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From: Subject: nrich.maths.org :: Mathematics Enrichment :: Ratio Sudoku 1 Date: Thu, 11 Jun 2009 10:43:58 +0100 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0000_01C9EA81.8633EB00" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.5579 This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C9EA81.8633EB00 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Location: http://nrich.maths.org/public/viewer.php?obj_id=4827&part=index&nomenu=1 nrich.maths.org :: = Mathematics Enrichment :: Ratio Sudoku 1

'Ratio Sudoku 1' printed from http://nrich.maths.org/

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Produced by Henry Kwok
3D""=20

Basic Rules of "Ratio Sudoku"

The special clue-numbers in this Sudoku variant are fractions or = ratios in=20 the lowest terms, hence the name for the puzzle.

The clue-numbers are always placed on the border lines between = selected pairs=20 of neighbouring cells of the grid.

Each clue-number is the fraction of the two numbers in adjacent cells = (to the=20 left and right). Each fraction is written in its lowest terms, with the = smaller=20 number denoted as the numerator. Thus $1/2$ can stand for the following=20 combinations of numbers in the two adjacent cells: $1$ and $2$, $2$ and = $1$, $2$=20 and $4$, $4$ and $2$, $3$ and $6$, $6$ and $3$, $4$ and $8$, $8$ and = $4$.

Suppose the answers in the two adjacent cells of a puzzle are = actually $7$=20 and $5$, the clue-number will be written in the form $5/7$ instead of = $7/5$,=20 otherwise this would give away the answer and the puzzle would be too = easy to=20 solve!

When fractions are placed on consecutive border lines, a number of=20 combinations are possible. For example:

if two fractions, $2/3$ and $1/4$, are placed consecutively from = left to=20 right, the possible combinations of answers in the three neighbouring = cells=20 would be the following two sets of numbers $3$, $2$ and $8$, and $6$, = $4$ and=20 $1$;
if three fractions are placed on the consecutive border lines in = the=20 following order $2/3, 1/4$ and $1/2$, the combinations of answers in = the four=20 neighbouring cells would be $3, 2, 8$ and $4$, and $6, 4, 1$ and=20 $2$.

The remaining rules are as in a "standard" Sudoku, and the object of = the=20 puzzle is to fill in the whole $9 \times 9$ grid with numbers $1$ = through $9$=20 (one number per cell) so that each horizontal line, each vertical line, = and each=20 of the nine $3 \times 3$ squares (outlined with the bold lines) must = contain all=20 the nine different numbers $1$ through $9$.

=20
Download=20 Mathematical Fonts for better quality.

=20

NRICH is part of the family of activities in the Millennium Mathematics Project, which = also=20 includes the Plus and Motivate sites.

