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The date today is 7th June 2019, or 07/06/2019 (using the English DD/MM/YYYY ordering).

When is the next date that when written in this way has all eight digits different?

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Rather remarkably, I wrote down this exact puzzle in my notebook a couple of years ago to which I think the answer is

17/06/2345 in DD/MM/YYYY format.

Reasoning

Notice that the first M will either be $0$ or $1$.
If it is $0$ then the first D will either be $1$ or $2$ or DD will be $31$.
If it is $1$ then either the second M will be $0$ or the second M will be $2$ and the day will contain a $0$.

Overall, this means that $0$ and either $1$ or $2$ must be used in the DD/MM part. If we don't want to skip to the next millenium, we need the $2$ for the beginning of the year.
Hence the DD/MM part requires both $0$ and $1$.
After that, we focus on the nearest year possible which comes from assigning the digits $3,4,5$ in order to century, decade and digit of the year.
It makes more sense to assign the $0$ to the month instead of $1$ but we cannot assign both since we cannot have a day without any of $0$, $1$ or $2$. Hence, we assign $6$ to the month and then $7$ to create the day.

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    $\begingroup$ I can confirm that this is also the result of writing a dumb Python program to try future dates until it finds one satisfying the given condition. $\endgroup$ – Gareth McCaughan Jun 7 at 15:04
  • $\begingroup$ Pleasingly (but also somewhat obviously) the two days following also satisfy the condition. $\endgroup$ – IanF1 Jun 16 at 21:06

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