
<value val="15"/>
<symbolic val="true" />
<show object="true" label="true"/>
<objColor r="255" g="102" b="153" alpha="0.10000000149011612"/>
<bgColor r="192" g="192" b="192" alpha="255"/>
<layer val="0"/>
<labelOffset x="-16" y="-7"/>
<labelMode val="1"/>
<slider min="1" max="20" width="4" x="0.20999999999999952" y="-2.069999999999998" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="30" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<javascript onUpdate="updateSquares();"/>
</element>
<element type="numeric" label="n">
<value val="9"/>
<symbolic val="true" />
<show object="true" label="true"/>
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
<bgColor r="192" g="192" b="192" alpha="255"/>
<layer val="0"/>
<labelOffset x="-70" y="2"/>
<labelMode val="1"/>
<slider min="1" max="m" width="4" x="-2.0380000000000007" y="1.8640000000000037" fixed="true" horizontal="false" showAlgebra="true"/>
<lineStyle thickness="30" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<javascript onUpdate="updateSquares();"/>
</element>
<element type="point" label="PinX">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.19" y="-0.17" z="1"/>
<pointSize val="5"/>
<pointStyle val="3"/>
</element>
<command name="Sequence">
<input a0="Text[k, (k, 0) + PinX, true, true]" a1="k" a2="0" a3="m - 1"/>
<output a0="labelx"/>
</command>
<element type="list" label="labelx">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="102" b="153" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<font serif="false" sizeM="1.4" size="7" style="1"/>
<angleStyle val="0"/>
</element>
<element type="point" label="PinY">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-0.63" y="0.81" z="1"/>
<pointSize val="5"/>
<pointStyle val="3"/>
</element>
<command name="Sequence">
<input a0="Text[k, (0, k) + PinY, true, true]" a1="k" a2="0" a3="n - 1"/>
<output a0="labely"/>
</command>
<element type="list" label="labely">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<font serif="false" sizeM="1.4" size="7" style="1"/>
<angleStyle val="0"/>
<symbolic val="true" />
</element>
<command name="Sequence">
<input a0="Segment[(k, 0), (k, n)]" a1="k" a2="0" a3="m"/>
<output a0="list_1"/>
</command>
<element type="list" label="list_1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Segment[(0, k), (m, k)]" a1="k" a2="0" a3="n"/>
<output a0="list_2"/>
</command>
<element type="list" label="list_2">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<expression label="a" exp="23"/>
<element type="numeric" label="a">
<value val="23"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="2"/>
<animation speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
</element>
<expression label="A" exp="(Mod[a, m], Mod[a, n])" type="point" />
<element type="point" label="A">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="8" y="5" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="B" exp="(Div[a, m], Div[a, n])" type="point" />
<element type="point" label="B">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="1" y="2" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="PinA">
<show object="false" label="false" ev="4"/>
<objColor r="102" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.03" y="0.93" z="1"/>
<pointSize val="5"/>
<pointStyle val="3"/>
</element>
<element type="numeric" label="dx">
<value val="0.4"/>
<symbolic val="true" />
<show object="true" label="true"/>
<objColor r="255" g="102" b="153" alpha="0.10000000149011612"/>
<bgColor r="128" g="128" b="128" alpha="255"/>
<layer val="0"/>
<labelMode val="2"/>
<slider min="0.1" max="0.5" width="1" x="-4.273310971777028" y="7.110344514111492" fixed="false" horizontal="false" showAlgebra="false"/>
<lineStyle thickness="30" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="dy">
<value val="0.3"/>
<symbolic val="true" />
<show object="true" label="true"/>
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
<bgColor r="128" g="128" b="128" alpha="255"/>
<layer val="0"/>
<labelMode val="2"/>
<slider min="0.1" max="0.5" width="1" x="-3.165046238240424" y="7.097563009407662" fixed="false" horizontal="false" showAlgebra="false"/>
<lineStyle thickness="30" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<command name="LCM">
<input a0="m" a1="n"/>
<output a0="lcm"/>
</command>
<element type="numeric" label="lcm">
<value val="45"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<auxiliary val="true"/>
</element>
<element type="numeric" label="r">
<value val="3"/>
<symbolic val="true" />
<show object="true" label="true"/>
<objColor r="102" g="153" b="0" alpha="0.10000000149011612"/>
<bgColor r="128" g="128" b="128" alpha="255"/>
<layer val="0"/>
<labelMode val="2"/>
<slider min="3" max="6" width="1" x="-2.0959999999999943" y="7.108000000000002" fixed="false" horizontal="false" showAlgebra="false"/>
<lineStyle thickness="30" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<caption val="%v rows"/>
</element>
<command name="Polygon">
<input a0="{A, A + (1, 0), A + (1, 1), A + (0, 1)}"/>
<output a0="rect"/>
</command>
<element type="polygon" label="rect">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="0"/>
<show object="false" label="false" ev="4"/>
<objColor r="102" g="153" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
</element>
<expression label="text1" exp=""\mathbf{gcd}(m,n)=" + (LaTeX[GCD[m, n]]) + """ />
