Happy Pi Day! Pizza Hut's Math Competition for free pizza for 3.14 years

Three problems, if you are the first to solve any of them, then you win the grand pizza prize.

Calling all math experts and Pizza Hut fans alike! National Pi Day is here and this is your chance to win free “pie,” that’s 3.14 years of Pizza Hut pizza (awarded in Pizza Hut® gift cards)! Take a look at the math problems below and provide your answer to Option A, B, or C in the comments section. Please be sure to note which you are trying to solve. Answers will be time stamped to determine the potential winner and participants can only win once.
Best of luck!
– Pizza Hut & John H. Conway
The three problems:

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?

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?”

OPTION C: YET TO BE SOLVED, No one has gotten this one exactly right yet! Hint: It helps to show your work!
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?

Since the first two are solved already in the comments. I chose to focus and solve 'Option C'.


The solution to Option C I arrived at is 7 + n, where n is any positive integer. Here's the picture I drew to convince myself:

The numbers represent how many rings each circle represents and 'n' represents any number of set of rings or chain of rings (like a chain of 1 rings). It got a bit messy when I tried to draw each of them individually. ;)

The two at the bottom don't need to be interlinking, especially when n=2. There are other combination of rings, but this seems to provide the simplest answer and largest possible answer space.

The "pairs" are indistinguishable. The triplet is distinguishable from any other triplet. This meets all the requirements of the puzzle.

Thanks for the opportunity to solve it. Pizza Hut and Mr. Conway

The Pizza Hut blog post: National Pi Day Math Contest Problems

~ Danial Goodwin ~

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