Pigeon Hole explanation












2












$begingroup$


I understand that the pigeonhole principle is supposedly a quite simple concept. However could you please explain to me the reasoning of how you reach this answer. Thank you.



Question: A basket cannot contain more than $24$ apples. What is the minimum amount of baskets you must have, to ensure you have at least $5$ baskets with the same number of apples in them (all baskets have at least $1$ apple contained within).



Answer of this question being $97$ baskets.






pigeonhole-principle







share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    I understand that the pigeonhole principle is supposedly a quite simple concept. However could you please explain to me the reasoning of how you reach this answer. Thank you.



    Question: A basket cannot contain more than $24$ apples. What is the minimum amount of baskets you must have, to ensure you have at least $5$ baskets with the same number of apples in them (all baskets have at least $1$ apple contained within).



    Answer of this question being $97$ baskets.






    pigeonhole-principle







    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      I understand that the pigeonhole principle is supposedly a quite simple concept. However could you please explain to me the reasoning of how you reach this answer. Thank you.



      Question: A basket cannot contain more than $24$ apples. What is the minimum amount of baskets you must have, to ensure you have at least $5$ baskets with the same number of apples in them (all baskets have at least $1$ apple contained within).



      Answer of this question being $97$ baskets.






      pigeonhole-principle







      share|cite|improve this question











      $endgroup$




      I understand that the pigeonhole principle is supposedly a quite simple concept. However could you please explain to me the reasoning of how you reach this answer. Thank you.



      Question: A basket cannot contain more than $24$ apples. What is the minimum amount of baskets you must have, to ensure you have at least $5$ baskets with the same number of apples in them (all baskets have at least $1$ apple contained within).



      Answer of this question being $97$ baskets.






      pigeonhole-principle




      pigeonhole-principle






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 18 hours ago









      Peter

      49.2k1240138




      49.2k1240138










      asked 18 hours ago









      LaykenLayken

      324




      324






















          4 Answers
          4






          active

          oldest

          votes


















          3












          $begingroup$

          The basic idea of the pigeonhole principle is trivial , but the application can be much more difficult.



          Main idea : If we distribute $n+1$ pigeons among $n$ cages, at least one cage must have more than one pigeon.



          The problem here :



          There are $24$ possibilities for the number of apples in a basket.



          Therefore, $96$ baskets cannot be enough because every number from $1$ to $24$ can appear exactly four times.



          But if we add another basket, it is not possible anymore that all the numbers appear at most $4$ times because then, at most $96$ baskets would be possible.






          share|cite|improve this answer









          $endgroup$





















            3












            $begingroup$

            Why $96$ is not enought:



            For every number $x$ between $1$ and $24$ you take $4$ baskets with $x$ apples inside. Then you have $4cdot 24 = 96$ baskets. By construction, there is no number such that $5$ baskets have this number of apples.



            On the other hand if you have $97$ baskets and assume there are maximum $4$ baskets with the same number of apples. Then again the number of baskets is limited to $4cdot 24$ which is less than your number of baskets. So you have a contradiction. Hence, there is no configuration with max $4$ baskets with the same number of apples for $97$ baskets.






            share|cite|improve this answer









            $endgroup$





















              2












              $begingroup$

              Since a basket cannot contain more than $24$ apples, we consider a basket to be a collection of $24$ pigeonholes. Hence if we have $4$ baskets, there are altogether $24 times 4 = 96$ pigeonholes. Indeed $96$ pigeonholes can contain $96$ pigeons (apples). But if we have $97$ pigeons, by pigeonhole principle, at least one of the pigeonhole must have two pigeons, which is not allowed (the $4$ baskets are full, so we need an extra one). Hence $97$ is the answer.






              share|cite|improve this answer









              $endgroup$





















                2












                $begingroup$

                The number of apples in a basket is between 1 and 24. That's 24 different values.



                Suppose that in our collection of baskets, none of these numbers occurs at least five times.



                This means that in our collection of baskets each of these numbers occurs at most four times.



                But then we can have at most 24 times 4 baskets.



