How many times do all 3 clock hands align?











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I have seen this question on several different websites and mathematics journals, but I haven't seen any concrete answers. For instance, some sources say it will happen only twice, while others say it will happen 22 times. Upon researching it, it seems as if all three hands will line up some amount higher than 2 times, but I am unsure of how many times it would occur.










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  • Presumably you mean how many times per 24 hours.
    – Travis
    Nov 26 at 16:42










  • Look here (second answer) for a clear explanation.
    – rogerl
    Nov 26 at 16:56










  • Maybe the 22 answer was just for the hour and minute hands aligning.
    – GEdgar
    Dec 6 at 23:47















up vote
0
down vote

favorite
1












I have seen this question on several different websites and mathematics journals, but I haven't seen any concrete answers. For instance, some sources say it will happen only twice, while others say it will happen 22 times. Upon researching it, it seems as if all three hands will line up some amount higher than 2 times, but I am unsure of how many times it would occur.










share|cite|improve this question






















  • Presumably you mean how many times per 24 hours.
    – Travis
    Nov 26 at 16:42










  • Look here (second answer) for a clear explanation.
    – rogerl
    Nov 26 at 16:56










  • Maybe the 22 answer was just for the hour and minute hands aligning.
    – GEdgar
    Dec 6 at 23:47













up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





I have seen this question on several different websites and mathematics journals, but I haven't seen any concrete answers. For instance, some sources say it will happen only twice, while others say it will happen 22 times. Upon researching it, it seems as if all three hands will line up some amount higher than 2 times, but I am unsure of how many times it would occur.










share|cite|improve this question













I have seen this question on several different websites and mathematics journals, but I haven't seen any concrete answers. For instance, some sources say it will happen only twice, while others say it will happen 22 times. Upon researching it, it seems as if all three hands will line up some amount higher than 2 times, but I am unsure of how many times it would occur.







permutations






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asked Nov 26 at 16:18









Kenneth Sweezy

32




32












  • Presumably you mean how many times per 24 hours.
    – Travis
    Nov 26 at 16:42










  • Look here (second answer) for a clear explanation.
    – rogerl
    Nov 26 at 16:56










  • Maybe the 22 answer was just for the hour and minute hands aligning.
    – GEdgar
    Dec 6 at 23:47


















  • Presumably you mean how many times per 24 hours.
    – Travis
    Nov 26 at 16:42










  • Look here (second answer) for a clear explanation.
    – rogerl
    Nov 26 at 16:56










  • Maybe the 22 answer was just for the hour and minute hands aligning.
    – GEdgar
    Dec 6 at 23:47
















Presumably you mean how many times per 24 hours.
– Travis
Nov 26 at 16:42




Presumably you mean how many times per 24 hours.
– Travis
Nov 26 at 16:42












Look here (second answer) for a clear explanation.
– rogerl
Nov 26 at 16:56




Look here (second answer) for a clear explanation.
– rogerl
Nov 26 at 16:56












Maybe the 22 answer was just for the hour and minute hands aligning.
– GEdgar
Dec 6 at 23:47




Maybe the 22 answer was just for the hour and minute hands aligning.
– GEdgar
Dec 6 at 23:47










3 Answers
3






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up vote
1
down vote



accepted










This happens exactly once every $12$ hours. If $$tmapsto e^{it}qquad(0leq t< 2pi)$$ describes the position of the hour hand on the dial $S^1$ during the $12$ hour interval beginning at 12.00 noon then $tmapsto e^{12it}$ describes the position of the minute hand, and $tmapsto e^{720it}$ describes the position of the second hand. The hour hand and the minute hand coincide when $e^{it}=e^{12it}$, i.e., when $e^{11it}=1$. This is the case when $t={2pi kover11}$ $(0leq k<11)$. Similarly, the hour hand and the second hand coincide when $t={2pi lover 719}$ $(0leq l<719)$. As ${rm gcd}(11,719)=1$ there is no $tin[0,2pi[>$ fulfilling both conditions other than $t=0$.



It follows that all three hands will line up exactly two times every $24$ hours.






share|cite|improve this answer






























    up vote
    1
    down vote













    The answer depends on which hands' motions are continuous and which are discrete, and if so, how they are discretized. For simplicity, I'll address the case that all motions are continuous, so that all three hands have constant velocity.