email: nrich@damtp.cam.ac.uk=20

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Copyright (c) 2008 John Resig (jquery.com)=0A= * Dual licensed under the MIT (MIT-LICENSE.txt)=0A= * and GPL (GPL-LICENSE.txt) licenses.=0A= *=0A= * $Date: 2008/06/09 08:59:32 $=0A= * $Rev: 4663 $=0A= */=0A= =0A= // Map over jQuery in case of overwrite=0A= if ( window.jQuery )=0A= var _jQuery =3D window.jQuery;=0A= =0A= var jQuery =3D window.jQuery =3D function( selector, context ) {=0A= // The jQuery object is actually just the init constructor 'enhanced'=0A= return new jQuery.prototype.init( selector, context );=0A= };=0A= =0A= // Map over the $ in case of overwrite=0A= if ( window.$ )=0A= var _$ =3D window.$;=0A= =0A= // Map the jQuery namespace to the '$' one=0A= window.$ =3D jQuery;=0A= =0A= // A simple way to check for HTML strings or ID strings=0A= // (both of which we optimize for)=0A= var quickExpr =3D /^[^<]*(<(.|\s)+>)[^>]*$|^#(\w+)$/;=0A= =0A= // Is it a simple selector=0A= var isSimple =3D /^.[^:#\[\.]*$/;=0A= =0A= jQuery.fn =3D jQuery.prototype =3D {=0A= init: function( selector, context ) {=0A= // Make sure that a selection was provided=0A= selector =3D selector || document;=0A= =0A= // Handle $(DOMElement)=0A= if ( selector.nodeType ) {=0A= this[0] =3D selector;=0A= this.length =3D 1;=0A= return this;=0A= =0A= // Handle HTML strings=0A= } else if ( typeof selector =3D=3D "string" ) {=0A= // Are we dealing with HTML string or an ID?=0A= var match =3D quickExpr.exec( selector );=0A= =0A= // Verify a match, and that no context was specified for #id=0A= if ( match && (match[1] || !context) ) {=0A= =0A= // HANDLE: $(html) -> $(array)=0A= if ( match[1] )=0A= selector =3D jQuery.clean( [ match[1] ], context );=0A= =0A= // HANDLE: $("#id")=0A= else {=0A= var elem =3D document.getElementById( match[3] );=0A= =0A= // Make sure an element was located=0A= if ( elem )=0A= // Handle the case where IE and Opera return items=0A= // by name instead of ID=0A= if ( elem.id !=3D match[3] )=0A= return jQuery().find( selector );=0A= =0A= // Otherwise, we inject the element directly into the jQuery object=0A= else {=0A= this[0] =3D elem;=0A= this.length =3D 1;=0A= return this;=0A= }=0A= =0A= else=0A= selector =3D [];=0A= }=0A= =0A= // HANDLE: $(expr, [context])=0A= // (which is just equivalent to: $(content).find(expr)=0A= } else=0A= return new jQuery( context ).find( selector );=0A= =0A= // HANDLE: $(function)=0A= // Shortcut for document ready=0A= } else if ( jQuery.isFunction( selector ) )=0A= return new jQuery( document )[ jQuery.fn.ready ? "ready" : "load" ]( = selector );=0A= =0A= return this.setArray(=0A= // HANDLE: $(array)=0A= selector.constructor =3D=3D Array && selector ||=0A= =0A= // HANDLE: $(arraylike)=0A= // Watch for when an array-like object, contains DOM nodes, is passed = in as the selector=0A= (selector.jquery || selector.length && selector !=3D window && = !selector.nodeType && selector[0] !=3D undefined && = selector[0].nodeType) && jQuery.makeArray( selector ) ||=0A= =0A= // HANDLE: $(*)=0A= [ selector ] );=0A= },=0A= =0A= // The current version of jQuery being used=0A= jquery: "1.2.3",=0A= =0A= // The number of elements contained in the matched element set=0A= size: function() {=0A= return this.length;=0A= },=0A= =0A= // The number of elements contained in the matched element set=0A= length: 0,=0A= =0A= // Get the Nth element in the matched element set OR=0A= // Get the whole matched element set as a clean array=0A= get: function( num ) {=0A= return num =3D=3D undefined ?=0A= =0A= // Return a 'clean' array=0A= jQuery.makeArray( this ) :=0A= =0A= // Return just the object=0A= this[ num ];=0A= },=0A= =0A= // Take an array of elements and push it onto the stack=0A= // (returning the new matched element set)=0A= pushStack: function( elems ) {=0A= // Build a new jQuery matched element set=0A= var ret =3D jQuery( elems );=0A= =0A= // Add the old object onto the stack (as a reference)=0A= ret.prevObject =3D this;=0A= =0A= // Return the newly-formed element set=0A= return ret;=0A= },=0A= =0A= // Force the current matched set of elements to become=0A= // the specified array of elements (destroying the stack in the process)=0A= // You should use pushStack() in order to do this, but maintain the = stack=0A= setArray: function( elems ) {=0A= // Resetting the length to 0, then using the native Array push=0A= // is a super-fast way to populate an object with array-like properties=0A= this.length =3D 0;=0A= Array.prototype.push.apply( this, elems );=0A= =0A= return this;=0A= },=0A= =0A= // Execute a callback for every element in the matched set.=0A= // (You can seed the arguments with an array of args, but this is=0A= // only used internally.)=0A= each: function( callback, args ) {=0A= return jQuery.each( this, callback, args );=0A= },=0A= =0A= // Determine the position of an element within =0A= // the matched set of elements=0A= index: function( elem ) {=0A= var ret =3D -1;=0A= =0A= // Locate the position of the desired element=0A= this.each(function(i){=0A= if ( this =3D=3D elem )=0A= ret =3D i;=0A= });=0A= =0A= return ret;=0A= },=0A= =0A= attr: function( name, value, type ) {=0A= var options =3D name;=0A= =0A= // Look for the case where we're accessing a style value=0A= if ( name.constructor =3D=3D String )=0A= if ( value =3D=3D undefined )=0A= return this.length && jQuery[ type || "attr" ]( this[0], name ) || = undefined;=0A= =0A= else {=0A= options =3D {};=0A= options[ name ] =3D value;=0A= }=0A= =0A= // Check to see if we're setting style values=0A= return this.each(function(i){=0A= // Set all the styles=0A= for ( name in options )=0A= jQuery.attr(=0A= type ?