                Now, $24 times 4 = 96$. So if we have more than that number of baskets, that is if we have at least 97 baskets, then one of those numbers between 1 and 24 must occur at least five times, that is then we have at least five baskets with the same number of apples.






                share|cite|improve this answer











                $endgroup$













                • $begingroup$
                  Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                  $endgroup$
                  – L. F.
                  17 hours ago












                Your Answer








                StackExchange.ready(function() {
                var channelOptions = {
                tags: "".split(" "),
                id: "69"
                };
                initTagRenderer("".split(" "), "".split(" "), channelOptions);

                StackExchange.using("externalEditor", function() {
                // Have to fire editor after snippets, if snippets enabled
                if (StackExchange.settings.snippets.snippetsEnabled) {
                StackExchange.using("snippets", function() {
                createEditor();
                });
                }
                else {
                createEditor();
                }
                });

                function createEditor() {
                StackExchange.prepareEditor({
                heartbeatType: 'answer',
                autoActivateHeartbeat: false,
                convertImagesToLinks: true,
                noModals: true,
                showLowRepImageUploadWarning: true,
                reputationToPostImages: 10,
                bindNavPrevention: true,
                postfix: "",
                imageUploader: {
                brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
                contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
                allowUrls: true
                },
                noCode: true, onDemand: true,
                discardSelector: ".discard-answer"
                ,immediatelyShowMarkdownHelp:true
                });


                }
                });














                draft saved

                draft discarded


















                StackExchange.ready(
                function () {
                StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3187132%2fpigeon-hole-explanation%23new-answer', 'question_page');
                }
                );

                Post as a guest















                Required, but never shown

























                4 Answers
                4






                active

                oldest

                votes








                4 Answers
                4






                active

                oldest

                votes









                active

                oldest

                votes






                active

                oldest

                votes









                3












                $begingroup$

                The basic idea of the pigeonhole principle is trivial , but the application can be much more difficult.



                Main idea : If we distribute $n+1$ pigeons among $n$ cages, at least one cage must have more than one pigeon.



                The problem here :



                There are $24$ possibilities for the number of apples in a basket.



                Therefore, $96$ baskets cannot be enough because every number from $1$ to $24$ can appear exactly four times.



                But if we add another basket, it is not possible anymore that all the numbers appear at most $4$ times because then, at most $96$ baskets would be possible.






                share|cite|improve this answer









                $endgroup$


















                  3












                  $begingroup$

                  The basic idea of the pigeonhole principle is trivial , but the application can be much more difficult.



                  Main idea : If we distribute $n+1$ pigeons among $n$ cages, at least one cage must have more than one pigeon.



                  The problem here :



                  There are $24$ possibilities for the number of apples in a basket.



                  Therefore, $96$ baskets cannot be enough because every number from $1$ to $24$ can appear exactly four times.



                  But if we add another basket, it is not possible anymore that all the numbers appear at most $4$ times because then, at most $96$ baskets would be possible.






                  share|cite|improve this answer









                  $endgroup$
















                    3












                    3








                    3





                    $begingroup$

                    The basic idea of the pigeonhole principle is trivial , but the application can be much more difficult.



                    Main idea : If we distribute $n+1$ pigeons among $n$ cages, at least one cage must have more than one pigeon.



                    The problem here :



                    There are $24$ possibilities for the number of apples in a basket.



                    Therefore, $96$ baskets cannot be enough because every number from $1$ to $24$ can appear exactly four times.



                    But if we add another basket, it is not possible anymore that all the numbers appear at most $4$ times because then, at most $96$ baskets would be possible.






                    share|cite|improve this answer









                    $endgroup$



                    The basic idea of the pigeonhole principle is trivial , but the application can be much more difficult.



                    Main idea : If we distribute $n+1$ pigeons among $n$ cages, at least one cage must have more than one pigeon.



                    The problem here :



                    There are $24$ possibilities for the number of apples in a basket.



                    Therefore, $96$ baskets cannot be enough because every number from $1$ to $24$ can appear exactly four times.



                    But if we add another basket, it is not possible anymore that all the numbers appear at most $4$ times because then, at most $96$ baskets would be possible.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 18 hours ago









                    PeterPeter

                    49.2k1240138




                    49.2k1240138























                        3












                        $begingroup$

                        Why $96$ is not enought:



                        For every number $x$ between $1$ and $24$ you take $4$ baskets with $x$ apples inside. Then you have $4cdot 24 = 96$ baskets. By construction, there is no number such that $5$ baskets have this number of apples.