    Per 12-hour period ($1$ rotation of the hour hand), the minute hand rotates $12$ times, so over this time the angular distance between the hour and minute hands is zero $12 - 1 = 11$ times. (This is where the figure of $22$ comes from: This is the number of times two hands align per 24 hours.) These 11 times are equally spaced, and include $0:00$, so they occur precisely when the hour hand has completed $frac{k}{11}$ of a rotation, $k = 0, ldots, 10$.



    The second hand moves $12 cdot 60$ times as fast as the large hand, so in the time between successive alignments of the hour and minute hands, the second hand has completed $frac{720}{11} = n + frac{5}{11}$ rotations for some integer $n$, so the accumulated angular distance between the second hand and the common angle of the other two hands has increased by $frac{5}{11} - frac{1}{11} = frac{4}{11}$ of a rotation. It follows from the fact that $4$ and $11$ are coprime that the only times at which all three hands are aligned is when they all point upward. Each day, this happens only at $0:00$ and $12:00$.






    share|cite|improve this answer






























      up vote
      1
      down vote













      We can answer this by determining the times when the hour and minute hands coincide, then checking whether the second hand will also match.



      In $12$ hours, the hour hand rotates once while the minute hand rotates $12$ times. Therefore the minute hand does $12-1 = 11$ rotations relative to the hour hand in this time. So the interval between the times when it overtakes the hour hand is $frac{12}{11}$ hours. ($=1$h $5$m $27 frac{3}{11}$s).



      All three hands are aligned at $12:00:00$. Adding integer multiples of $1:5:27frac{3}{11}$ to this gives the times when the hour and minute hands are aligned (in $12$ hour format) as



      $12:00:00$
      $1:05:27frac{3}{11}$
      $2:10:54frac{6}{11}$
      $3:16:21frac{9}{11}$
      $4:21:49frac{1}{11}$
      $5:27:16frac{4}{11}$
      $6:32:43frac{7}{11}$
      $7:38:10frac{10}{11}$
      $8:43:38frac{2}{11}$
      $9:49:05frac{5}{11}$
      $10:54:31frac{8}{11}$



      and it turns out that the seconds only match the minutes at $12:00:00$. [Edit: Travis's answer shows why this is.]



      So the answer is: all three hands coincide twice a day.






      share|cite|improve this answer























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        3 Answers
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        This happens exactly once every $12$ hours. If $$tmapsto e^{it}qquad(0leq t< 2pi)$$ describes the position of the hour hand on the dial $S^1$ during the $12$ hour interval beginning at 12.00 noon then $tmapsto e^{12it}$ describes the position of the minute hand, and $tmapsto e^{720it}$ describes the position of the second hand. The hour hand and the minute hand coincide when $e^{it}=e^{12it}$, i.e., when $e^{11it}=1$. This is the case when $t={2pi kover11}$ $(0leq k<11)$. Similarly, the hour hand and the second hand coincide when $t={2pi lover 719}$ $(0leq l<719)$. As ${rm gcd}(11,719)=1$ there is no $tin[0,2pi[>$ fulfilling both conditions other than $t=0$.



        It follows that all three hands will line up exactly two times every $24$ hours.






        share|cite|improve this answer



























          up vote
          1
          down vote



          accepted










          This happens exactly once every $12$ hours. If $$tmapsto e^{it}qquad(0leq t< 2pi)$$ describes the position of the hour hand on the dial $S^1$ during the $12$ hour interval beginning at 12.00 noon then $tmapsto e^{12it}$ describes the position of the minute hand, and $tmapsto e^{720it}$ describes the position of the second hand. The hour hand and the minute hand coincide when $e^{it}=e^{12it}$, i.e., when $e^{11it}=1$. This is the case when $t={2pi kover11}$ $(0leq k<11)$. Similarly, the hour hand and the second hand coincide when $t={2pi lover 719}$ $(0leq l<719)$. As ${rm gcd}(11,719)=1$ there is no $tin[0,2pi[>$ fulfilling both conditions other than $t=0$.



