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At noon, all three hands are at $0$ degrees from each other.

Let's say the Moment of Truth is after the hour hand has moved around $x$ degrees. In that time, the minute hand moves $12x$ degrees and the second hand moves $720x$ degrees. So we want $x,12x(\mathrm{mod }360),720x(\mathrm{mod }360)$ to be separated by $120$ from each other.

From the minute and second hands we know $720x-12x\pm120$ is a multiple of $360$, which means $59x\pm10$ is a multiple of $30$. From the minute and hour hands we know $11x\pm120$ is a multiple of $360$. So $11x$ and $59x$ are both integers, which means $x$ is an integer (since 11 and 59 are coprime). But then $720x$ is a multiple of $360$, which means $x$ must be 120 or 240 (mod 360) and so $12x$ is another multiple of 360, contradiction. So

there is no solution!

At noon, all three hands are at $0$ degrees from each other.

Let's say the Moment of Truth is after the hour hand has moved around $x$ degrees. In that time, the minute hand moves $12x$ degrees and the second hand moves $720x$ degrees. So we want $x, 12x\pmod{360},$ $720x \pmod {360}$ to be separated by $120$ from each other.

From the minute and second hands we know $720x-12x\pm120$ is a multiple of $360$, which means $59x\pm10$ is a multiple of $30$. From the minute and hour hands we know $11x\pm120$ is a multiple of $360$. So $11x$ and $59x$ are both integers, which means $x$ is an integer (since 11 and 59 are coprime). But then $720x$ is a multiple of $360$, which means $x$ must be $120$ or $240 \pmod {360}$ and so $12x$ is another multiple of $360$ contradiction. So

there is no solution!

At noon, all three hands are at $0$ degrees from each other.

Let's say the Moment of Truth is after the hour hand has moved around $x$ degrees. In that time, the minute hand moves $12x$ degrees and the second hand moves $720x$ degrees. So we want $x,12x(\mathrm{mod }360),720x(\mathrm{mod }360)$ to be separated by $120$ from each other.

From the minute and second hands we know $720x-12x\pm120$ is a multiple of $360$, which means $59x\pm10$ is a multiple of $30$. From the minute and hour hands we know $11x\pm120$ is a multiple of $360$. So $11x$ and $59x$ are both integers, which means $x$ is an integer (since 11 and 59 are coprime). But then $720x$ is a multiple of $360$, which means $x$ must be 120 or 240 (mod 360) and so $12x$ is 4 or 8 (mod 360), contradiction. So

there is no solution!

At noon, all three hands are at $0$ degrees from each other.

Let's say the Moment of Truth is after the hour hand has moved around $x$ degrees. In that time, the minute hand moves $12x$ degrees and the second hand moves $720x$ degrees. So we want $x,12x(\mathrm{mod }360),720x(\mathrm{mod }360)$ to be separated by $120$ from each other.

From the minute and second hands we know $720x-12x\pm120$ is a multiple of $360$, which means $59x\pm10$ is a multiple of $30$. From the minute and hour hands we know $11x\pm120$ is a multiple of $360$. So $11x$ and $59x$ are both integers, which means $x$ is an integer (since 11 and 59 are coprime). But then $720x$ is a multiple of $360$, which means $x$ must be 120 or 240 (mod 360) and so $12x$ is another multiple of 360, contradiction. So

there is no solution!

At noon, all three hands are at $0$ degrees from each other.

Let's say the Moment of Truth is after the hour hand has moved around $x$ degrees. In that time, the minute hand moves $60x$ degrees and the second hand moves $3600x$ degrees. So we want $x,60x(\mathrm{mod }360),3600x(\mathrm{mod }360)$ to be separated by $120$ from each other.

From the minute and second hands we know $3600x-60x\pm120$ is a multiple of $360$, which means $59x\pm2$ is a multiple of $6$. From the minute and hour hands we know $59x$ is a multiple of $360$. So we have a multiple of $6$ and a multiple of $360$ differing by just $2$, which means

there is no solution!

At noon, all three hands are at $0$ degrees from each other.

Let's say the Moment of Truth is after the hour hand has moved around $x$ degrees. In that time, the minute hand moves $12x$ degrees and the second hand moves $720x$ degrees. So we want $x,12x(\mathrm{mod }360),720x(\mathrm{mod }360)$ to be separated by $120$ from each other.

From the minute and second hands we know $720x-12x\pm120$ is a multiple of $360$, which means $59x\pm10$ is a multiple of $30$. From the minute and hour hands we know $11x\pm120$ is a multiple of $360$. So $11x$ and $59x$ are both integers, which means $x$ is an integer (since 11 and 59 are coprime). But then $720x$ is a multiple of $360$, which means $x$ must be 120 or 240 (mod 360) and so $12x$ is 4 or 8 (mod 360), contradiction. So

there is no solution!