                        On the other hand if you have $97$ baskets and assume there are maximum $4$ baskets with the same number of apples. Then again the number of baskets is limited to $4cdot 24$ which is less than your number of baskets. So you have a contradiction. Hence, there is no configuration with max $4$ baskets with the same number of apples for $97$ baskets.






                        share|cite|improve this answer









                        $endgroup$


















                          3












                          $begingroup$

                          Why $96$ is not enought:



                          For every number $x$ between $1$ and $24$ you take $4$ baskets with $x$ apples inside. Then you have $4cdot 24 = 96$ baskets. By construction, there is no number such that $5$ baskets have this number of apples.



                          On the other hand if you have $97$ baskets and assume there are maximum $4$ baskets with the same number of apples. Then again the number of baskets is limited to $4cdot 24$ which is less than your number of baskets. So you have a contradiction. Hence, there is no configuration with max $4$ baskets with the same number of apples for $97$ baskets.






                          share|cite|improve this answer









                          $endgroup$
















                            3












                            3








                            3





                            $begingroup$

                            Why $96$ is not enought:



                            For every number $x$ between $1$ and $24$ you take $4$ baskets with $x$ apples inside. Then you have $4cdot 24 = 96$ baskets. By construction, there is no number such that $5$ baskets have this number of apples.



                            On the other hand if you have $97$ baskets and assume there are maximum $4$ baskets with the same number of apples. Then again the number of baskets is limited to $4cdot 24$ which is less than your number of baskets. So you have a contradiction. Hence, there is no configuration with max $4$ baskets with the same number of apples for $97$ baskets.






                            share|cite|improve this answer









                            $endgroup$



                            Why $96$ is not enought:



                            For every number $x$ between $1$ and $24$ you take $4$ baskets with $x$ apples inside. Then you have $4cdot 24 = 96$ baskets. By construction, there is no number such that $5$ baskets have this number of apples.



                            On the other hand if you have $97$ baskets and assume there are maximum $4$ baskets with the same number of apples. Then again the number of baskets is limited to $4cdot 24$ which is less than your number of baskets. So you have a contradiction. Hence, there is no configuration with max $4$ baskets with the same number of apples for $97$ baskets.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 18 hours ago









                            Nathanael SkrepekNathanael Skrepek

                            1,7921615




                            1,7921615























                                2












                                $begingroup$

                                Since a basket cannot contain more than $24$ apples, we consider a basket to be a collection of $24$ pigeonholes. Hence if we have $4$ baskets, there are altogether $24 times 4 = 96$ pigeonholes. Indeed $96$ pigeonholes can contain $96$ pigeons (apples). But if we have $97$ pigeons, by pigeonhole principle, at least one of the pigeonhole must have two pigeons, which is not allowed (the $4$ baskets are full, so we need an extra one). Hence $97$ is the answer.






                                share|cite|improve this answer









                                $endgroup$


















                                  2












                                  $begingroup$

                                  Since a basket cannot contain more than $24$ apples, we consider a basket to be a collection of $24$ pigeonholes. Hence if we have $4$ baskets, there are altogether $24 times 4 = 96$ pigeonholes. Indeed $96$ pigeonholes can contain $96$ pigeons (apples). But if we have $97$ pigeons, by pigeonhole principle, at least one of the pigeonhole must have two pigeons, which is not allowed (the $4$ baskets are full, so we need an extra one). Hence $97$ is the answer.






                                  share|cite|improve this answer









                                  $endgroup$
















                                    2












                                    2








                                    2





                                    $begingroup$

                                    Since a basket cannot contain more than $24$ apples, we consider a basket to be a collection of $24$ pigeonholes. Hence if we have $4$ baskets, there are altogether $24 times 4 = 96$ pigeonholes. Indeed $96$ pigeonholes can contain $96$ pigeons (apples). But if we have $97$ pigeons, by pigeonhole principle, at least one of the pigeonhole must have two pigeons, which is not allowed (the $4$ baskets are full, so we need an extra one). Hence $97$ is the answer.






