思维私塾——一行代码解决算法

<div class="output_wrapper" id="output_wrapper_id" style="font-size: 15px; color: rgb(62, 62, 62); line-height: 1.8; word-spacing: 2px; letter-spacing: 2px; font-family: 'Helvetica Neue', Helvetica, 'Hiragino Sans GB', 'Microsoft YaHei', Arial, sans-serif; background-image: linear-gradient(90deg, rgba(50, 0, 0, 0.05) 3%, rgba(0, 0, 0, 0) 3%), linear-gradient(360deg, rgba(50, 0, 0, 0.05) 3%, rgba(0, 0, 0, 0) 3%); background-size: 20px 20px; background-position: center center;"><p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">有些算法题目，只要掌握了思路就可以用很短的代码来实现它。比如下面这几道题目：</p>
<h3 id="h" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">二的幂</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3>
<h4 id="h-1" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">问题</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">判断一个数字是否是2的n次方</p>
<h4 id="h-2" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">解答</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">遇到2的幂次方，要建立位移操作的思想，如果n是二的幂，即最高位为1其他位置为0，那么n-1就是最高位为0，其余位置为1，那么n&(n-1)就是0</p>
<pre style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;"><span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">1</span><span class="hljs-function" style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; word-wrap: inherit !important; word-break: inherit !important;"><span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">boolean</span> <span class="hljs-title" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(153, 0, 0); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">isPowerOfTow</span><span class="hljs-params" style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; word-wrap: inherit !important; word-break: inherit !important;">(<span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">int</span> n)</span></span>{
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">2</span>    <span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">return</span> (n><span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">0</span>)%%(n&(n-<span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">1</span>))==<span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">0</span>;
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">3</span>}
</pre>
<h3 id="h-3" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">三的幂</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3>
<h4 id="h-4" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">问题</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">判断一个数字是否为3的n次方</p>
<h4 id="h-5" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">解答</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">int类型中不溢出情况下3的最高次方即为1162261467，那么只要这个数去约n为0，那么说明n的约数只有三，即为3的n次方.</p>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">同样的做法也可以适用于5的幂,7的幂,X的幂（X是一个素数）之类的题目。</p>
<pre style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;"><span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">1</span><span class="hljs-function" style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; word-wrap: inherit !important; word-break: inherit !important;"><span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">boolean</span> <span class="hljs-title" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(153, 0, 0); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">isPowerOfThree</span><span class="hljs-params" style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; word-wrap: inherit !important; word-break: inherit !important;">(<span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">int</span> n)</span></span>{
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">2</span>    <span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">return</span> (n><span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">0</span>)&&(<span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">116261467</span>%n)==<span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">0</span>;
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">3</span>}
</pre>
<h3 id="h1" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">二进制1的个数</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3>
<h4 id="h-6" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">问题</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">给定一个数字n，求n转换为二进制表示中1的个数</p>
<h4 id="h-7" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">解答</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">JDK大法好，直接使用自带的方法：</p>
<ul style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; padding-left: 32px; list-style-type: disc;">
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">bitCount：返回二进制中1的个数</span></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">lowestOneBit：返回二进制表示中最低位1的位置</span></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">highestOneBit：返回二进制表示中最高位1的位置</span></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">numberOfLeadingZeros：返回二进制表示中高位0的数目</span></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">numberOfTrailingZeros：返回二进制表示中低位0的数目</span></li>
</ul>
<pre style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;"><span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">1</span> <span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">int</span> bitCount(<span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">int</span> <span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">num</span>) {
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">2</span>        <span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">return</span> Integer.bitCount(<span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">num</span>);
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">3</span>    }
</pre>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">或者也可以不使用jdk类库的解法</p>
<pre style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;"><span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">1</span><span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">int</span> bitOfOne(<span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">int</span> <span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">num</span>) {
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">2</span>        <span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">return</span> <span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">num</span> == <span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">0</span> ? <span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">0</span> : <span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">1</span> + bitOfOne(<span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">num</span> & (<span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">num</span> - <span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">1</span>));
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">3</span>    }
</pre>
<h3 id="h-8" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">阶乘后的零</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3>
<h4 id="h-9" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">问题</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">给定一个数字n，求n！位数中的0的个数。</p>
<h4 id="h-10" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">解答</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">这种问题我们要分析0的产生原因，因为10=52，对于一个n！的数字而言，末尾为0的数字即n！的约数中包含的52的次数，因为2的次数是足够的，也就是说这道题等价于求n！中约数5的个数（注意诸如25这种数字，约数中包括了2个5）。</p>
<pre style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;"><span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">1</span><span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">int</span> zeroAtTheEndCount(<span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">int</span> <span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">num</span>) {
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">2</span>        <span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">return</span> <span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">num</span> / <span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">5</span> + zeroAtTheEndCount(<span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">num</span> / <span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">5</span>);
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">3</span>    }
</pre>
<h3 id="h-11" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">熄灭的灯</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3>
<h4 id="h-12" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">问题</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">有x个灯，一个开关控制一个灯，按下开关会使灯熄灭或者打开，现在灯都是熄灭的状态，我们进行n次操作，每次操作都把编号为i的倍数的开关全部按下，即</p>
<ul style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; padding-left: 32px; list-style-type: disc;">
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">将编号为1的倍数的开关全部按下</span></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">将编号为2的倍数的开关全部按下</span></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">…</span></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">将编号为x的倍数的开关全部按下</span></li>
</ul>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">已知灯的总数为x，求最后灯亮起来状态的数量</p>
<h4 id="h-13" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">解答</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">首先任意取灯的编号n，会有几次操作会影响这个灯的状态呢？答案是n的所有约数的个数次。</p>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">好比说编号为12的灯，12的约数有1，2，3，4，6，12。也就是说编号为12的灯的状态会被切换6次，分别在第1，2，3，4，6，12次切换开关的时候。</p>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">那么从下面2种情况考虑：</p>
<ul style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; padding-left: 32px; list-style-type: disc;">
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">假设n是一个非完全平方的数：因为为一个非平方的数的约数个数为偶数个，也就是说会切换偶数次状态，也就是说状态不会变。</span></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">假设n是一个完全平方数：完全平方数的约数是奇数个，其状态会发生改变。</span></li>
</ul>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">那么按照题目的要求，一开始全是熄灭状态的，求亮起来状态的灯数目，就是说求会发生变化的编号的数目，也就是相当于求1～x之间完全平方数的个数。</p>
<pre style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;"><span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">1</span><span class="hljs-function" style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; word-wrap: inherit !important; word-break: inherit !important;"><span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">int</span> <span class="hljs-title" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(153, 0, 0); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">lightCount</span><span class="hljs-params" style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; word-wrap: inherit !important; word-break: inherit !important;">(<span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">int</span> num)</span> </span>{
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">2</span>        <span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">return</span> (<span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">int</span>) Math.<span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">sqrt</span>(num);
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">3</span>    }
</pre>
<h3 id="h-14" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">奇怪的数</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3>
<h4 id="h-15" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">问题</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">给定一个非空数组，其中除了一个元素只出现一次外，其他元素都出现了两次，求这个只出现一次的元素</p>
<h4 id="h-16" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">解答</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">这里的一个关键运算符是^ 异或运算符，主要是利用到了异或运算a^b满足的如下规律：</p>
<ul style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; padding-left: 32px; list-style-type: disc;">
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">a^b=b^a 交换律</span></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">a^b^c=a^(b^c)结合律</span></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">a^a=0,同时0^0=0</span></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">0^a=a</span></li>
</ul>
<pre style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;"><span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">1</span><span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">int</span> loseNumber(<span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">int</span>[] <span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">num</span>) {
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">2</span>        <span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">return</span> Arrays.stream(<span class="hljs-built_in" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 134, 179); word-wrap: inherit !important; word-break: inherit !important;">num</span>).reduce(<span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">0</span>, (n1, n2) -> n1 ^ n2);
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">3</span>    }
</pre>
<h3 id="h-17" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; border-bottom: 2px solid rgb(239, 112, 96); font-size: 1.3em;"><span style="font-size: inherit; line-height: inherit; margin: 0px; display: inline-block; font-weight: normal; background: rgb(239, 112, 96); color: rgb(255, 255, 255); padding: 3px 10px 1px; border-top-right-radius: 3px; border-top-left-radius: 3px; margin-right: 3px;">约瑟夫环问题</span><span style="display: inline-block; vertical-align: bottom; border-bottom: 36px solid rgb(239, 235, 233); border-right: 20px solid transparent;"> </span></h3>
<h4 id="h-18" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">问题</span></h4>
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">编号为1-n的n个战士围成一圈，从编号1的战士开始依次报数，数到m的战士会淘汰，以后的战士再重新报数，直到剩下一个士兵，求这个士兵的编号</p>
<h4 id="h-19" style="color: inherit; line-height: inherit; padding: 0px; margin: 1.6em 0px; font-weight: bold; font-size: 1.2em;"><span style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;">解答</span></h4>

