Clock It

Solution

Let \(t\) be the time on the first clock, measured in hours modulo \(12\).

The second clock therefore shows \(t+x\), where \(0 < x < 12\). For a clock showing time \(t\), \[ \text{minute hand angle} = 360t, \qquad \text{hour hand angle} = 30t. \] The directed angle between the hands is \[ 360t - 30t = 330t = 30(11t) \] degrees, taken modulo \(360\). An acute angle means the smaller angle between the hands is less than \(90^\circ\). Thus we require \[ 30(11t) \pmod{360} \in (0,90)\cup(270,360). \] Dividing by \(30\) gives \[ 11t \pmod{12} \in (0,3)\cup(9,12). \] Define \[ u = 11t \pmod{12}. \] Since multiplication by \(11\) permutes the residues modulo \(12\), a uniformly random \(t\) implies \(u\) is uniformly distributed on \([0,12)\). Thus one clock has an acute angle exactly when \[ u \in S := (0,3)\cup(9,12). \] The total length of \(S\) is \[ 3+3=6, \] so for a single clock \[ P(\text{acute})=\frac{6}{12}=\frac{1}{2}. \] Now consider the second clock. Its time is \(t+x\), so \[ 11(t+x)\equiv 11t+11x \pmod{12}. \] Because \[ 11 \equiv -1 \pmod{12}, \] this becomes \[ 11(t+x)\equiv u-x \pmod{12}. \] Thus the second clock has an acute angle when \[ u-x \in S \pmod{12}. \] Therefore both clocks are acute exactly when \[ u\in S \quad \text{and} \quad u-x \in S. \] This means \(u\) must lie in the overlap of \(S\) and \(S+x\) on the circle of circumference \(12\). The set \(S\) is a semicircle of length \(6\). If two semicircles are shifted by circular distance \[ d=\min(x,12-x), \] their overlap has length \[ 6-d. \] Since \(u\) is uniform on a circle of length \(12\), the probability equals \[ \frac{6-d}{12}. \] Substituting \(d\) gives \[ P(x)=\frac{6-\min(x,12-x)}{12}. \] Equivalently, \[ P(x)= \begin{cases} \dfrac{6-x}{12}, & 0 < x\le 6, \\ \dfrac{x-6}{12}, & 6\le x<12. \end{cases} \] As checks, \[ x\to 0 \Rightarrow P(x)\to \frac12, \] since the clocks become identical, and \[ x=6 \Rightarrow P(6)=0, \] because a \(6\)-hour shift reverses the relative angle, forcing one clock to be obtuse whenever the other is acute.