The 41 second clock failure syndrome

[jedishrfu]

So I used simple factoring to show that 42 days has 10! seconds

42 days in seconds is 42 . 24 . 36 . 100: $2^8 . 3^4 . 5^2 . 7$

10! is 10 . 9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1 which is $2^8 . 3^4 . 5^2 . 7$

How did you come to learn this arcane factoid?

 

[DaveC426913]

42 days in seconds is 42 . 24 . 36 . 100: $2^8 . 3^4 . 5^2 . 7$

OK, I get this, but...

10! is 10 . 9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1 which is $2^8 . 3^4 . 5^2 . 7$

How do you get this? That doesn't seem to be a shortcut or anything: 256.81.25.7

 

[DrGreg]

How did you come to learn this arcane factoid?

I can't remember when I first saw this, but it stuck in my mind as another candidate for the Ultimate Question, "What is the smallest integer number of days that is an exact factorial number of seconds?"

$$\begin{align*}
&\frac
{42 \times 24 \times 60 \times 60}
{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \\
&= \frac{60}{10} \times \frac{1}{9} \times \frac{24}{8} \times \frac{42}{7 \times 6} \times \frac{60}{5 \times 4} \times \frac{1}{3 \times 2} \\
&= 6 \times \left( \frac{1}{9} \times 3 \times 1 \times 3 \right) \times \frac{1}{6} \\
&= 1
\end{align*}$$

 

[jedishrfu]

OK, I get this, but...

How do you get this? That doesn't seem to be a shortcut or anything: 256.81.25.7

I just prime factored the 10! Factors and counted up the 2s 3s 5s and 7s

2.5.3.3.2.2.2.7.2.3.5.2.2.3.2.1

You can think there are 5 even factors in 10! Hence five 2s But 8 gives you two more and 4 one more for a total of 8 2s or $2^8$ …

 

[billm]

I've not got an answer to your 41 second problem but it may be not dissimilar to the Thursday effect for grandfather clocks. This causes longcase clocks to stop on Thursdays* i.e. when the pendulum is the same length as the (hanging) weights causing a transfer of energy to the (swinging) weights if there is the slightest tendency for the clock to sway.
The standard cure is to put a match stick under the appropriate corner of the clock for which the clockmaker (or repairer) will charge five guineas.
*This, of course, is based on the normal rules of the household i.e. that the clock is wound up on a Sunday evening.

 

[DaveC426913]

So there's your answer @anorlunda : tilt your clock:

 

[berkeman]

So there's your answer @anorlunda : tilt your clock:

That'll be five guineas please...

 

[Andrea_Dubai]

my experience (about 30 years) with many clocks is that the stuck occurred at 44 seconds when the effort from 44 to 45 requires more energy and tend to increase. The effort from 45 to 46 tends to use the same energy but it moves to decreasing power needs. If the clock is able to move from 44 it is not guaranteed to achieve 45, but if it is able to move from 45, for sure it will reach a 46 and it will be able to 1 more clockwise round. However, and influencing parameter could be the threshold move of the second clock hand. I had a very 'smooth like' hand which stopped at 47 seconds in the past.