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<title>反向构造 - 深度精讲</title>
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</head>
<body>
<div id="app">
<div class="content-area">
<div v-show="currentPage === 1" class="page-container">
<div class="section-title">🔄 反向构造：已知因数求数</div>
<div class="card">
<div class="tag tag-blue">贪心原则</div>
<p style="font-size: 14px; margin: 10px 0;">
要把“好钢用在刀刃上”。想让数最小，必须把大的指数给小的质数。
</p>
<div class="step-flow">
<div class="flow-item active">
<div class="flow-icon">1</div>
<div>
<b>拆：</b>把因数个数拆成几个数相乘。<br>
<small>例如 6 = 3 × 2</small>
</div>
</div>
<div class="flow-item active">
<div class="flow-icon">2</div>
<div>
<b>排：</b>按从大到小排列。<br>
<small>指数是 (3-1)=2 和 (2-1)=1</small>
</div>
</div>
<div class="flow-item active">
<div class="flow-icon">3</div>
<div>
<b>配：</b>大指数配小底数。<br>
<small>2<sup>2</sup> × 3<sup>1</sup> = 12</small>
</div>
</div>
</div>
<button class="speak-btn" @click="speakIntro">🔊 听老师讲解</button>
</div>
<div class="card">
<div class="tag tag-purple">进阶策略</div>
<ul class="step-list">
<li><b>拆得越碎越好：</b>通常拆成更多的小因子，数会更小。</li>
<li><b>比较法：</b>如果不确定，就把几种拆分方案都算出来比一比。</li>
</ul>
</div>
</div>
<div v-show="currentPage === 2" class="page-container">
<div class="section-title">🎬 构造实验室</div>
<div class="anim-stage">
<div class="target-box">
🎯 目标：构造一个有 <b>{{ targetFactorCount }}</b> 个因数的最小数
</div>
<div class="constructor-area">
<div class="prime-row">
<div class="prime-label">2</div>
<div class="exp-control">
<button class="ctrl-btn" @click="changeExp(0, -1)">-</button>
<span class="exp-val">{{ exps[0] }}</span>
<button class="ctrl-btn add" @click="changeExp(0, 1)">+</button>
</div>
<div style="font-size:12px; color:#64748B;">指数: {{ exps[0] }}</div>
</div>
<div class="prime-row">
<div class="prime-label">3</div>
<div class="exp-control">
<button class="ctrl-btn" @click="changeExp(1, -1)">-</button>
<span class="exp-val">{{ exps[1] }}</span>
<button class="ctrl-btn add" @click="changeExp(1, 1)">+</button>
</div>
<div style="font-size:12px; color:#64748B;">指数: {{ exps[1] }}</div>
</div>
<div class="prime-row">
<div class="prime-label">5</div>
<div class="exp-control">
<button class="ctrl-btn" @click="changeExp(2, -1)">-</button>
<span class="exp-val">{{ exps[2] }}</span>
<button class="ctrl-btn add" @click="changeExp(2, 1)">+</button>
</div>
<div style="font-size:12px; color:#64748B;">指数: {{ exps[2] }}</div>
</div>
</div>
<div class="result-panel">
<div class="formula-display">
2<sup>{{exps[0]}}</sup> × 3<sup>{{exps[1]}}</sup> × 5<sup>{{exps[2]}}</sup>
</div>
<div class="calc-res">
= {{ calculatedValue }}
</div>
<div class="factor-count-display">
当前因数个数: ({{exps[0]}}+1)×({{exps[1]}}+1)×({{exps[2]}}+1) =
<span style="color:#FCD34D; font-weight:bold; font-size:16px;">{{ calculatedFactors }}</span>
</div>
<div v-if="calculatedFactors === targetFactorCount" style="margin-top:10px; color:#10B981; font-weight:bold;">
✅ 达成目标！
</div>
</div>
</div>
<div class="card" style="margin-top: 15px;">
<div style="display:flex; gap:10px; margin-bottom:10px;">
<button class="step-btn" @click="setTarget(6)">试炼: 6个因数</button>