=0A= this.style :=0A= this,=0A= name, jQuery.prop( this, options[ name ], type, i, name )=0A= );=0A= });=0A= },=0A= =0A= css: function( key, value ) {=0A= // ignore negative width and height values=0A= if ( (key =3D=3D 'width' || key =3D=3D 'height') && parseFloat(value) = < 0 )=0A= value =3D undefined;=0A= return this.attr( key, value, "curCSS" );=0A= },=0A= =0A= text: function( text ) {=0A= if ( typeof text !=3D "object" && text !=3D null )=0A= return this.empty().append( (this[0] && this[0].ownerDocument || = document).createTextNode( text ) );=0A= =0A= var ret =3D "";=0A= =0A= jQuery.each( text || this, function(){=0A= jQuery.each( this.childNodes, function(){=0A= if ( this.nodeType !=3D 8 )=0A= ret +=3D this.nodeType !=3D 1 ?=0A= this.nodeValue :=0A= jQuery.fn.text( [ this ] );=0A= });=0A= });=0A= =0A= return ret;=0A= },=0A= =0A= wrapAll: function( html ) {=0A= if ( this[0] 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Calling detachEvent on the=0A= // clone will also remove the events from the orignal=0A= // In order to get around this, we use innerHTML.=0A= // Unfortunately, this means some modifications to =0A= // attributes in IE that are actually only stored =0A= // as properties will not be copied (such as the=0A= // the name attribute on an input).=0A= var clone =3D this.cloneNode(true),=0A= container =3D document.createElement("div");=0A= container.appendChild(clone);=0A= return jQuery.clean([container.innerHTML])[0];=0A= } else=0A= return this.cloneNode(true);=0A= });=0A= =0A= // Need to set the expando to null on the cloned set if it exists=0A= // removeData doesn't work here, IE removes it from the original as = well=0A= // this is primarily for IE but the data expando shouldn't be copied = over in any browser=0A= var clone =3D ret.find("*").andSelf().each(function(){=0A= if ( this[ expando ] !=3D undefined )=0A= this[ expando ] =3D null;=0A= });=0A= =0A= // Copy the events from the original to the clone=0A= if ( events =3D=3D=3D true )=0A= this.find("*").andSelf().each(function(i){=0A= if (this.nodeType =3D=3D 3)=0A= return;=0A= var events =3D jQuery.data( this, "events" );=0A= =0A= for ( var type in events )=0A= for ( var handler in events[ type ] )=0A= jQuery.event.add( clone[ i ], type, events[ type ][ handler ], = events[ type ][ handler ].data );=0A= });=0A= =0A= // Return the cloned set=0A= return ret;=0A= },=0A= =0A= filter: function( selector ) {=0A= return this.pushStack(=0A= jQuery.isFunction( selector ) &&=0A= jQuery.grep(this, function(elem, i){=0A= return selector.call( elem, i );=0A= }) ||=0A= =0A= jQuery.multiFilter( selector, this ) );=0A= },=0A= =0A= not: function( selector ) {=0A= if ( selector.constructor =3D=3D String )=0A= // test special case where just one selector is passed in=0A= if ( isSimple.test( selector ) )=0A= return this.pushStack( jQuery.multiFilter( selector, this, true ) );=0A= else=0A= selector =3D jQuery.multiFilter( selector, this );=0A= =0A= var isArrayLike =3D selector.length && selector[selector.length - 1] = !=3D=3D undefined && !selector.nodeType;=0A= return this.filter(function() {=0A= return isArrayLike ? jQuery.inArray( this, selector ) < 0 : this !=3D = selector;=0A= });=0A= },=0A= =0A= add: function( selector ) {=0A= return !selector ? this : this.pushStack( jQuery.merge( =0A= this.get(),=0A= selector.constructor =3D=3D String ? =0A= jQuery( selector ).get() :=0A= selector.length !=3D undefined && (!selector.nodeName || = jQuery.nodeName(selector, "form")) ?=0A= selector : [selector] ) );=0A= },=0A= =0A= is: function( selector ) {=0A= return selector ?=0A= jQuery.multiFilter( selector, this ).length > 0 :=0A= false;=0A= },=0A= =0A= hasClass: function( selector ) {=0A= return this.is( "." + selector );=0A= },=0A= =0A= val: function( value ) {=0A= if ( value =3D=3D undefined ) {=0A= =0A= if ( this.length ) {=0A= var elem =3D this[0];=0A= =0A= // We need to handle select boxes special=0A= if ( jQuery.nodeName( elem, "select" ) ) {=0A= var index =3D elem.selectedIndex,=0A= values =3D [],=0A= options =3D elem.options,=0A= one =3D elem.type =3D=3D "select-one";=0A= =0A= // Nothing was selected=0A= if ( index < 0 )=0A= return null;=0A= =0A= // Loop through all the selected options=0A= for ( var i =3D one ? index : 0, max =3D one ? index + 1 : = options.length; i < max; i++ ) {=0A= var option =3D options[ i ];=0A= =0A= if ( option.selected ) {=0A= // Get the specifc value for the option=0A= value =3D jQuery.browser.msie && = !option.attributes.value.specified ? option.text : option.value;=0A= =0A= // We don't need an array for one selects=0A= if ( one )=0A= return value;=0A= =0A= // Multi-Selects return an array=0A= values.push( value );=0A= }=0A= }=0A= =0A= return values;=0A= =0A= // Everything else, we just grab the value=0A= } else=0A= return (this[0].value || "").replace(/\r/g, "");=0A= =0A= }=0A= =0A= return undefined;=0A= }=0A= =0A= return this.each(function(){=0A= if ( this.nodeType !=3D 1 )=0A= return;=0A= =0A= if ( value.constructor =3D=3D Array && /radio|checkbox/.test( = this.type ) )=0A= this.checked =3D (jQuery.inArray(this.value, value) >=3D 0 ||=0A= jQuery.inArray(this.name, value) >=3D 0);=0A= =0A= else if ( jQuery.nodeName( this, "select" ) ) {=0A= var values =3D value.constructor =3D=3D Array ?