The Famous Watch Story
Here we have the proof developed by “KZ.” This was the first analytical proof developed. It’s interesting to note both the similarities and differences between KZ’s proof and the MO’ Proof.
Symmetry Of Time
There is a time when all of a clock’s hands (hour, minute, and second) are in such a position that angles between neighboring hands are equal to a perfect 120 degrees.
T Time it takes hour hand of the clock to make a full circle
t independent variable denoting any time between 0 and T
H Number of hours past between 0 and t
M Number of minutes past since the beginning of hour
Let’s define functions expressing the location of the clock’s hands as time progresses from 0 to T:
hA(t) := 2pi
~~~ * t
mA(t) := 2pi
~~~ * 12 * t
sA(t) := 2pi
~~~ * 720 * t
Notice that functions mA(t) and sA(t) are continuously growing and assuming values greater than 360 degrees. Now let’s define functions expressing the angles between the clock’s hands:
mhA(t) := mA(t) – hA(t)
mhA(t) := 2pi
~~~ * 11 * t Angle between minute and hr hands
shA(t) := sA(t) – hA(t)
shA(t) := 2pi
~~~ * 719 * t Angle between second and hr hands
The angle between the hour and minute hands is equal to 120 degrees when the value of function mhA(t) is equal to 120 degrees plus H times 360 degrees, or 240 degrees plus H times 360 degrees.
mhA(t) := 2pi mhA(t) := 4pi
~~~ + H*2*pi or ~~~ + H*2*pi
It means that t must be equal:
T 1 T 2
tm120 := ~~~ * H + ~~~ Or Tm240 := ~~~ * H + ~~~
11 3 11 3
First conclusion: There are 24 times in a twelve hour period when the angle between the hour and minute hands is equal to 120 degrees.
For every of these points in time, the angle between the second hand and hour hand must be equal to either 120 degrees or 240 degrees bisecting the remaining 240 degree angle between the minute and hour hand.
shA(t) := ~~~ + (60H + M)*2pi or shA(t) := ~~~ + (60H + M)*2pi
T 2 T 1
ts240 := ~~~ * 60H + M + ~~~ Or ts120 := ~~~ * 60H + M + ~~~
719 3 719 3
tm120 := ts240 Or tm240 := ts120
the following must be true:
59H – 11M := -697/3 OR 59H – 11M := -1427/3
Because H and M are integers and 697/3 and 1427/3 are not, that proves that the original hypothesis was false. The ordinary clock is an asymmetrical time device (when you look at it)!