## A Very Special Ten-Digit Number

This puzzle was given to me by John H. Conway, and he heard it from someone else:

Find a ten-digit number with all distinct digits such that the string formed by the first k digits is divisible by k for any k ≤ 10.

Surprisingly, there is a unique solution to this puzzle. Can you find this very special ten-digit number?

For the contrast, consider ten-digit numbers with all distinct digits such that the string formed by the last k digits is divisible by k for any k ≤ 10. These numbers are not so special: there are 202 of them. My puzzle is: find the smallest not-so-special number.

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1. #### An Answer To “A Very Special Ten-Digit Number” | Finer Recliner:

[…] Khovanova poses an interesting math question on her blog: This puzzle was given to me by John H. Conway, and he heard it from someone […]

2. #### Ken Roberts:

Fun puzzle. Bit of reasoning deals with all but the div-by-7 case, and then used brute force trying all 8 possibilities.
Has anyone found a way to reason thru to the solution without need for division in the div-by-7 test?

3. #### pianowow:

Took me about an hour of eliminating results based on divisibility rules here:
https://mathforum.org/dr.math/faq/faq.divisibility.html
I got 3816547290 for Conway’s puzzle.
For your puzzle, I had to write a computer program, and got 9123567480.
Thanks for the intriguing puzzle!

4. #### Walking Randomly » The 46th Carnival of Mathematics - the last one of 2008.:

[…] No Carnival is complete without a puzzle to solve and Tanya Khovanova gave us a great one back in […]

5. #### Tanya Khovanova’s Math Blog » Blog Archive » It Has Been Two Years:

[…] A Very Special Ten-Digit Number […]

6. #### Håkan Olsson:

Another question could be whether the number of solutions depend on which numerical system we choose.