=0A= value :=0A= [ value ];=0A= =0A= jQuery( "option", this ).each(function(){=0A= this.selected =3D (jQuery.inArray( this.value, values ) >=3D 0 ||=0A= jQuery.inArray( this.text, values ) >=3D 0);=0A= });=0A= =0A= if ( !values.length )=0A= this.selectedIndex =3D -1;=0A= =0A= } else=0A= this.value =3D value;=0A= });=0A= },=0A= =0A= html: function( value ) {=0A= return value =3D=3D undefined ?=0A= (this.length ?=0A= this[0].innerHTML :=0A= null) :=0A= this.empty().append( value );=0A= },=0A= =0A= replaceWith: function( value ) {=0A= return this.after( value ).remove();=0A= },=0A= =0A= eq: function( i ) {=0A= return this.slice( i, i + 1 );=0A= },=0A= =0A= slice: function() {=0A= return this.pushStack( Array.prototype.slice.apply( this, arguments ) = );=0A= },=0A= =0A= map: function( callback ) {=0A= return this.pushStack( jQuery.map(this, function(elem, i){=0A= return callback.call( elem, i, elem );=0A= }));=0A= },=0A= =0A= andSelf: function() {=0A= return this.add( this.prevObject );=0A= },=0A= =0A= data: function( key, value ){=0A= var parts =3D key.split(".");=0A= parts[1] =3D parts[1] ? "." + parts[1] : "";=0A= =0A= if ( value =3D=3D null ) {=0A= var data =3D this.triggerHandler("getData" + parts[1] + "!", = [parts[0]]);=0A= =0A= if ( data =3D=3D undefined && this.length )=0A= data =3D jQuery.data( this[0], key );=0A= =0A= return data =3D=3D null && parts[1] ?=0A= this.data( parts[0] ) :=0A= data;=0A= } else=0A= return this.trigger("setData" + parts[1] + "!", [parts[0], = value]).each(function(){=0A= jQuery.data( this, key, value );=0A= });=0A= },=0A= =0A= removeData: function( key ){=0A= return this.each(function(){=0A= jQuery.removeData( this, key );=0A= });=0A= },=0A= =0A= domManip: function( args, table, reverse, callback ) {=0A= var clone =3D this.length > 1, elems; =0A= =0A= return this.each(function(){=0A= if ( !elems ) {=0A= elems =3D jQuery.clean( args, this.ownerDocument );=0A= =0A= if ( reverse )=0A= elems.reverse();=0A= }=0A= =0A= var obj =3D this;=0A= =0A= if ( table && jQuery.nodeName( this, "table" ) && jQuery.nodeName( = elems[0], "tr" ) )=0A= obj =3D this.getElementsByTagName("tbody")[0] || this.appendChild( = this.ownerDocument.createElement("tbody") );=0A= =0A= var scripts =3D jQuery( [] );=0A= =0A= jQuery.each(elems, function(){=0A= var elem =3D clone ?=0A= jQuery( this ).clone( true )[0] :=0A= this;=0A= =0A= // execute all scripts after the elements have been injected=0A= if ( jQuery.nodeName( elem, "script" ) ) {=0A= scripts =3D scripts.add( elem );=0A= } else {=0A= // Remove any inner scripts for later evaluation=0A= if ( elem.nodeType =3D=3D 1 )=0A= scripts =3D scripts.add( jQuery( "script", elem ).remove() );=0A= =0A= // Inject the elements into the document=0A= callback.call( obj, elem );=0A= }=0A= });=0A= =0A= scripts.each( evalScript );=0A= });=0A= }=0A= };=0A= =0A= // Give the init function the jQuery prototype for later instantiation=0A= jQuery.prototype.init.prototype =3D jQuery.prototype;=0A= =0A= function evalScript( i, elem ) {=0A= if ( elem.src )=0A= jQuery.ajax({=0A= url: elem.src,=0A= async: false,=0A= dataType: "script"=0A= });=0A= =0A= else=0A= jQuery.globalEval( elem.text || elem.textContent || elem.innerHTML || = "" );=0A= =0A= if ( elem.parentNode )=0A= elem.parentNode.removeChild( elem );=0A= }=0A= =0A= jQuery.extend =3D jQuery.fn.extend =3D function() {=0A= // copy reference to target object=0A= var target =3D arguments[0] || {}, i =3D 1, length =3D = arguments.length, deep =3D false, options;=0A= =0A= // Handle a deep copy situation=0A= if ( target.constructor =3D=3D Boolean ) {=0A= deep =3D target;=0A= target =3D arguments[1] || {};=0A= // skip the boolean and the target=0A= i =3D 2;=0A= }=0A= =0A= // Handle case when target is a string or something (possible in deep = copy)=0A= if ( typeof target !=3D "object" && typeof target !=3D "function" )=0A= target =3D {};=0A= =0A= // extend jQuery itself if only one argument is passed=0A= if ( length =3D=3D 1 ) {=0A= target =3D this;=0A= i =3D 0;=0A= }=0A= =0A= for ( ; i < length; i++ )=0A= // Only deal with non-null/undefined values=0A= if ( (options =3D arguments[ i ]) !=3D null )=0A= // Extend the base object=0A= for ( var name in options ) {=0A= // Prevent never-ending loop=0A= if ( target =3D=3D=3D options[ name ] )=0A= continue;=0A= =0A= // Recurse if we're merging object values=0A= if ( deep && options[ name ] && typeof options[ name ] =3D=3D = "object" && target[ name ] && !options[ name ].nodeType )=0A= target[ name ] =3D jQuery.extend( target[ name ], options[ name ] );=0A= =0A= // Don't bring in undefined values=0A= else if ( options[ name ] !=3D undefined )=0A= target[ name ] =3D options[ name ];=0A= =0A= }=0A= =0A= // Return the modified object=0A= return target;=0A= };=0A= =0A= var expando =3D "jQuery" + (new Date()).getTime(), uuid =3D 0, = windowData =3D {};=0A= =0A= // exclude the following css properties to add px=0A= var exclude =3D /z-?index|font-?weight|opacity|zoom|line-?height/i;=0A= =0A= jQuery.extend({=0A= noConflict: function( deep ) {=0A= window.$ =3D _$;=0A= =0A= if ( deep )=0A= window.jQuery =3D _jQuery;=0A= =0A= return jQuery;=0A= },=0A= =0A= // See test/unit/core.js for details concerning this function.=0A= isFunction: function( fn ) {=0A= return !!fn && typeof fn !=3D "string" && !fn.nodeName && =0A= fn.constructor !