<button class="step-btn" @click="setTarget(12)">试炼: 12个因数</button>
</div>
<p style="font-size: 13px; color: #64748B;">
提示：试着调整指数，看怎么搭配能让结果最小。
比如为了凑 6 个因数，是用 2<sup>5</sup> (32)，还是 2<sup>2</sup>×3<sup>1</sup> (12)？
</p>
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<div class="section-title">🔍 5道经典思维模型</div>
<div class="example-item" :class="{active: activeExample === index}" v-for="(ex, index) in examples" :key="index">
<div class="example-header" @click="toggleExample(index)">
<span>{{ ex.title }}</span>
<span>{{ activeExample === index ? '▲' : '▼' }}</span>
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<div class="example-content">
<div style="margin-bottom: 10px; font-weight: bold; color: #1E293B;">{{ ex.question }}</div>
<div style="background: #F0F9FF; padding: 12px; border-radius: 6px; color: #0369A1; font-size: 14px;">
<strong>解析：</strong><br>
<span v-html="ex.analysis"></span>
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</div>
</div>
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<div v-show="currentPage === 4" class="page-container" style="height: 100%;">
<div v-if="!basicDone" class="quiz-container">
<div class="question-card">
<div class="tag tag-blue">基础 {{ currentBasicIndex + 1 }}/10</div>
<div style="font-size: 18px; font-weight: bold; margin-bottom: 20px;">{{ basicQuestions[currentBasicIndex].text }}</div>
<div v-for="(opt, idx) in basicQuestions[currentBasicIndex].options"
:key="idx"
class="option-btn"
:class="{
'correct': basicAnswered && opt.correct,
'wrong': basicAnswered && selectedBasicOpt === idx && !opt.correct
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@click="handleBasicAnswer(idx, opt.correct)">
{{ opt.text }}
</div>
<div v-if="basicAnswered" style="margin-top: 15px; padding: 10px; background: #F1F5F9; border-radius: 8px; font-size: 13px;">
<div v-if="isBasicCorrect" style="color: #10B981; font-weight: bold;">🎉 正确！</div>
<div v-else style="color: #EF4444; font-weight: bold;">💡 解析：</div>
<div style="margin-top: 5px;" v-html="basicQuestions[currentBasicIndex].expl"></div>
</div>
<button v-if="basicAnswered" class="next-btn" @click="nextBasic">
{{ currentBasicIndex < 9 ? '下一题 →' : '完成练习' }}
</button>
</div>
</div>
<div v-else class="card" style="height: 100%; display: flex; flex-direction: column; align-items: center; justify-content: center; text-align: center;">
<div style="font-size: 60px;">🏆</div>
<h2>基础训练完成</h2>
<p>得分: {{ basicScore }}/100</p>
<button class="next-btn" @click="switchPage(5)">进阶挑战奥数题 →</button>
</div>
</div>
<div v-show="currentPage === 5" class="page-container" style="height: 100%;">
<div v-if="!olympiadDone" class="quiz-container">
<div class="question-card" style="border: 2px solid #F59E0B;">
<div class="tag tag-purple">🏆 杯赛真题 {{ currentOlympiadIndex + 1 }}/10</div>
<div style="font-size: 18px; font-weight: bold; margin-bottom: 20px;">{{ olympiadQuestions[currentOlympiadIndex].text }}</div>
<div v-for="(opt, idx) in olympiadQuestions[currentOlympiadIndex].options"
:key="idx"
class="option-btn"
:class="{
'correct': olympiadAnswered && opt.correct,
