Finding the nine...
This challenging puzzle comes from our good friend James Grime — thanks James!
Find a nine digit numbers, using the numbers 1 to 9, and using each number once without repeats, such that; the first digit is a number divisible by 1. The first two digits form a number divisible by 2; the first three digits form a number divisible by 3 and so on until we get a nine digit number divisible by 9.
You might try, for example, the number 923,156,784. But this number doesn't work — the first three digit number, 923, is not divisible by 3. Can you find a nine digit number that works?
Hint: you don't need a computer to find it. Try looking at your clock instead....
James Grime is a lecturer and public speaker on mathematics, and can be mostly found touring the country on behalf of the Millennium Mathematics Project carting his trusty Enigma Machine. If you'd like James and the machine to visit your school, visit the Enigma website.
You can also read more from James in his article, Curious Dice.
Solution link:
Finding the nine - solution 
Comments
Generalizing to other bases
If we generalize this problem to other bases, these are what you get.
For Base 2 (binary), there are no solutions.
For Base 4 (quaternary), there are two solutions: 123 and 321.
For Base 6 (senary), there are two solutions: 14325 and 54321.
For Base 8 (octal), there are six solutions: 1274563, 3254167, 5234761, 5614723, 5674321, and 7234561.
I have yet to do the odd bases or the bases higher than 10 (decimal).
Solution
I found this in 15 seconds, mentally. 14 here.
243,157,896.
Wrong solution
2431 is not divisible by 4.
answer
here i have one possible solution.
123654321
Not correct
Sorry this is not correct because all the number 1..9 should occur once
986754321
986754321
Not correct
Sorry not correct because 986 is not divisible by 3
123456789 is not a correct
123456789 is not a correct answer I'm afraid :(
anonymous
123456789 is also a correct answer .
Sorry 1234 is not divisible
Sorry 1234 is not divisible by 4
ANSWER (not going to tell you how :P)
381654729
my solution
E=5 because multiples of 5 end in 5 or 0
Alternate digits must be even, so the rest have to be odd.
Digits C and D have to go "odd,even", and make a number which is a multiple of 4. So D has to be 2 or 6.
Ditto position H- must be 2 or 6.
The only even numbers left are 4 and 8, and these must go in B and F.
Looking at the first three digits, whose digital root must be 3,6 or 9, there are 9 options for filling these given the conditions we've already worked out.
We tried each of these in turn and worked out the digital root up to F. This also has to be 3,6 or 9 to make it divisible by 6. So you can work out in each case whether D is 2 or 6.
From this we can see what H is as one of its options has been used.
Only two digits remain- we test whether either makes a multiple of 7 when put in position G.
Finding that one of these works, check that the first 8 digits are divisible by 8.
After this only one answer remains:
381654729
(Also, has anyone noticed the patterns this and other suggestions make on the calculator buttons? They are symmetrical or generally interesting.)
If you followed that, I'm impressed. :)
not unique
I found at least two numbers 921,252,564 and 987,654,564
did you listen?
I don't think you heard the question correctly. go back and listen taking note about how many times each digit can appear
The nine digit number
Let the number be abcd5fghi. It's clear that the fifth digit has to be 5. b, d, f and h are elements of {2,4,6,8} and the remaining a, c, g and i are elements of {1,3,7,9}. So there are at most 24 * 24 = 576 possibilities. But we can limit these possibilities drastic. 2c + d has to be divisible by 4, 4d + 20 + 4f by 6 and 2g + h by 4. Now you will find only one possibility: 381654729
cool
Best answer by far.
1-9
987654321
sol
number=987654321
division by 7
This doesnt work on division by 7 : 9876543/7 = 1410934.714... hence not evenly divisible, the only solution is 381654729
Finding the nine digit number
381654729 is not unique.
963258147 also works.
peterh@nipltd.com
Finding the awnser to the nine digit number
381654729 WORKS!!!
Finding the nine digit number
Sorry 963258147 fails on division by 8
Can somebody explain the reference to looking at the clock
Stuart Barker dandsb@ntlworld.com
Finding the nine digit number
I think "looking at the clock" is meant to suggest using modular arithmetic. I have heard modular arithmetic explained that way before (i.e. for an analog clock, the hour is incremented modulo 12).
btw, I think the answer is unique.
solution
number is 987654321
No it's not. It fails at 7th
No it's not. It fails at 7th digit, 9876543 is not divisible by 7.
Smart Alec
Sure it is.
9876543/7 = 1410934.714285714....
Did you mean "evenly divisible"?
In that case, every number is
In that case, every number is divisible by every number.. hence, all 9 digit numbers would be solutions.. :- ) But, you seem to missed the point.
Anil Sharma
pedantic
you have missed the 'spirit' of the question - as well you know!