=3D Array && /function/i.test( fn + "" );=0A= },=0A= =0A= // check if an element is in a (or is an) XML document=0A= isXMLDoc: function( elem ) {=0A= return elem.documentElement && !elem.body ||=0A= elem.tagName && elem.ownerDocument && !elem.ownerDocument.body;=0A= },=0A= =0A= // Evalulates a script in a global context=0A= globalEval: function( data ) {=0A= data =3D jQuery.trim( data );=0A= =0A= if ( data ) {=0A= // Inspired by code by Andrea Giammarchi=0A= // = http://webreflection.blogspot.com/2007/08/global-scope-evaluation-and-dom= .html=0A= var head =3D document.getElementsByTagName("head")[0] || = document.documentElement,=0A= script =3D document.createElement("script");=0A= =0A= script.type =3D "text/javascript";=0A= if ( jQuery.browser.msie )=0A= script.text =3D data;=0A= else=0A= script.appendChild( document.createTextNode( data ) );=0A= =0A= head.appendChild( script );=0A= head.removeChild( script );=0A= }=0A= },=0A= =0A= nodeName: function( elem, name ) {=0A= return elem.nodeName && elem.nodeName.toUpperCase() =3D=3D = name.toUpperCase();=0A= },=0A= =0A= cache: {},=0A= =0A= data: function( elem, name, data ) {=0A= elem =3D elem =3D=3D window ?=0A= windowData :=0A= elem;=0A= =0A= var id =3D elem[ expando ];=0A= =0A= // Compute a unique ID for the element=0A= if ( !id ) =0A= id =3D elem[ expando ] =3D ++uuid;=0A= =0A= // Only generate the data cache if we're=0A= // trying to access or manipulate it=0A= if ( name && !jQuery.cache[ id ] )=0A= jQuery.cache[ id ] =3D {};=0A= =0A= // Prevent overriding the named cache with undefined values=0A= if ( data !=3D undefined )=0A= jQuery.cache[ id ][ name ] =3D data;=0A= =0A= // Return the named cache data, or the ID for the element =0A= return name ?=0A= jQuery.cache[ id ][ name ] :=0A= id;=0A= },=0A= =0A= removeData: function( elem, name ) {=0A= elem =3D elem =3D=3D window ?=0A= windowData :=0A= elem;=0A= =0A= var id =3D elem[ expando ];=0A= =0A= // If we want to remove a specific section of the element's data=0A= if ( name ) {=0A= if ( jQuery.cache[ id ] ) {=0A= // Remove the section of cache data=0A= delete jQuery.cache[ id ][ name ];=0A= =0A= // If we've removed all the data, remove the element's cache=0A= name =3D "";=0A= =0A= for ( name in jQuery.cache[ id ] )=0A= break;=0A= =0A= if ( !name )=0A= jQuery.removeData( elem );=0A= }=0A= =0A= // Otherwise, we want to remove all of the element's data=0A= } else {=0A= // Clean up the element expando=0A= try {=0A= delete elem[ expando ];=0A= } catch(e){=0A= // IE has trouble directly removing the expando=0A= // but it's ok with using removeAttribute=0A= if ( elem.removeAttribute )=0A= elem.removeAttribute( expando );=0A= }=0A= =0A= // Completely remove the data cache=0A= delete jQuery.cache[ id ];=0A= }=0A= },=0A= =0A= // args is for internal usage only=0A= each: function( object, callback, args ) {=0A= if ( args ) {=0A= if ( object.length =3D=3D undefined ) {=0A= for ( var name in object )=0A= if ( callback.apply( object[ name ], args ) =3D=3D=3D false )=0A= break;=0A= } else=0A= for ( var i =3D 0, length =3D object.length; i < length; i++ )=0A= if ( callback.apply( object[ i ], args ) =3D=3D=3D false )=0A= break;=0A= =0A= // A special, fast, case for the most common use of each=0A= } else {=0A= if ( object.length =3D=3D undefined ) {=0A= for ( var name in object )=0A= if ( callback.call( object[ name ], name, object[ name ] ) = =3D=3D=3D false )=0A= break;=0A= } else=0A= for ( var i =3D 0, length =3D object.length, value =3D object[0]; =0A= i < length && callback.call( value, i, value ) !=3D=3D false; value = =3D object[++i] ){}=0A= }=0A= =0A= return object;=0A= },=0A= =0A= prop: function( elem, value, type, i, name ) {=0A= // Handle executable functions=0A= if ( jQuery.isFunction( value ) )=0A= value =3D value.call( elem, i );=0A= =0A= // Handle passing in a number to a CSS property=0A= return value && value.constructor =3D=3D Number && type =3D=3D = "curCSS" && !exclude.test( name ) ?=0A= value + "px" :=0A= value;=0A= },=0A= =0A= className: {=0A= // internal only, use addClass("class")=0A= add: function( elem, classNames ) {=0A= jQuery.each((classNames || "").split(/\s+/), function(i, className){=0A= if ( elem.nodeType =3D=3D 1 && !jQuery.className.has( = elem.className, className ) )=0A= elem.className +=3D (elem.className ? " " : "") + className;=0A= });=0A= },=0A= =0A= // internal only, use removeClass("class")=0A= remove: function( elem, classNames ) {=0A= if (elem.nodeType =3D=3D 1)=0A= elem.className =3D classNames !=3D undefined ?=0A= jQuery.grep(elem.className.split(/\s+/), function(className){=0A= return !jQuery.className.has( classNames, className ); =0A= }).join(" ") :=0A= "";=0A= },=0A= =0A= // internal only, use is(".class")=0A= has: function( elem, className ) {=0A= return jQuery.inArray( className, (elem.className || = elem).toString().split(/\s+/) ) > -1;=0A= }=0A= },=0A= =0A= // A method for quickly swapping in/out CSS properties to get correct = calculations=0A= swap: function( elem, options, callback ) {=0A= var old =3D {};=0A= // Remember the old values, and insert the new ones=0A= for ( var name in options ) {=0A= old[ name ] =3D elem.style[ name ];=0A= elem.style[ name ] =3D options[ name ];=0A= }=0A= =0A= callback.call( elem );=0A= =0A= // Revert the old values=0A= for ( var name in options )=0A= elem.style[ name ] =3D old[ name ];=0A= },=0A= =0A= css: function( elem, name, force ) {=0A= if ( name =3D=3D "width" || name =3D=3D "height" ) {=0A= var val, props =3D { position: "absolute", visibility: "hidden", = display:"block" }, which =3D name =3D=3D "width" ? [ "Left", "Right" ] : = [ "Top", "Bottom" ];=0A= =0A= function getWH() {=0A= val =3D name =3D=3D "width" ? elem.offsetWidth : elem.offsetHeight;=0A= var padding =3D 0, border =3D 0;=0A= jQuery.each( which, function() {=0A= padding +=3D parseFloat(jQuery.curCSS( elem, "padding" + this, = true)) || 0;=0A= border +=3D parseFloat(jQuery.curCSS( elem, "border" + this + = "Width", true)) || 0;=0A= });=0A= val -=3D Math.round(padding + border);=0A= }=0A= =0A= if ( jQuery(elem).is(":visible") )=0A= getWH();=0A= else=0A= jQuery.swap( elem, props, getWH );=0A= =0A= return Math.max(0, val);=0A= }=0A= =0A= return jQuery.curCSS( elem, name, force );=0A= },=0A= =0A= curCSS: function( elem, name, force ) {=0A= var ret;=0A= =0A= // A helper method for determining if an element's values are broken=0A= function color( elem ) {=0A= if ( !jQuery.browser.safari )=0A= return false;=0A= =0A= var ret =3D document.defaultView.getComputedStyle( elem, null );=0A= return !ret || ret.getPropertyValue("color") =3D=3D "";=0A= }=0A= =0A= // We need to handle opacity special in IE=0A= if ( name =3D=3D "opacity" && jQuery.browser.msie ) {=0A= ret =3D jQuery.attr( elem.style, "opacity" );=0A= =0A= return ret =3D=3D "" ?=0A= "1" :=0A= ret;=0A= }=0A= // Opera sometimes will give the wrong display answer, this fixes it, = see #2037=0A= if ( jQuery.browser.opera && name =3D=3D "display" ) {=0A= var save =3D elem.style.outline;=0A= elem.style.outline =3D "0 solid black";=0A= elem.style.outline =3D save;=0A= }=0A= =0A= // Make sure we're using the right name for getting the float value=0A= if ( name.match( /float/i ) )=0A= name =3D styleFloat;=0A= =0A= if ( !force && elem.style && elem.style[ name ] )=0A= ret =3D elem.style[ name ];=0A= =0A= else if ( document.defaultView && = document.defaultView.getComputedStyle ) {=0A= =0A= // Only "float" is needed here=0A= if ( name.match( /float/i ) )=0A= name =3D "float";=0A= =0A= name =3D name.replace( /([A-Z])/g, "-$1" ).toLowerCase();=0A= =0A= var getComputedStyle =3D document.defaultView.getComputedStyle( elem, = null );=0A= =0A= if ( getComputedStyle && !color( elem ) )=0A= ret =3D getComputedStyle.getPropertyValue( name );=0A= =0A= // If the element isn't reporting its values properly in Safari=0A= // then some display: none elements are involved=0A= else {=0A= var swap =3D [], stack =3D [];=0A= =0A= // Locate all of the parent display: none elements=0A= for ( var a =3D elem; a && color(a); a =3D a.parentNode )=0A= stack.unshift(a);=0A= =0A= // Go through and make them visible, but in reverse=0A= // (It would be better if we knew the exact display type that they = had)=0A= for ( var i =3D 0; i < stack.length; i++ )=0A= if ( color( stack[ i ] ) ) {=0A= swap[ i ] =3D stack[ i ].style.display;=0A= stack[ i ].style.display =3D "block";=0A= }=0A= =0A= // Since we flip the display style, we have to handle that=0A= // one special, otherwise get the value=0A= ret =3D name =3D=3D "display" && swap[ stack.length - 1 ] !=3D null ?=0A= "none" :=0A= ( getComputedStyle && getComputedStyle.getPropertyValue( name ) ) = || "";=0A= =0A= // Finally, revert the display styles back=0A= for ( var i =3D 0; i < swap.length; i++ )=0A= if ( swap[ i ] !=3D null )=0A= stack[ i ].style.display =3D swap[ i ];=0A= }=0A= =0A= // We should always get a number back from opacity=0A= if ( name =3D=3D "opacity" && ret =3D=3D "" )=0A= ret =3D "1";=0A= =0A= } else if ( elem.currentStyle ) {=0A= var camelCase =3D name.replace(/\-(\w)/g, function(all, letter){=0A= return letter.toUpperCase();=0A= });=0A= =0A= ret =3D elem.currentStyle[ name ] || elem.currentStyle[ camelCase ];=0A= =0A= // From the awesome hack by Dean Edwards=0A= // http://erik.eae.net/archives/2007/07/27/18.54.15/#comment-102291=0A= =0A= // If we're not dealing with a regular pixel number=0A= // but a number that has a weird ending, we need to convert it to = pixels=0A= if ( !/^\d+(px)?$/i.test( ret ) && /^\d/.test( ret ) ) {=0A= // Remember the original values=0A= var style =3D elem.style.left, runtimeStyle =3D = elem.runtimeStyle.left;=0A= =0A= // Put in the new values to get a computed value out=0A= elem.runtimeStyle.left =3D elem.currentStyle.left;=0A= elem.style.left =3D ret || 0;=0A= ret =3D elem.style.pixelLeft + "px";=0A= =0A= // Revert the changed values=0A= elem.style.left =3D style;=0A= elem.runtimeStyle.left =3D runtimeStyle;=0A= }=0A= }=0A= =0A= return ret;=0A= },=0A= =0A= clean: function( elems, context ) {=0A= var ret =3D [];=0A= context =3D context || document;=0A= // !context.createElement fails in IE with an error but returns typeof = 'object'=0A= if (typeof context.createElement =3D=3D 'undefined') =0A= context =3D context.ownerDocument || context[0] && = context[0].ownerDocument || document;=0A= =0A= jQuery.each(elems, function(i, elem){=0A= if ( !elem )=0A= return;=0A= =0A= if ( elem.constructor =3D=3D Number )=0A= elem =3D elem.toString();=0A= =0A= // Convert html string into DOM nodes=0A= if ( typeof elem =3D=3D "string" ) {=0A= // Fix "XHTML"-style tags in all browsers=0A= elem =3D elem.replace(/(<(\w+)[^>]*?)\/>/g, function(all, front, = tag){=0A= return = tag.match(/^(abbr|br|col|img|input|link|meta|param|hr|area|embed)$/i) ?=0A= all :=0A= front + ">";=0A= });=0A= =0A= // Trim whitespace, otherwise indexOf won't work as expected=0A= var tags =3D jQuery.trim( elem ).toLowerCase(), div =3D = context.createElement("div");=0A= =0A= var wrap =3D=0A= // option or optgroup=0A= !tags.indexOf("", "" ] ||=0A= =0A= !tags.indexOf("", "" ] ||=0A= =0A= tags.match(/^<(thead|tbody|tfoot|colg|cap)/) &&=0A= [ 1, "", "