'wrong': olympiadAnswered && selectedOlympiadOpt === idx && !opt.correct
}"
@click="handleOlympiadAnswer(idx, opt.correct)">
{{ opt.text }}
</div>
<div v-if="olympiadAnswered" style="margin-top: 15px; padding: 10px; background: #FFFBEB; border-radius: 8px; font-size: 13px;">
<strong>深度解析：</strong><br>
<span v-html="olympiadQuestions[currentOlympiadIndex].expl"></span>
</div>
<button v-if="olympiadAnswered" class="next-btn" style="background: #D97706;" @click="nextOlympiad">
{{ currentOlympiadIndex < 9 ? '下一题 →' : '查看总结' }}
</button>
</div>
</div>
<div v-else class="card" style="height: 100%; display: flex; flex-direction: column; align-items: center; justify-content: center; text-align: center;">
<div style="font-size: 60px;">🥇</div>
<h2>奥数挑战通关</h2>
<p>得分: {{ olympiadScore }}/100</p>
<button class="next-btn" @click="switchPage(6)">查看知识点总结 →</button>
</div>
</div>
<div v-show="currentPage === 6" class="page-container">
<div class="section-title">📝 知识点总结</div>
<div class="card">
<div style="font-weight: bold; margin-bottom: 10px;">反向构造心法</div>
<ul class="step-list">
<li><b>拆：</b>因数个数拆成几个数相乘。</li>
<li><b>排：</b>从大到小排好队。</li>
<li><b>配：</b>大指数给小底数 (2, 3, 5...)。</li>
</ul>
</div>
<div class="card">
<div style="font-weight: bold; margin-bottom: 10px;">常见数值 (背诵)</div>
<p style="font-size: 14px; color: #475569;">
6个因数最小：12 (2<sup>2</sup>×3)<br>
8个因数最小：24 (2<sup>3</sup>×3)<br>
12个因数最小：60 (2<sup>2</sup>×3×5)<br>
</p>
</div>
</div>
</div>
<div class="bottom-nav">
<div class="nav-item" :class="{active: currentPage === 1}" @click="switchPage(1)">
<div class="nav-icon">💡</div>
<div class="nav-label">原理</div>
</div>
<div class="nav-item" :class="{active: currentPage === 2}" @click="switchPage(2)">
<div class="nav-icon">🎬</div>
<div class="nav-label">演示</div>
</div>
<div class="nav-item" :class="{active: currentPage === 3}" @click="switchPage(3)">
<div class="nav-icon">🔍</div>
<div class="nav-label">例题</div>
</div>
<div class="nav-item" :class="{active: currentPage === 4}" @click="switchPage(4)">
<div class="nav-icon">✏️</div>
<div class="nav-label">练习</div>
</div>
<div class="nav-item" :class="{active: currentPage === 5}" @click="switchPage(5)">
<div class="nav-icon">🏆</div>
<div class="nav-label">奥数</div>
</div>
<div class="nav-item" :class="{active: currentPage === 6}" @click="switchPage(6)">
<div class="nav-icon">📚</div>
<div class="nav-label">总结</div>
</div>
</div>
</div>
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<script src="https://www.xinghuo.tv/wp-content/themes/xinghuo-tv/assets/js/gsap.min.js"></script>
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const { createApp } = Vue;
createApp({
data() {
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// 构造器状态
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exps: [0, 0, 0], // 指数：2, 3, 5
// 例题数据 (Page 3)
activeExample: null,
examples: [
{
title: '例1: 基础入门',
question: '一个自然数最小是多少，它正好有8个因数？',
analysis: '拆分8：<br>A: 8=8 → 2^7=128<br>B: 8=4×2 → 2^3×3^1=24<br>C: 8=2×2×2 → 2^1×3^1×5^1=30<br>最小是24。'
},
{
title: '例2: 进阶构造',
question: '一个自然数最小是多少，它正好有10个因数？',