<element type="text" label="text1">
<show object="true" label="false" ev="8"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint x="-4.2399999999999975" y="0.29000000000000214" z="1"/>
</element>
<expression label="text2" exp=""\mathbf{lcm}(m,n)=" + (LaTeX[LCM[m, n]]) + """ />
<element type="text" label="text2">
<show object="true" label="false" ev="8"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint x="-4.219999999999997" y="-0.6299999999999996" z="1"/>
</element>
<expression label="text3" exp=""\phi(m)=" + (LaTeX[phi[m]]) + """ />
<element type="text" label="text3">
<show object="true" label="false" ev="40"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint x="5.220000000000018" y="-1.7900000000000007" z="1"/>
</element>
<expression label="text4" exp=""\phi(n)=" + (LaTeX[phi[n]]) + """ />
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint x="7.900000000000024" y="-1.7700000000000016" z="1"/>
</element>
<expression label="text5" exp=""\phi(mn)=" + (LaTeX[phi[(m * n)]]) + """ />
<element type="text" label="text5">
<show object="true" label="false" ev="40"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint x="10.60000000000003" y="-1.7100000000000006" z="1"/>
</element>
<command name="Sequence">
<input a0="If[GCD[k, m] ≟ 1, Frame[(k, -1), 0.1], NaN]" a1="k" a2="1" a3="m - 1"/>
<output a0="coprimesX"/>
</command>
<element type="list" label="coprimesX">
<show object="true" label="false" ev="4"/>
<condition showObject="b"/>
<objColor r="0" g="0" b="0" alpha="0.25"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="If[GCD[k, n] ≟ 1, Frame[(-1, k), 0.1], NaN]" a1="k" a2="1" a3="n - 1"/>
<output a0="coprimesY"/>
</command>
<element type="list" label="coprimesY">
<show object="true" label="false" ev="4"/>
<condition showObject="b"/>
<objColor r="0" g="0" b="0" alpha="0.30000001192092896"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<element type="boolean" label="b">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="52" y="704"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="show coprimes"/>
</element>
<command name="Sequence">
<input a0="If[GCD[k, (m * n)] ≟ 1, Frame[Remainders[k, m, n], 0.08], NaN]" a1="k" a2="1" a3="(m * n) - 1"/>
<output a0="coprimesA"/>
</command>
<element type="list" label="coprimesA">
<show object="true" label="false" ev="4"/>
<condition showObject="b"/>
<objColor r="0" g="0" b="0" alpha="0.15000000596046448"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="13" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Text[k, Remainders[k, m, n] + PinA + Prod[(Div[Div[k, lcm], r], Mod[Div[k, lcm], r]), (dx, (-dy))], true, true]" a1="k" a2="0" a3="(m * n) - 1"/>
<output a0="numbers"/>
</command>
<element type="list" label="numbers">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<font serif="true" sizeM="0.7" size="-5" style="1"/>
<angleStyle val="0"/>
</element>
<expression label="phantom" exp="0"/>
<element type="numeric" label="phantom">
<value val="0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="2"/>
<labelMode val="2"/>
<animation speed="1" type="0" playing="false"/>
</element>
<command name="MySquare">
<input a0="0 + phantom" a1="m" a2="n"/>
<output a0="sq0"/>
</command>
<element type="polygon" label="sq0">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="1 + phantom" a1="m" a2="n"/>
<output a0="sq1"/>
</command>
<element type="polygon" label="sq1">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="2 + phantom" a1="m" a2="n"/>
<output a0="sq2"/>
</command>
<element type="polygon" label="sq2">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="3 + phantom" a1="m" a2="n"/>
<output a0="sq3"/>
</command>
<element type="polygon" label="sq3">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="4 + phantom" a1="m" a2="n"/>
<output a0="sq4"/>
</command>
<element type="polygon" label="sq4">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="5 + phantom" a1="m" a2="n"/>
<output a0="sq5"/>
</command>
<element type="polygon" label="sq5">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="6 + phantom" a1="m" a2="n"/>
<output a0="sq6"/>
</command>
<element type="polygon" label="sq6">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="7 + phantom" a1="m" a2="n"/>
<output a0="sq7"/>
</command>
<element type="polygon" label="sq7">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="8 + phantom" a1="m" a2="n"/>
<output a0="sq8"/>
</command>
<element type="polygon" label="sq8">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="9 + phantom" a1="m" a2="n"/>
<output a0="sq9"/>
</command>
<element type="polygon" label="sq9">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="10 + phantom" a1="m" a2="n"/>
<output a0="sq10"/>
</command>
<element type="polygon" label="sq10">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="11 + phantom" a1="m" a2="n"/>
<output a0="sq11"/>
</command>
<element type="polygon" label="sq11">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="12 + phantom" a1="m" a2="n"/>
<output a0="sq12"/>
</command>
<element type="polygon" label="sq12">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="13 + phantom" a1="m" a2="n"/>
<output a0="sq13"/>
</command>
<element type="polygon" label="sq13">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="14 + phantom" a1="m" a2="n"/>
<output a0="sq14"/>
</command>