" ] ||=0A= =0A= !tags.indexOf("", "" ] ||=0A= =0A= // matched above=0A= (!tags.indexOf("", "" ] ||=0A= =0A= !tags.indexOf("", "" ] ||=0A= =0A= // IE can't serialize and =0A= * =0A= * (If you are including this file into your page via Server-Side=0A= * Includes, you should remove line above.)=0A= * =0A= * You can make copies of this file with different settings=0A= * if you need to have several different configurations.=0A= * =0A= **********************************************************************/=0A= =0A= if (!window.jsMath) {window.jsMath =3D {}}=0A= =0A= jsMath.Easy =3D {=0A= //=0A= // The URL of the root jsMath directory on your server=0A= // (it must be in the same domain as the HTML page).=0A= // It should include "http://yoursite.com/", or should=0A= // be relative to the root of your server. It is possible=0A= // to be a relative URL, but it will be relative to the=0A= // HTML page loading this file.=0A= // =0A= // If you leave this blank, jsMath will try to look it up from=0A= // the URL where it loaded this file, but that may not work.=0A= //=0A= root: "",=0A= =0A= //=0A= // The default scaling factor for mathematics compared to the=0A= // surrounding text.=0A= //=0A= scale: 120,=0A= =0A= //=0A= // 1 means use the autoload plug-in to decide if jsMath should be = loaded=0A= // 0 means always load jsMath=0A= //=0A= autoload: 0,=0A= =0A= //=0A= // Setting any of these will cause the tex2math plugin to be used=0A= // to add the