          It follows that all three hands will line up exactly two times every $24$ hours.






          share|cite|improve this answer

























            up vote
            1
            down vote



            accepted







            up vote
            1
            down vote



            accepted






            This happens exactly once every $12$ hours. If $$tmapsto e^{it}qquad(0leq t< 2pi)$$ describes the position of the hour hand on the dial $S^1$ during the $12$ hour interval beginning at 12.00 noon then $tmapsto e^{12it}$ describes the position of the minute hand, and $tmapsto e^{720it}$ describes the position of the second hand. The hour hand and the minute hand coincide when $e^{it}=e^{12it}$, i.e., when $e^{11it}=1$. This is the case when $t={2pi kover11}$ $(0leq k<11)$. Similarly, the hour hand and the second hand coincide when $t={2pi lover 719}$ $(0leq l<719)$. As ${rm gcd}(11,719)=1$ there is no $tin[0,2pi[>$ fulfilling both conditions other than $t=0$.



            It follows that all three hands will line up exactly two times every $24$ hours.






            share|cite|improve this answer














            This happens exactly once every $12$ hours. If $$tmapsto e^{it}qquad(0leq t< 2pi)$$ describes the position of the hour hand on the dial $S^1$ during the $12$ hour interval beginning at 12.00 noon then $tmapsto e^{12it}$ describes the position of the minute hand, and $tmapsto e^{720it}$ describes the position of the second hand. The hour hand and the minute hand coincide when $e^{it}=e^{12it}$, i.e., when $e^{11it}=1$. This is the case when $t={2pi kover11}$ $(0leq k<11)$. Similarly, the hour hand and the second hand coincide when $t={2pi lover 719}$ $(0leq l<719)$. As ${rm gcd}(11,719)=1$ there is no $tin[0,2pi[>$ fulfilling both conditions other than $t=0$.



            It follows that all three hands will line up exactly two times every $24$ hours.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Nov 27 at 9:50

























            answered Nov 26 at 16:58









            Christian Blatter

            171k7111325




            171k7111325






















                up vote
                1
                down vote













                The answer depends on which hands' motions are continuous and which are discrete, and if so, how they are discretized. For simplicity, I'll address the case that all motions are continuous, so that all three hands have constant velocity.



                Per 12-hour period ($1$ rotation of the hour hand), the minute hand rotates $12$ times, so over this time the angular distance between the hour and minute hands is zero $12 - 1 = 11$ times. (This is where the figure of $22$ comes from: This is the number of times two hands align per 24 hours.) These 11 times are equally spaced, and include $0:00$, so they occur precisely when the hour hand has completed $frac{k}{11}$ of a rotation, $k = 0, ldots, 10$.



                The second hand moves $12 cdot 60$ times as fast as the large hand, so in the time between successive alignments of the hour and minute hands, the second hand has completed $frac{720}{11} = n + frac{5}{11}$ rotations for some integer $n$, so the accumulated angular distance between the second hand and the common angle of the other two hands has increased by $frac{5}{11} - frac{1}{11} = frac{4}{11}$ of a rotation. It follows from the fact that $4$ and $11$ are coprime that the only times at which all three hands are aligned is when they all point upward. Each day, this happens only at $0:00$ and $12:00$.






                share|cite|improve this answer



























                  up vote
                  1
                  down vote













                  The answer depends on which hands' motions are continuous and which are discrete, and if so, how they are discretized. For simplicity, I'll address the case that all motions are continuous, so that all three hands have constant velocity.



                  Per 12-hour period ($1$ rotation of the hour hand), the minute hand rotates $12$ times, so over this time the angular distance between the hour and minute hands is zero $12 - 1 = 11$ times. (This is where the figure of $22$ comes from: This is the number of times two hands align per 24 hours.) These 11 times are equally spaced, and include $0:00$, so they occur precisely when the hour hand has completed $frac{k}{11}$ of a rotation, $k = 0, ldots, 10$.