删除前（old）	删除后（new）
m-2	n-2
m-1	n-1
m	无
m+1	1
m+2	2

<ul style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; padding-left: 32px; list-style-type: disc;">
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">假设old为删除前的节点编号，new为删除后的节点编号</p></li>
<li style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; margin-bottom: 0.5em;"><p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 1.7em 0px;">- 关系：old=(new+m-1)%n+1</p></li>
</ul>
<blockquote style="line-height: inherit; display: block; padding: 15px 15px 15px 1rem; font-size: 0.9em; margin: 1em 0px; color: rgb(0, 0, 0); border-left: 5px solid rgb(239, 112, 96); background: rgb(239, 235, 233); overflow: auto; overflow-wrap: normal; word-break: normal;">
<p style="font-size: inherit; color: inherit; line-height: inherit; padding: 0px; margin: 0px;">注意：并不是old=(new+m)%n，因为编号是从1开始的而不是从0 开始的，如果new+m==n的话，会导致最后的计算结果old=0.</p>
</blockquote>
<pre style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px;"><span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">1</span><span class="hljs-function" style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; word-wrap: inherit !important; word-break: inherit !important;"><span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">int</span> <span class="hljs-title" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(153, 0, 0); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">f</span><span class="hljs-params" style="font-size: inherit; color: inherit; line-height: inherit; margin: 0px; padding: 0px; word-wrap: inherit !important; word-break: inherit !important;">(<span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">int</span> n,<span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">int</span> m)</span></span>{
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">2</span>    <span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">if</span>(n==<span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">1</span>) 
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">3</span>        <span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">return</span> n;
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">4</span>    <span class="hljs-keyword" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(51, 51, 51); font-weight: bold; word-wrap: inherit !important; word-break: inherit !important;">return</span> (f(n<span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">-1</span>,m)+m<span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">-1</span>)%n+<span class="hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); word-wrap: inherit !important; word-break: inherit !important;">1</span>;
<span class="linenum hljs-number" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; color: rgb(0, 128, 128); padding-right: 20px; word-spacing: 0px; word-wrap: inherit !important; word-break: inherit !important;">5</span>}
</pre>
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