and tags that jsMath needs. See the=0A= // documentation for the tex2math plugin for more information.=0A= //=0A= processSlashParens: 1, // process $...$ in text?=0A= processSlashBrackets: 1, // process \[...\] in text?=0A= processDoubleDollars: 1, // process $$...$$ in text?=0A= processSingleDollars: 1, // process $...$ in text?=0A= processLaTeXenvironments: 1, // process \begin{xxx}...\end{xxx} = outside math mode?=0A= fixEscapedDollars: 1, // convert \$ to $ outside of math mode?=0A= doubleDollarsAreInLine: 0, // make $$...$$ be in-line math?=0A= allowDisableTag: 1, // allow ID=3D"tex2math_off" to disable = tex2math?=0A= //=0A= // If you want to use your own custom delimiters for math instead=0A= // of the usual ones, then uncomment the following four lines and=0A= // insert your own delimiters within the quotes. You may want to=0A= // turn off processing of the dollars and other delimiters above=0A= // as well, though you can use them in combination with the=0A= // custom delimiters if you wish. See the tex2math documentation=0A= // for more details.=0A= // =0A= //customDelimiters: [=0A= // '[math]','[/math]', // to begin and end in-line math=0A= // '[display]','[/display]' // to begin and end display math=0A= //],=0A= =0A= //=0A= // Show TeX source when mathematics is double-clicked?=0A= //=0A= allowDoubleClicks: 1,=0A= =0A= //=0A= // Show jsMath font warning messages? (Disabling this prevents yours=0A= // users from finding out that they can have a better experience on = your=0A= // site by installing some fonts, so don't disable this).=0A= //=0A= showFontWarnings: 0,=0A= =0A= //=0A= // Use "Process" or "ProcessBeforeShowing". See the jsMath=0A= // author's documentation for the difference between these=0A= // two routines.=0A= //=0A= method: "Process",=0A= =0A= //=0A= // List of plug-ins and extensions that you want to be=0A= // loaded automatically. E.g.=0A= // ["plugins/mimeTeX.js","extensions/AMSsymbols.js"]=0A= // =0A= loadFiles: ["plugins/MSIE-zoom-fix.js", "extensions/AMSsymbols.js"],=0A= =0A= //=0A= // List of fonts to load automatically. E.g.=0A= // ["cmmib10"]=0A= //=0A= loadFonts: ["msam10","msbm10"],=0A= =0A= //=0A= // Allow jsMath to enter global mode?=0A= // (Uses frames, so may not always work with complex web sites)=0A= //=0A= allowGlobal: 0,=0A= =0A= //=0A= // Disable image fonts? (In case you don't load them on your server.)=0A= //=0A= noImageFonts: 0=0A= =0A= };=0A= =0A= /****************************************************************/=0A= /****************************************************************/=0A= //=0A= // DO NOT MAKE CHANGES BELOW THIS=0A= //=0A= /****************************************************************/=0A= /****************************************************************/=0A= =0A= if (jsMath.Easy.root =3D=3D "") {=0A= jsMath.Easy.root =3D document.getElementsByTagName("script");=0A= jsMath.Easy.root =3D jsMath.Easy.root[jsMath.Easy.root.length-1].src=0A= if (jsMath.Easy.root.match(/\/easy\/[^\/]*$/)) {=0A= jsMath.Easy.root =3D jsMath.Easy.root.replace(/\/easy\/[^\/]*$/,"");=0A= } else {=0A= jsMath.Easy.root =3D = jsMath.Easy.root.replace(/\/(jsMath\/(easy\/)?)?[^\/]*$/,"/jsMath");=0A= }=0A= }=0A= jsMath.Easy.root =3D jsMath.Easy.root.replace(/\/$/,""); // trim = trailing "/" if any=0A= =0A= document.write('