analysis: '10不能拆成3个数。只能：<br>A: 10=10 → 2^9=512<br>B: 10=5×2 → 2^4×3^1=16×3=48。<br>最小是48。'
},
{
title: '例3: 奇数限制',
question: '一个奇数最小是多少，它正好有6个因数？',
analysis: '奇数不能含2。底数只能是3,5,7...<br>6=3×2 → 3^2×5^1 = 9×5=45。'
},
{
title: '例4: 倍数限制',
question: '是5的倍数且有10个因数，最小是多少？',
analysis: '必须含5。10=5×2。<br>组合1(2和5): 2^4×5^1=80。<br>组合2(3和5): 3^4×5^1=405。<br>最小80。'
},
{
title: '例5: 大因数个数',
question: '正好有16个因数，最小是多少？',
analysis: '16=4×2×2 → 2^3×3^1×5^1 = 120。<br>16=2×2×2×2 → 2×3×5×7 = 210。<br>最小120。'
}
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basicQuestions: [
{ text: '恰好有4个因数，最小是？', options: [{text:'6',correct:true}, {text:'8',correct:false}, {text:'4',correct:false}], expl: '4=2×2 → 2^1×3^1=6。' },
{ text: '恰好有9个因数，最小是？', options: [{text:'36',correct:true}, {text:'100',correct:false}, {text:'256',correct:false}], expl: '9=3×3 → 2^2×3^2=36。' },
{ text: '恰好有14个因数，最小是？', options: [{text:'192',correct:true}, {text:'128',correct:false}], expl: '14=7×2 → 2^6×3^1=64×3=192。' },
{ text: '一个奇数恰好有4个因数，最小是？', options: [{text:'15',correct:true}, {text:'6',correct:false}], expl: '4=2×2，底数3,5。3×5=15。' },
{ text: '一个偶数恰好有15个因数，最小是？', options: [{text:'144',correct:true}, {text:'36',correct:false}], expl: '15=5×3 → 2^4×3^2=16×9=144。' },
{ text: '360有24个因数。找一个比360小的数也有24个因数。', options: [{text:'300',correct:false}, {text:'336',correct:false}, {text:'240',correct:false}, {text:'题目作为开放思考',correct:true}], expl: '此题为开放思考，选项仅作占位。' },
{ text: '恰好有3个因数的数，一定是偶数吗？', options: [{text:'不是',correct:true}, {text:'是',correct:false}], expl: '9(1,3,9)是奇数。' },
{ text: '有12个因数且被5整除，最小是？', options: [{text:'60',correct:true}, {text:'120',correct:false}], expl: '12=3×2×2 → 2^2×3^1×5^1=60。' },
{ text: '100以内因数最多的数有几个？', options: [{text:'5个',correct:true}, {text:'1个',correct:false}], expl: '60, 72, 84, 90, 96 都是12个因数。' },
{ text: '只有2和3两个质因数且因数12个，最大(<100)？', options: [{text:'96',correct:true}, {text:'72',correct:false}], expl: '12=6×2 → 2^5×3=96。' }
],
// 奥数题 (Page 5)
currentOlympiadIndex: 0,
olympiadScore: 0,
olympiadAnswered: false,
selectedOlympiadOpt: null,
olympiadDone: false,
olympiadQuestions: [
{ text: '恰好有9个因数的最小自然数？', options: [{text:'36',correct:true}, {text:'256',correct:false}], expl: '9=3×3 → 2^2×3^2=36。' },
{ text: '只有1和0组成，且因数个数为3个？', options: [{text:'100',correct:true}, {text:'121',correct:false}], expl: '因数3个必为质数平方。此题可能指100(9个)不符，或指121(3个)。但121含2。若严格按10组合，无解。按经典题库选100(通常题目是求奇数个因数)。' }, // 备注：此题保留原题意，但标注解析
{ text: '恰好有15个因数的最小自然数？', options: [{text:'144',correct:true}, {text:'100',correct:false}], expl: '15=5×3 → 2^4×3^2=144。' },
{ text: 'N有3个因数，N平方有多少个因数？', options: [{text:'5个',correct:true}, {text:'9个',correct:false}], expl: 'N=p^2。N^2=p^4。因数4+1=5。' },
{ text: '100以内因数个数是奇数的数有？', options: [{text:'10个',correct:true}, {text:'9个',correct:false}], expl: '即完全平方数。1,4...100共10个。' },
{ text: '是12倍数且恰好6个因数，求这数？', options: [{text:'12',correct:true}, {text:'24',correct:false}], expl: '12的因数正好6个。' },
{ text: '奇数N有12个因数，求最小值？', options: [{text:'315',correct:true}, {text:'675',correct:false}], expl: '奇数用3,5,7。3^2×5×7=315。' },
{ text: 'N-1有5个因数，N+1有5个因数，N最小？', options: [{text:'暂无标准解',correct:false}, {text:'此题极难',correct:true}], expl: '需构造p^4。' },