                                    share|cite|improve this answer









                                    $endgroup$



                                    Since a basket cannot contain more than $24$ apples, we consider a basket to be a collection of $24$ pigeonholes. Hence if we have $4$ baskets, there are altogether $24 times 4 = 96$ pigeonholes. Indeed $96$ pigeonholes can contain $96$ pigeons (apples). But if we have $97$ pigeons, by pigeonhole principle, at least one of the pigeonhole must have two pigeons, which is not allowed (the $4$ baskets are full, so we need an extra one). Hence $97$ is the answer.







                                    share|cite|improve this answer












                                    share|cite|improve this answer



                                    share|cite|improve this answer










                                    answered 18 hours ago









                                    tonychow0929tonychow0929

                                    42137




                                    42137























                                        2












                                        $begingroup$

                                        The number of apples in a basket is between 1 and 24. That's 24 different values.



                                        Suppose that in our collection of baskets, none of these numbers occurs at least five times.



                                        This means that in our collection of baskets each of these numbers occurs at most four times.



                                        But then we can have at most 24 times 4 baskets.



                                        Now, $24 times 4 = 96$. So if we have more than that number of baskets, that is if we have at least 97 baskets, then one of those numbers between 1 and 24 must occur at least five times, that is then we have at least five baskets with the same number of apples.






                                        share|cite|improve this answer











                                        $endgroup$













                                        • $begingroup$
                                          Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                          $endgroup$
                                          – L. F.
                                          17 hours ago
















                                        2












                                        $begingroup$

                                        The number of apples in a basket is between 1 and 24. That's 24 different values.



                                        Suppose that in our collection of baskets, none of these numbers occurs at least five times.



                                        This means that in our collection of baskets each of these numbers occurs at most four times.



                                        But then we can have at most 24 times 4 baskets.



                                        Now, $24 times 4 = 96$. So if we have more than that number of baskets, that is if we have at least 97 baskets, then one of those numbers between 1 and 24 must occur at least five times, that is then we have at least five baskets with the same number of apples.






                                        share|cite|improve this answer











                                        $endgroup$













                                        • $begingroup$
                                          Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                          $endgroup$
                                          – L. F.
                                          17 hours ago














                                        2












                                        2








                                        2





                                        $begingroup$

                                        The number of apples in a basket is between 1 and 24. That's 24 different values.



                                        Suppose that in our collection of baskets, none of these numbers occurs at least five times.



                                        This means that in our collection of baskets each of these numbers occurs at most four times.



                                        But then we can have at most 24 times 4 baskets.



                                        Now, $24 times 4 = 96$. So if we have more than that number of baskets, that is if we have at least 97 baskets, then one of those numbers between 1 and 24 must occur at least five times, that is then we have at least five baskets with the same number of apples.






                                        share|cite|improve this answer











                                        $endgroup$



                                        The number of apples in a basket is between 1 and 24. That's 24 different values.



                                        Suppose that in our collection of baskets, none of these numbers occurs at least five times.



                                        This means that in our collection of baskets each of these numbers occurs at most four times.



                                        But then we can have at most 24 times 4 baskets.



                                        Now, $24 times 4 = 96$. So if we have more than that number of baskets, that is if we have at least 97 baskets, then one of those numbers between 1 and 24 must occur at least five times, that is then we have at least five baskets with the same number of apples.







                                        share|cite|improve this answer














                                        share|cite|improve this answer



                                        share|cite|improve this answer








                                        edited 5 hours ago

























                                        answered 18 hours ago









                                        jflippjflipp

                                        3,7711711




                                        3,7711711












                                        • $begingroup$
                                          Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                          $endgroup$
                                          – L. F.
                                          17 hours ago


















                                        • $begingroup$
                                          Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                          $endgroup$
                                          – L. F.
                                          17 hours ago
















                                        $begingroup$
                                        Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                        $endgroup$
                                        – L. F.
                                        17 hours ago




                                        $begingroup$
                                        Please don't use $24 * 4 = 96$, it looks bad ... Use $24 times 4$ or $24 cdot 4$.
                                        $endgroup$
                                        – L. F.
                                        17 hours ago


















                                        draft saved

                                        draft discarded




















































                                        Thanks for contributing an answer to Mathematics Stack Exchange!


                                        • Please be sure to answer the question. Provide details and share your research!

                                        But avoid …



                                        • Asking for help, clarification, or responding to other answers.

                                        • Making statements based on opinion; back them up with references or personal experience.


                                        Use MathJax to format equations. MathJax reference.


                                        To learn more, see our tips on writing great answers.




