Pizza Hut Meth Contest
 

Win pizza for 3.14 years:

1. I’m thinking of a ten-digit integer whose digits are all distinct. It happens that the number formed by the first n of them is divisible by n for each n from 1 to 10. What is my number?

2. Our school’s puzzle-club meets in one of the schoolrooms every Friday after school.

Last Friday, one of the members said, “I’ve hidden a list of numbers in this envelope that add up to the number of this room.” A girl said, “That’s obviously not enough information to determine the number of the room. If you told us the number of numbers in the envelope and their product, would that be enough to work them all out?”

He (after scribbling for some time): “No.” She (after scribbling for some more time): “well, at least I’ve worked out their product.”

What is the number of the school room we meet in?”

3. My key-rings are metal circles of diameter about two inches. They are all linked together in a strange jumble, so that try as I might, I can’t tell any pair from any other pair.

However, I can tell some triple from other triples, even though I’ve never been able to distinguish left from right. What are the possible numbers of key-rings in this jumble?

1 is trivial to brute force.

2 reminds me of the similar problem about the Russian mathematician's sons' age. Presumably it's a similar sort of answer.

I don't even understand 3.

[QUOTE=richs;429092]Win pizza for 3.14 years:

1. I’m thinking of a ten-digit integer whose digits are all distinct. It happens that the number formed by the first n of them is divisible by n for each n from 1 to 10. What is my number?
[/QUOTE]
Here is 1 answer, I am sure there are others:

9216543870

[QUOTE=a1call;429157]Here is 1 answer, I am sure there are others:

9216543870[/QUOTE]

92165438 is not divisible by 8

There is only one solution, and is of the format OEOE5EOE0, which means you have to brute force 4!*4! = 576 permutations.

[QUOTE=axn;429168]92165438 is not divisible by 8

There is only one solution, and is of the format OEOE5EOE0, which means you have to brute force 4!*4! = 576 permutations.[/QUOTE]

Well...
OEOE5EOEO0
But yes, 4!*4!

[QUOTE=0PolarBearsHere;429170]Well...
OEOE5EOEO0[/QUOTE]

:doh!:

[QUOTE=axn;429168]92165438 is not divisible by 8

There is only one solution, and is of the format OEOE5EOE0, which means you have to brute force 4!*4! = 576 permutations.[/QUOTE]

Excel roundung and late night calculations again,:smile:

How about

[B]3816547290[/B]

I verified with iPhone calculator.
Thanks for the reply and checking,

For the first problem a for a brute force you would need to check if the number is a multiples of 2520 (the least common multipler of 1 thru 9).

[QUOTE=WMHalsdorf;429187]For the first problem a for a brute force you would need to check if the number is a multiples of 2520 (the least common multipler of 1 thru 9).[/QUOTE]

3816547290 is not a multiple of 2520, yet it satisfies the constraints of the puzzle.

[QUOTE=a1call;429198]3816547290 is not a multiple of 2520, yet it satisfies the constraints of the puzzle.[/QUOTE]
3816547290 is not divisible by 7

1234759680 meets all constraints. Note there may be larger numbers.

[QUOTE=WMHalsdorf;429202]3816547290 is not divisible by 7

1234759680 meets all constraints. Note there may be larger numbers.[/QUOTE]


for the first thing only the first 7 digits need to be admittedly they aren't either but it's a key distinction. no your answer doesn't a number ending in 7 does not divide by 5. look at the new post a1call made it seems i was confusing your new number and their old one.