## Eketahuna’s Party

Eketahuna is having a party. The party was to be held in his private mansion in the city of Puzzlaria. The city, conveniently, is laid out in a perfect grid. Every house’s address is composed of two positive numbers, indicating its relative position to the Town-hall in the southwest corner. For example, House 3-8 is two houses east of the Town Hall and seven houses north.

Currently Puzzlaria is one hundred by one hundred houses in area, and that includes the Town Hall – 1-1.

Two of his esteemed guests, Cam and Chris, were both given the address weeks in advance. And while both were quick-witted, they also were short memoried.

Only a few hours before the party the following dialog occurred:

Cam: I’m afraid I’ve forgotten the address. I can only remember the product of the two numbers, and that the first number wasn’t greater than the second.

Chris: I’ve forgotten it too, but can only remember the sum of the two numbers, and that neither number was 1.

Cam: I can’t figure out where the party is.

Chris: I knew that.

Cam: OK, I know where the party is.

Chris: OK, so do I.

Cam and Chris were being perfectly truthful, and no other information was exchanged apart from this dialog.

What was the party’s address?

October 12th, 2010 at 12:55 pm

This is a rehash of the sum product puzzle.

October 12th, 2010 at 11:47 pm

2-3

October 13th, 2010 at 12:52 am

heard this one before and still can’t get it!

brain chash

October 13th, 2010 at 4:22 am

Brain chash indeed – but at least you splet your name rite

October 13th, 2010 at 7:45 pm

Ohh man, I must have been partying to hard … I don’t even know where I live myself ! Lucky I live there and don’t need to find my way there, but if I did I’d just walk around the 8 places I’ve narrowed it down to until I found it.

I actually think my place is too small, and the party should be in the town hall so that everyone can make it, the teeming throngs can just pick up Chris and Cam as they pass.

October 13th, 2010 at 7:46 pm

oops, type – *too

October 13th, 2010 at 7:46 pm

oops, typo – *typo

Right, that’s it – where did I leave my drink ?

October 14th, 2010 at 4:40 am

Answer – 1-24 or 1-23 or 1-25

October 14th, 2010 at 12:07 pm

If one of them can only remember the product, which is the number followed by a multiplication problem, and the first number was < the second. The other says they only remember the sum, answer to an addition problem, and neither number was 1. with these two peices of info, i can gather that the party cant be on the west well or the south wall, for that would mean the house address being 3-1 or 1-3. as well as the house address cant be a repeating number such as 3-3 or 4-4 etc. with this in mind we can narrow it down to these addresses:

2-3, 2-4, 2-5, 2-6, 2-7, 2-8, 2-9, 2-10, 2-11

3-4, 3-5, 3-6, 3-7, 3-8, 3-9, 3-10,3-11

4-5, 4-6, 4-7, 4-8, 4-9, 4-10,4-11

5-6, 5-7, 5-8, 5-9, 5-10,5-11

6-7, 6-8, 6-9, 6-10,6-11

7-8, 7-9, 7-10,7-11

8-9, 8-10,8-11

9-10,9-11

10-11

and thats as far as I got. If anyone can refine or expand on what ive said, please do.

October 15th, 2010 at 12:09 am

It is easy for Cam because he can remember the product of the two numbers, we can’t …

October 15th, 2010 at 3:51 am

From Chris and Cam’s first two statements, we know the product, let’s call it P, isn’t prime (because neither number was 1):

But much more informative is Cam’s second statement that he can’t figure out where the party is with the information Chris supplied. If P was the multiple of exactly two prime numbers, Cam would know it by now (he’d simply factor his number into its unique prime components). So he knows the address isn’t two prime numbers (although it could be one prime number and a composite number). By stating this, he’s telling Chris as much.

Chris’s next statement is more telling still; he already knew Cam couldn’t figure it out. So whatever sum, let’s call it S, Chris remembers, it can’t be made by adding two prime numbers together- if it could, then there would have been the possibility Cam knew the answer.

When Chris announces his prescience, Cam then knows the answer. This means that out of all the possible addresses that multiply to make P, only one has the property of S described above.

Chris makes the same logical deductions we have, and solves the problem. once you’ve figured out the logical steps needed to solve the problem, described above, it’s still quite a bit of work to actually find the unique pair of numbers that satisfies these properties.

Incidentally, the party’s address is 4-13.

October 17th, 2010 at 12:47 pm

No wonder I can’t find my house !

I took a perfect grid of 100 houses to be 10 x 10, which means there will be no 4-13

October 17th, 2010 at 5:06 pm

Its not 100 houses, but 100 by 100 houses.

October 17th, 2010 at 7:53 pm

Ah, yes, thanks. Obvious when you re-read it (properly)

October 21st, 2010 at 12:22 am

I still don’t get the result that Karl Sharman explained above.

I created an Excel table with all possible addresses situated at a single row and calculated their sums and products.

Then I used the countif function to see how many products there are and removed all single result products since Cam would have remembered if it was a unique product.

Then I applied the same function for sums and removed all unique sums as well as houses on the first row and first column since neither value was 1.

I kept this loop until all sums and products were repeated at least once (non-unique). That still leaves me with 3146 possible addresses which does not include any address with both prime numbers.

I can’t see where to move on from here..

October 22nd, 2010 at 12:01 am

Ozzie – Only one of the address numbers needs to be prime?

October 22nd, 2010 at 3:48 am

I’ve read over the previous posts, but … I don’t think there’s enough info to be able to find out the answer. The wording of the puzzle leads to too many possible answers, I think. Just me? :/

October 30th, 2010 at 12:59 pm

it’s 11