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OT Analog clock math problems

Posted By: Daniel Murphy
Date: Tuesday, 4 May 2010, at 6:47 p.m.

In Response To: OT Analog clock math problems (Jason Lee)


So often there is a brute force solution that reveals shortcuts that reveal an elegant solution. Not getting from brute to elegant is uncool. Getting from brute to elegant is cool. Starting with elegant is coolest of all. But brute force has its charms. 

 

   

T(hrs)  T(Min)                                          T(Seconds)                   Second and Minute hands 

                                                                                     form 120° angle? 

 

12	00.0000 (Clock starts)   Interior angle formed 

                                 by hour/minute hand 

------------------------------ 

12      21.8181                  120°               49.0909                             No               

12      43.6363                  120°               38.1818                             No 

 1       5.4545                    0°               27.2727                             No 

 1      27.2727                  120°               16.3636                             No 

 1      49.0909                  120°                5.4545                             No 

 2      10.9090                    0°               54.5454                             No 

 2      32.7272                  120°               43.6363                             No 

 2      54.5454                  120°               32.7272                             No 

 3      16.3636                    0°               21.8181                             No 

 3      38.1818                  120°               10.9090                             No 

 4      00.0000                  120°               00.0000                             No 

--------------- 

 4      21.8181                    0°               49.0909                             No 

 4      43.6363                  120°               38.1818                             No 

 5       5.4545                  120°               27.2727                             No 

 5      27.2727                    0°               16.3636                             No 

 5      49.0909                  120°                5.4545                             No 

 6      10.9090                  120°               54.5454                             No 

 6      32.7272                    0°               43.6363                             No 

 6      54.5454                  120°               32.7272                             No 

 7      16.3636                  120°               21.8181                             No 

 7      38.1818                    0°               10.9090                             No 

 8      00.0000                  120°               00.0000                             No 

--------------- 

 8      21.8181                  120°               49.0909                             No 

 8      43.6363                    0°               38.1818                             No 

 9       5.4545                  120°               27.2727                             No 

 9      27.2727                  120°               16.3636                             No 

 9      49.0909                    0°                5.4545                             No 

10      10.9090                  120°               54.5454                             No 

10      32.7272                  120°               43.6363                             No 

10      54.5454                    0°               32.7272                             No 

11      16.3636                  120°               21.8181                             No 

11      38.1818                  120°               10.9090                             No 

12      00.0000                    0°               00.0000                             No 

--------------- 

(decimal position                Hour and minute hand form 

 of hour hand                    120° angle 22 times in 12 hours 

 omitted)                        44 times in 24 hours


There are a few times per day when the three hands come quite close to dividing the clock face into three equal areas, but never exactly.


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