四年级奥数难题（2023.3.10）：页面编号 标签：速算与巧算
<p>优学奥数难题以小学4-6年级的杯赛题为来源，试题挑选、答案详解准确性均经优学奥数名师鉴证；根据对历年杯赛真题的研究、总结及归纳，结合了赛题中的高频考点、难点、易错点、以及最近几年命题趋势所得；适合志在杯赛中夺取佳绩的学生。</p><p><strong><strong>一本书共有500页，编上页码：1，2，3，4……499，500.问：数字2在页码中一共出现了多少次?</strong></strong></p><p>沈丽娟老师</p><p>选题编辑：沈丽娟老师</p><p>毕业于华南师范大学数学与应用数学 (师范)专业，优学专职教师，中国数学奥林匹克二级教练员。在大学期间修读“竞赛数学”，成绩优异。对中小学奥数知识体系了解透彻，重难点把握到位。辅导的学生中多人获得“华杯赛”奖项。</p><p><strong>教学特色：</strong></p><p>1、语言生动幽默，十分有亲和力，易于学生接受。2、拥有很强的数学功底，同时善于解题和总结。3、上课思路清晰、讲解透彻，注重知识及思维的发生、发展过程，深入浅出进行引导，善于联系学生的生活经验为学生构建形象生动的情境，帮助学生理解题目。</p><p><strong>老师教你解难题-试题详解</strong></p><p>数字2在页码中一共出现了200次。</p><p>这道题目看上去好像不太复杂。例如，在2，25和52这些数中“2”分别出现一次;在22、242中又分别出现了二次;而在222中，它出现了三次。但像这样盲目地东找一个，西找一个，要准确求出答案仍然是非常困难的。</p><p>因此，在解这道题目的时候，我们可以把1-500按一百一百地分成五段并按数位进行统计。</p><p>统计表如下：</p><p><table border="1" cellpadding="0" cellspacing="0" style="border-right: medium none; border-top: medium none; border-left: medium none; border-bottom: medium none; border-collapse: collapse"><tbody><tr style="height: 48.25pt"><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; border-top: windowtext 1pt solid; padding-left: 5.4pt; padding-bottom: 0cm; border-left: windowtext 1pt solid; width: 4cm; padding-top: 0cm; border-bottom: windowtext 1pt solid; height: 48.25pt; background-color: transparent" valign="top" width="151"> 数位 “2”出现 的次数 分段</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; border-top: windowtext 1pt solid; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 72pt; padding-top: 0cm; border-bottom: windowtext 1pt solid; height: 48.25pt; background-color: transparent" valign="top" width="96"> 百位</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; border-top: windowtext 1pt solid; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 70.2pt; padding-top: 0cm; border-bottom: windowtext 1pt solid; height: 48.25pt; background-color: transparent" valign="top" width="94"> ʮλ</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; border-top: windowtext 1pt solid; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; padding-top: 0cm; border-bottom: windowtext 1pt solid; height: 48.25pt; background-color: transparent" valign="top" width="114"> 个位</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; border-top: windowtext 1pt solid; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; padding-top: 0cm; border-bottom: windowtext 1pt solid; height: 48.25pt; background-color: transparent" valign="top" width="114"> 小计</td></tr><tr><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; padding-bottom: 0cm; border-left: windowtext 1pt solid; width: 4cm; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="151"> 1-100</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 72pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="96"> /</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 70.2pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="94"> 10</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 10</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 20</td></tr><tr><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; padding-bottom: 0cm; border-left: windowtext 1pt solid; width: 4cm; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="151"> 101-200</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 72pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="96"> 1</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 70.2pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="94"> 10</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 10</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 21</td></tr><tr><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; padding-bottom: 0cm; border-left: windowtext 1pt solid; width: 4cm; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="151"> 201-300</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 72pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="96"> 99</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 70.2pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="94"> 10</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 10</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 119</td></tr><tr><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; padding-bottom: 0cm; border-left: windowtext 1pt solid; width: 4cm; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="151"> 301-400</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 72pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="96"> /</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 70.2pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="94"> 10</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 10</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 20</td></tr><tr><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; padding-bottom: 0cm; border-left: windowtext 1pt solid; width: 4cm; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="151"> 401-500</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 72pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="96"> /</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 70.2pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="94"> 10</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 10</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 20</td></tr><tr><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; padding-bottom: 0cm; border-left: windowtext 1pt solid; width: 4cm; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="151"> 合计</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 72pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="96"> 100</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 70.2pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="94"> 50</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 50</td><td style="border-right: windowtext 1pt solid; padding-right: 5.4pt; padding-left: 5.4pt; border-left-color: #d4d0c8; padding-bottom: 0cm; width: 85.25pt; border-top-color: #d4d0c8; padding-top: 0cm; border-bottom: windowtext 1pt solid; background-color: transparent" valign="top" width="114"> 200</td></tr></tbody></table> 所以，数字2在页码中一共出现了200次。</p><p>小结：在解答页码问题的数学题主要有两类：知道一本书的页码，求共有多少个数码，反过来，知道一本书的数码，求这本书的页码。</p>
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