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...