                  The second hand moves $12 cdot 60$ times as fast as the large hand, so in the time between successive alignments of the hour and minute hands, the second hand has completed $frac{720}{11} = n + frac{5}{11}$ rotations for some integer $n$, so the accumulated angular distance between the second hand and the common angle of the other two hands has increased by $frac{5}{11} - frac{1}{11} = frac{4}{11}$ of a rotation. It follows from the fact that $4$ and $11$ are coprime that the only times at which all three hands are aligned is when they all point upward. Each day, this happens only at $0:00$ and $12:00$.






                  share|cite|improve this answer

























                    up vote
                    1
                    down vote










                    up vote
                    1
                    down vote









                    The answer depends on which hands' motions are continuous and which are discrete, and if so, how they are discretized. For simplicity, I'll address the case that all motions are continuous, so that all three hands have constant velocity.



                    Per 12-hour period ($1$ rotation of the hour hand), the minute hand rotates $12$ times, so over this time the angular distance between the hour and minute hands is zero $12 - 1 = 11$ times. (This is where the figure of $22$ comes from: This is the number of times two hands align per 24 hours.) These 11 times are equally spaced, and include $0:00$, so they occur precisely when the hour hand has completed $frac{k}{11}$ of a rotation, $k = 0, ldots, 10$.



                    The second hand moves $12 cdot 60$ times as fast as the large hand, so in the time between successive alignments of the hour and minute hands, the second hand has completed $frac{720}{11} = n + frac{5}{11}$ rotations for some integer $n$, so the accumulated angular distance between the second hand and the common angle of the other two hands has increased by $frac{5}{11} - frac{1}{11} = frac{4}{11}$ of a rotation. It follows from the fact that $4$ and $11$ are coprime that the only times at which all three hands are aligned is when they all point upward. Each day, this happens only at $0:00$ and $12:00$.






                    share|cite|improve this answer














                    The answer depends on which hands' motions are continuous and which are discrete, and if so, how they are discretized. For simplicity, I'll address the case that all motions are continuous, so that all three hands have constant velocity.



                    Per 12-hour period ($1$ rotation of the hour hand), the minute hand rotates $12$ times, so over this time the angular distance between the hour and minute hands is zero $12 - 1 = 11$ times. (This is where the figure of $22$ comes from: This is the number of times two hands align per 24 hours.) These 11 times are equally spaced, and include $0:00$, so they occur precisely when the hour hand has completed $frac{k}{11}$ of a rotation, $k = 0, ldots, 10$.



                    The second hand moves $12 cdot 60$ times as fast as the large hand, so in the time between successive alignments of the hour and minute hands, the second hand has completed $frac{720}{11} = n + frac{5}{11}$ rotations for some integer $n$, so the accumulated angular distance between the second hand and the common angle of the other two hands has increased by $frac{5}{11} - frac{1}{11} = frac{4}{11}$ of a rotation. It follows from the fact that $4$ and $11$ are coprime that the only times at which all three hands are aligned is when they all point upward. Each day, this happens only at $0:00$ and $12:00$.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited Nov 26 at 17:07

























                    answered Nov 26 at 17:02









                    Travis

                    59.1k766144




                    59.1k766144






















                        up vote
                        1
                        down vote













                        We can answer this by determining the times when the hour and minute hands coincide, then checking whether the second hand will also match.



                        In $12$ hours, the hour hand rotates once while the minute hand rotates $12$ times. Therefore the minute hand does $12-1 = 11$ rotations relative to the hour hand in this time. So the interval between the times when it overtakes the hour hand is $frac{12}{11}$ hours. ($=1$h $5$m $27 frac{3}{11}$s).



                        All three hands are aligned at $12:00:00$. Adding integer multiples of $1:5:27frac{3}{11}$ to this gives the times when the hour and minute hands are aligned (in $12$ hour format) as



                        $12:00:00$
                        $1:05:27frac{3}{11}$
                        $2:10:54frac{6}{11}$
                        $3:16:21frac{9}{11}$
                        $4:21:49frac{1}{11}$
                        $5:27:16frac{4}{11}$
                        $6:32:43frac{7}{11}$
                        $7:38:10frac{10}{11}$
                        $8:43:38frac{2}{11}$
                        $9:49:05frac{5}{11}$
                        $10:54:31frac{8}{11}$



                        and it turns out that the seconds only match the minutes at $12:00:00$. [Edit: Travis's answer shows why this is.]



                        So the answer is: all three hands coincide twice a day.