<element type="polygon" label="sq14">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="15 + phantom" a1="m" a2="n"/>
<output a0="sq15"/>
</command>
<element type="polygon" label="sq15">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="16 + phantom" a1="m" a2="n"/>
<output a0="sq16"/>
</command>
<element type="polygon" label="sq16">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="17 + phantom" a1="m" a2="n"/>
<output a0="sq17"/>
</command>
<element type="polygon" label="sq17">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="18 + phantom" a1="m" a2="n"/>
<output a0="sq18"/>
</command>
<element type="polygon" label="sq18">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="19 + phantom" a1="m" a2="n"/>
<output a0="sq19"/>
</command>
<element type="polygon" label="sq19">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="20 + phantom" a1="m" a2="n"/>
<output a0="sq20"/>
</command>
<element type="polygon" label="sq20">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="21 + phantom" a1="m" a2="n"/>
<output a0="sq21"/>
</command>
<element type="polygon" label="sq21">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="22 + phantom" a1="m" a2="n"/>
<output a0="sq22"/>
</command>
<element type="polygon" label="sq22">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="23 + phantom" a1="m" a2="n"/>
<output a0="sq23"/>
</command>
<element type="polygon" label="sq23">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="24 + phantom" a1="m" a2="n"/>
<output a0="sq24"/>
</command>
<element type="polygon" label="sq24">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="25 + phantom" a1="m" a2="n"/>
<output a0="sq25"/>
</command>
<element type="polygon" label="sq25">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="26 + phantom" a1="m" a2="n"/>
<output a0="sq26"/>
</command>
<element type="polygon" label="sq26">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="255" g="127" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="27 + phantom" a1="m" a2="n"/>
<output a0="sq27"/>
</command>
<element type="polygon" label="sq27">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="28 + phantom" a1="m" a2="n"/>
<output a0="sq28"/>
</command>
<element type="polygon" label="sq28">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="29 + phantom" a1="m" a2="n"/>
<output a0="sq29"/>
</command>
<element type="polygon" label="sq29">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="30 + phantom" a1="m" a2="n"/>
<output a0="sq30"/>
</command>
<element type="polygon" label="sq30">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="31 + phantom" a1="m" a2="n"/>
<output a0="sq31"/>
</command>
<element type="polygon" label="sq31">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="32 + phantom" a1="m" a2="n"/>
<output a0="sq32"/>
</command>
<element type="polygon" label="sq32">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="33 + phantom" a1="m" a2="n"/>
<output a0="sq33"/>
</command>
<element type="polygon" label="sq33">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="34 + phantom" a1="m" a2="n"/>
<output a0="sq34"/>
</command>
<element type="polygon" label="sq34">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="35 + phantom" a1="m" a2="n"/>
<output a0="sq35"/>
</command>
<element type="polygon" label="sq35">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="36 + phantom" a1="m" a2="n"/>
<output a0="sq36"/>
</command>
<element type="polygon" label="sq36">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="255" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="37 + phantom" a1="m" a2="n"/>
<output a0="sq37"/>
</command>
<element type="polygon" label="sq37">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="255" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="38 + phantom" a1="m" a2="n"/>
<output a0="sq38"/>
</command>
<element type="polygon" label="sq38">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="255" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="39 + phantom" a1="m" a2="n"/>
<output a0="sq39"/>
</command>
<element type="polygon" label="sq39">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="255" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="40 + phantom" a1="m" a2="n"/>
<output a0="sq40"/>
</command>
<element type="polygon" label="sq40">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="255" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="41 + phantom" a1="m" a2="n"/>
<output a0="sq41"/>
</command>
<element type="polygon" label="sq41">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="255" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="42 + phantom" a1="m" a2="n"/>
<output a0="sq42"/>
</command>
<element type="polygon" label="sq42">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="255" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="43 + phantom" a1="m" a2="n"/>
<output a0="sq43"/>
</command>
<element type="polygon" label="sq43">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="255" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="MySquare">
<input a0="44 + phantom" a1="m" a2="n"/>
<output a0="sq44"/>
</command>
<element type="polygon" label="sq44">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="0" g="255" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
</construction>
</geogebra>