## 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

kdigits is divisible bykfor anyk ≤ 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.

## 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 […]

4 September 2008, 7:50 am## 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.

4 September 2008, 10:25 amHas anyone found a way to reason thru to the solution without need for division in the div-by-7 test?

## pianowow:

Took me about an hour of eliminating results based on divisibility rules here:

5 September 2008, 7:26 amhttps://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!

## 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 […]

29 December 2008, 2:42 pm## Tanya Khovanova’s Math Blog » Blog Archive » It Has Been Two Years:

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

11 December 2009, 10:20 am## Håkan Olsson:

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

12 March 2024, 1:51 pm