I was given the following riddle:
You are given a clock which has identical dials for the hours and minutes. How many times a day you can't tell the time? (i.e. the dials are indistinguishable)?
I'm not asking for a solution, but rather for help with my analysis.
My first observation was that since the hours face completely determines the minutes dial, if $\alpha$ is the angle from 12 of the hours dial, $f(\alpha)$ is the angle of the minutes dial.
If $\alpha$ is measured in degrees, we can write $f$ as follows (each hour takes 30 degrees):
$$f(\alpha)=(\alpha \mod 30) \cdot 360$$
Two conditions has to hold so they may be confused:
- $\alpha\neq f(\alpha)$. This has to hold since otherwise we know the hours dial and we're done.
- $\alpha=f(f(\alpha))$. This is the condition that means we can't distinguish between the dials.
Writing (2) explicitly, assuming $\alpha$ is measured in degrees gives us: $$\{*\} \alpha = [((\alpha \mod 30) \cdot 360)\mod 30]\cdot 360$$
Now we can use the following simple equality: $$(x \cdot 360)\mod 30 =(x\mod \frac{1}{12})\cdot 360$$
Using it in $\{*\}$ we get:
$$ \alpha = ((\alpha \mod 30)\mod \frac{1}{12}) \cdot 360^2 = (\alpha \mod \frac{1}{12}) \cdot 360^2$$
But since $g(\alpha) = (\alpha \mod \frac{1}{12}) \cdot 360^2$ intersects $h(\alpha)=\alpha$ about $360\cdot 12$ times, it is tempting to say the answer for the question is $2 \cdot [ (360 \cdot 12) - 1]$ (where the (-1) is the 12 o'clock case). This is clearly wrong as choosing to measure $\alpha$ in degrees is arbitrary, and measuring it differently would yield different result.
But where is the mistake in the process?
I think $f(\alpha)=(\alpha \bmod 30)\cdot 12$ because, when $\alpha$ increases by $30$, the angle of the minute hand increases by $30\cdot 12$