                                        draft saved


                                        draft discarded














                                        StackExchange.ready(
                                        function () {
                                        StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3187132%2fpigeon-hole-explanation%23new-answer', 'question_page');
                                        }
                                        );

                                        Post as a guest















                                        Required, but never shown





















































                                        Required, but never shown














                                        Required, but never shown












                                        Required, but never shown







                                        Required, but never shown

































                                        Required, but never shown














                                        Required, but never shown












                                        Required, but never shown







                                        Required, but never shown







                                        -

                                        Popular posts from this blog

                                        GameSpot

                                        Image
                                        body.skin-minerva .mw-parser-output table.infobox caption{text-align:center} GameSpot 戰地風雲：惡名昭彰2在Gamespot的評論 网站类型 新聞 持有者 CBS 创始人 Pete Deemer Vince Broady Jon Epstein 网站 http://www.gamespot.com/ 注册 Optional (free and paid) 推出时间 1996年5月1日 [1] GameSpot （中国大陆：游戏基地），於1996年5月由Pete Deemer和Vince Broady創立，是一個提供新聞、評論、預告片、下載及其他的相關資訊的電子遊戲網站。GameSpot被一間後來被CNET收購的企業ZDNet所收購。根據Alexa，GameSpot.com是200個網路擁擠最嚴重的網站之一。 除了由GameSpot員工創作的內容，網站還允許用戶寫評論、網誌、之後在網路論壇分享。一些在CNET旗下的GameFAQs分享。 2004年， GameSpot被Spike TV的觀眾選上「電子遊戲賞節目」贏得「最傑出遊戲網站。 [2] 其他的遊戲網站還有IGN、1UP.com、GameSpy是它最大的競爭對手。2008年，根據Compete.com的統計，「gamespot.com」吸引了最少6000萬人的點擊率。 [3] GameSpot的主頁鏈結了到最近新聞、評論、預告、和一些有關遊戲機的入口：Wii、任天堂DS、電腦遊戲、Xbox 360、PSP、PlayStation 2、PlayStation 3。它還有一列「最受歡迎遊戲名單」，還有給用戶快速獲得遊戲資訊的搜尋器。GameSpot 還包括一些小範圍的遊戲機：任天堂64、GameCube、Game Boy Color、Game Boy Advance、Xbox、PlayStation、SEGA Saturn、Dreamcast、Neo Geo Pocket Color、N-Gage、手機遊戲。 目录 1 歷史 1.1 國際歷史 1.2 著名的員工 2 評論和分
                                        Read more

                                        connect to host localhost port 22: Connection refused

                                        Image
                                        .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0; } 1 1 I created a Ubuntu Vagrant box and ssh'd into it. Now, created a ssh key and trying to ssh to my localhost but I get the following error. vagrant@vagrant-ubuntu-trusty-64:~$ ssh -vvvv localhost OpenSSH_6.6.1, OpenSSL 1.0.1f 6 Jan 2014 debug1: Reading configuration data /etc/ssh/ssh_config debug1: /etc/ssh/ssh_config line 19: Applying options for * debug2: ssh_connect: needpriv 0 debug1: Connecting to localhost [127.0.0.1] port 22. debug1: connect to address 127.0.0.1 port 22: Connection refused ssh: connect to host localhost port 22: Connection refused When I check the list of ports opened as follows: vagrant@vagrant-ubuntu-trusty-64:~$ s
                                        Read more

                                        Getting a Wifi WPA2 wifi connection

                                        Image
                                        .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0; } 0 I've been trying to get a simple wifi connection ! bu I have to admit I just can't get it done ! here is what I've tried: root@MYONE:/etc# clear root@MYONE:/etc# vi wpa_supplicant.conf ctrl_interface=/var/run/wpa_supplicant ctrl_interface_group=0 update_config=1 network={ ssid="Mi Engine" psk="7407352688e4" } root@MYONE:/etc# iw wlan0 link Not connected. root@MYONE:/etc# ifconfig eth0 Link encap:Ethernet HWaddr 74:FE:48:3A:55:AB inet addr:192.168.178.88 Bcast:192.168.178.255 Mask:255.255.255.0 inet6 addr: fe80::76fe:48ff:fe3a:55ab%4804152/64 Scope:Link UP BRO
                                        Read more