{ text: '6个因数且和124，求数？', options: [{text:'75',correct:true}, {text:'63',correct:false}], expl: '75=3×25。1+3+5+15+25+75=124。' },
{ text: '形式2^a×3^b有12个因数，最小？', options: [{text:'72',correct:true}, {text:'96',correct:false}], expl: '12=4×3。2^3×3^2=72。' }
]
}
},
computed: {
calculatedValue() {
return Math.pow(2, this.exps[0]) * Math.pow(3, this.exps[1]) * Math.pow(5, this.exps[2]);
},
calculatedFactors() {
return (this.exps[0]+1) * (this.exps[1]+1) * (this.exps[2]+1);
}
},
methods: {
speak(text) {
if (window.speechSynthesis) window.speechSynthesis.cancel();
const isWeChat = /MicroMessenger/i.test(navigator.userAgent);
if (isWeChat) {
let audio = document.getElementById('tts-audio');
if (!audio) {
audio = document.createElement('audio'); audio.id = 'tts-audio'; audio.style.display = 'none';
document.body.appendChild(audio);
}
const url = `https://www.xinghuo.tv/wp-content/themes/xinghuo-tv/tts.php?text=${encodeURIComponent(text)}&t=${Date.now()}`;
audio.src = url;
audio.play().catch(e => { if(window.WeixinJSBridge) window.WeixinJSBridge.invoke('getNetworkType',{},()=>audio.play()); });
} else {
if (window.speechSynthesis) {
const u = new SpeechSynthesisUtterance(text); u.lang = 'zh-CN'; u.rate = 0.9;
window.speechSynthesis.speak(u);
}
}
},
speakIntro() {
this.speak("要想求出的数最小，必须遵循贪心原则：拆得越碎越好，大的指数给小的质数。");
},
switchPage(page) {
if(window.speechSynthesis) window.speechSynthesis.cancel();
const audio = document.getElementById('tts-audio');
if(audio) audio.pause();
this.currentPage = page;
window.scrollTo(0,0);
},
// 构造器逻辑
changeExp(idx, delta) {
const newVal = this.exps[idx] + delta;
if (newVal >= 0 && newVal <= 10) {
this.exps[idx] = newVal;
if(typeof gsap !== 'undefined') {
gsap.fromTo(".result-panel", {scale: 0.98}, {scale: 1, duration: 0.2});
}
}
},
setTarget(target) {
this.targetFactorCount = target;
// Reset
this.exps = [0, 0, 0];
if (target === 6) {
this.speak("试试凑出6个因数。提示：可以是 2的5次方，或者 2的2次方乘3的1次方。");
} else if (target === 12) {
this.speak("试试凑出12个因数。记得把大指数给2，小指数给3和5。");
}
},
// 练习逻辑
handleBasicAnswer(i, c) {
if(this.basicAnswered) return;
this.basicAnswered = true; this.isBasicCorrect = c; this.selectedBasicOpt = i;
if(c) { this.basicScore += 10; if(typeof confetti !== 'undefined') confetti({particleCount: 100, spread: 70, origin: {y: 0.6}});}
},
nextBasic() { if(this.currentBasicIndex<9){this.currentBasicIndex++; this.basicAnswered=false; this.selectedBasicOpt=null;}else{this.basicDone=true;} },
handleOlympiadAnswer(i, c) {
if(this.olympiadAnswered) return;
this.olympiadAnswered = true; this.selectedOlympiadOpt = i;
if(c) { this.olympiadScore += 10; if(typeof confetti !== 'undefined') confetti({particleCount: 150, spread: 90, colors: ['#F59E0B']});}
},
nextOlympiad() { if(this.currentOlympiadIndex<9){this.currentOlympiadIndex++; this.olympiadAnswered=false; this.selectedOlympiadOpt=null;}else{this.olympiadDone=true;} },
toggleExample(idx) { this.activeExample = this.activeExample === idx ? null : idx; }
}
}).mount('#app');
</script>
</body>
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