                        share|cite|improve this answer



























                          up vote
                          1
                          down vote













                          We can answer this by determining the times when the hour and minute hands coincide, then checking whether the second hand will also match.



                          In $12$ hours, the hour hand rotates once while the minute hand rotates $12$ times. Therefore the minute hand does $12-1 = 11$ rotations relative to the hour hand in this time. So the interval between the times when it overtakes the hour hand is $frac{12}{11}$ hours. ($=1$h $5$m $27 frac{3}{11}$s).



                          All three hands are aligned at $12:00:00$. Adding integer multiples of $1:5:27frac{3}{11}$ to this gives the times when the hour and minute hands are aligned (in $12$ hour format) as



                          $12:00:00$
                          $1:05:27frac{3}{11}$
                          $2:10:54frac{6}{11}$
                          $3:16:21frac{9}{11}$
                          $4:21:49frac{1}{11}$
                          $5:27:16frac{4}{11}$
                          $6:32:43frac{7}{11}$
                          $7:38:10frac{10}{11}$
                          $8:43:38frac{2}{11}$
                          $9:49:05frac{5}{11}$
                          $10:54:31frac{8}{11}$



                          and it turns out that the seconds only match the minutes at $12:00:00$. [Edit: Travis's answer shows why this is.]



                          So the answer is: all three hands coincide twice a day.






                          share|cite|improve this answer

























                            up vote
                            1
                            down vote










                            up vote
                            1
                            down vote









                            We can answer this by determining the times when the hour and minute hands coincide, then checking whether the second hand will also match.



                            In $12$ hours, the hour hand rotates once while the minute hand rotates $12$ times. Therefore the minute hand does $12-1 = 11$ rotations relative to the hour hand in this time. So the interval between the times when it overtakes the hour hand is $frac{12}{11}$ hours. ($=1$h $5$m $27 frac{3}{11}$s).



                            All three hands are aligned at $12:00:00$. Adding integer multiples of $1:5:27frac{3}{11}$ to this gives the times when the hour and minute hands are aligned (in $12$ hour format) as



                            $12:00:00$
                            $1:05:27frac{3}{11}$
                            $2:10:54frac{6}{11}$
                            $3:16:21frac{9}{11}$
                            $4:21:49frac{1}{11}$
                            $5:27:16frac{4}{11}$
                            $6:32:43frac{7}{11}$
                            $7:38:10frac{10}{11}$
                            $8:43:38frac{2}{11}$
                            $9:49:05frac{5}{11}$
                            $10:54:31frac{8}{11}$



                            and it turns out that the seconds only match the minutes at $12:00:00$. [Edit: Travis's answer shows why this is.]



                            So the answer is: all three hands coincide twice a day.






                            share|cite|improve this answer














                            We can answer this by determining the times when the hour and minute hands coincide, then checking whether the second hand will also match.



                            In $12$ hours, the hour hand rotates once while the minute hand rotates $12$ times. Therefore the minute hand does $12-1 = 11$ rotations relative to the hour hand in this time. So the interval between the times when it overtakes the hour hand is $frac{12}{11}$ hours. ($=1$h $5$m $27 frac{3}{11}$s).



                            All three hands are aligned at $12:00:00$. Adding integer multiples of $1:5:27frac{3}{11}$ to this gives the times when the hour and minute hands are aligned (in $12$ hour format) as



                            $12:00:00$
                            $1:05:27frac{3}{11}$
                            $2:10:54frac{6}{11}$
                            $3:16:21frac{9}{11}$
                            $4:21:49frac{1}{11}$
                            $5:27:16frac{4}{11}$
                            $6:32:43frac{7}{11}$
                            $7:38:10frac{10}{11}$
                            $8:43:38frac{2}{11}$
                            $9:49:05frac{5}{11}$
                            $10:54:31frac{8}{11}$



                            and it turns out that the seconds only match the minutes at $12:00:00$. [Edit: Travis's answer shows why this is.]



                            So the answer is: all three hands coincide twice a day.







                            share|cite|improve this answer














                            share|cite|improve this answer



                            share|cite|improve this answer








                            edited Dec 7 at 22:38

























                            answered Nov 26 at 17:05









                            timtfj

                            666214




                            666214






























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