I have a very strange clock. At first glance, it looks like a normal clock with three hands and the numbers 1 through 12 all around. The only differences are that the hands are indistinguishable from each other and they are faster. One hand completes a circle in 3 minutes, another in 4 minutes, and the last in 6 minutes. They all go clockwise.

One morning, when I looked at the clock, the hands were all pointing exactly at the numbers 1, 2, and 3.
Later that day, I saw that the three hands were pointing exactly at 6, 10, and 11.

Can you identify which hands I saw each time? Prove it.

The 3-minute, 4-minute and 6-minute hands advance the following number of number positions each minute:

0  0  0


4  3  2


8  6  4


0  9  6


4  0  8


8  3 10


0  6  0


4  9  2


8  0  4


0  3  6


4  6  8


8  9 10


0  0  0

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Then, showing all the possible initial positions of the three hands (6 permutations), we get the following. The amount of advance table above is repeated on the left, and then the six permutations of the hands appear in six major columns to the right. Within each column the subcolumns are 3-minute, 4-minute and 6-minute hands:

0  0  0:  1  2  3;  1  3  2;  2  1  3;  2  3  1;  3  1  2;  3  2  1


4  3  2:  5  5  5;  5  6  4;  6  4  5;  6  6  3;  7  4  4;  7  5  3


8  6  4:  9  8  7;  9  9  6; 10  7  7; 10  9  5; 11  7  6; 11  8  5


0  9  6:  1 11  9;  1 12  8;  2 10  9;  2 12  7;  3 10  8;  3 11  7


4  0  8:  5  2 11;  5  3 10;  6  1 11;  6  3  9;  7  1 10;  7  2  9


8  3 10:  9  5  1;  9  6 12; 10  4  1; 10  6 11; 11  4 12; 11  5 11


0  6  0:  1  8  3;  1  9  2;  2  7  3;  2  9  1;  3  7  2;  3  8  1


4  9  2:  5 11  5;  5 12  4;  6 10  5;  6 12  3;  7 10  4;  7 11  3


8  0  4:  9  2  7;  9  3  6; 10  1  7; 10  3  5; 11  1  6; 11  2  5


0  3  6:  1  5  9;  1  6  8;  2  4  9;  2  6  7;  3  4  8;  3  5  7


4  6  8:  5  8 11;  5  9 10;  6  7 11;  6  9  9;  7  7 10;  7  8  9


8  9 10:  9 11  1;  9 12 12; 10 10  1; 10 12 11; 11 10 12; 11 11 11


0  0  0:  1  2  3;  1  3  2;  2  1  3;  2  3  1;  3  1  2;  3  2  1

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The only column that also includes a 6, a 10 and an 11 in the same group is the column headed 2 3 1, meaning the 3-minute hand was on the 2, the 4-minute hand was on the 3 and the 6-minute hand was on the 1. After 5 minutes the 3-minute hand was on the 10, the 4-minute hand was on the 6 and the 6-minute hand was on the 11.
Edited on February 17, 2004, 11:48 am

Posted by Charlie on 2004-02-17 11:46:54