[bonanova]

Marbles again, but now of colors only two.

Bags of marbles, yes two or more.

Every bag has more than one marble of each color.

Each bag has the same number of marbles as all the others.

No bag has marbles in equal quantity of the two colors.

If I told you the total number of marbles, you'd know the number of bags.

So I'll just say: the number of marbles is between 200 and 300.

Now you know how many bags.

Or you will, after a little thought.

[Guest]

probly wrong but

24 groups of ten marbles

[Guest]

Definitely the hardest part of this puzzle is the wording, but if I understand it right...

Might be 17 bags with 17 marbles in each bag for a total of 289 marbles. Also, I think there might be more than 1 correct answer.

I'm not sure if I understood the prompt though.

EDIT: Clarified how many bags, and number of marbles in each.

Edited October 13, 2008 by rossbeemer

[Guest]

Marbles again, but now of colors only two.

Bags of marbles, yes two or more.

Every bag has more than one marble of each color.

Each bag has the same number of marbles as all the others.

No bag has marbles in equal quantity of the two colors.

If I told you the total number of marbles, you'd know the number of bags.

So I'll just say: the number of marbles is between 200 and 300.

Now you know how many bags.

Or you will, after a little thought.

I just have a question on the red text - would the following be allowed:

Bag 1: 2 Red Marbles, 3 Blue Marbles

Bag 2: 3 Red Marbles, 2 Blue Marbles

My interpretation is that yes, it would be allowed. In which case

13 bags, of 17 marbles, for a total of 221.

Basically, I was looking for a number in [200,300] that was the product of two primes. That being said, I'm not sure why there couldn't be

11 bags of 19

13 bags of 19

11 bags of 23

etc.

EDIT: typo

Edited October 13, 2008 by Chuck Rampart

[Prime]

I pondered on this one for a while and I don't see a unique solution, but rather many...

m marbles offer m-3 combinations of two colors where each color is represented by no less than 2 marbles.

So the number of marbles in each bag must be at least 3 greater than the number of bags.

This makes for one boundary condition: no more than 15 bags, as 16*19 > 300.

Since the problem states that if we knew the total number of marbles we'd guess the number of bags. Which means that the product must offer a unique combination of valid multipliers while falling within the 200 and 300 boundaries. Simplest way to achive such combination is to use prime numbers such as 13*17=221, or 13*19=247, or 13*23=299. But they don't have to be prime numbers to satisfy the conditions. For example 11 bags 22 marbles each offer a unique combination and 11*22=242 falls within the boundaries too.

[Guest]

Chuck,

I agree with your interpretation. That's why I though there might be several answers. However...

It says that "If I told you the number of marbles, you'd know the number of bags." So if you were told there were 221 marbles, how would you know if there were 13 bags or 17 bags? That's why I thought it would have to be 289, because then it would have to be 17 bags and 17 marbles. Of course, I might not be interpreting the prompt correctly. Can you have 2 bags with, say, 2 red and 3 blue marbles? I'm not sure, I think you can.

EDIT: typo

Edited October 13, 2008 by rossbeemer

[Guest]

Does "No bag has marbles in equal quantity of the two colors" mean that for every bag, the number of marbles of one color is greater than the number of marbles of the other color. Or does it mean that none of the bags have the same number of colored marbles?

If it means the latter then shouldn't the line read "no two bags have marbles in equal quantity of the colors"?

finding all numbers between 200 and 300 that have only two prime factors should give us the number of bags.

Belowe are the possible combinations

23 * 13

13 * 11

19 * 13

19 * 11

17 * 13

So the number of bags must be either 13 or 11. I cant seem to find a way to narrow it down further.

Edited October 14, 2008 by vimil

[Guest]

Chuck,

I agree with your interpretation. That's why I though there might be several answers. However...

It says that "If I told you the number of marbles, you'd know the number of bags." So if you were told there were 221 marbles, how would you know if there were 13 bags or 17 bags? That's why I thought it would have to be 289, because then it would have to be 17 bags and 17 marbles. Of course, I might not be interpreting the prompt correctly. Can you have 2 bags with, say, 2 red and 3 blue marbles? I'm not sure, I think you can.

EDIT: typo

There's only one way to get 221 marbles - 13 bags of 17. You can't get 17 bags of 13. Let's try it to see why. Every number must be greater than one, so you start at 2+11:

2+11

3+10

4+9

5+8

6+7

7+6

8+5

9+4

10+3

11+2

That's only 10 distinct bags - any other bag with 13 marbles in it has to be a duplicate of one of these, which is prohibited by the OP.

As prime said, if you have m marbles in a bag than you can have at most m-3 bags.

I should also point out that there more than one way to get 13 bags of 17 - there are 14 distinct combinations for the 17 marbles, so you can have 14 different combinations to get your 13 bags. Either way, though, 221 marbles means 13 bags.

[Guest]

There's only one way to get 221 marbles - 13 bags of 17. You can't get 17 bags of 13. Let's try it to see why. Every number must be greater than one, so you start at 2+11:

2+11

3+10

4+9

5+8

6+7

7+6

8+5

9+4

10+3

11+2

That's only 10 distinct bags - any other bag with 13 marbles in it has to be a duplicate of one of these, which is prohibited by the OP.

As prime said, if you have m marbles in a bag than you can have at most m-3 bags.

I should also point out that there more than one way to get 13 bags of 17 - there are 14 distinct combinations for the 17 marbles, so you can have 14 different combinations to get your 13 bags. Either way, though, 221 marbles means 13 bags.

So then, each bag has to have a different combination of marbles? I didn't think the prompt was clear about that- maybe I missed something, but I thought that you could have, for example, 5 bags all with 8+5

[Prime]

Does "No bag has marbles in equal quantity of the two colors" mean that for every bag, the number of marbles of one color is greater than the number of marbles of the other color. Or does it mean that none of the bags have the same number of colored marbles?

If it means the latter then shouldn't the line read "no two bags have marbles in equal quantity of the colors"?

finding all numbers between 200 and 300 that have only two prime factors should give us the number of bags.

Belowe are the possible combinations

23 * 13

13 * 11

19 * 13

19 * 11

17 * 13

So the number of bags must be either 13 or 11. I cant seem to find a way to narrow it down further.

Either way the problem does not have a unique solution.

If it means that no 2 bags have the same number of marbles of a certain color, then the prime number solutions are as following:

BAGS -- Marbles

2 ------- 101, 103, 107, 109, 113, 127, 131, 137, 139, 143, 149.

3 ------- 67, 71, 73, 79, 83, 89, 97.

5 ------- 41, 43, 47, 53, 59.

7 ------- 29, 31, 37, 41.

11 ------ 19, 23.

13 ------ 17, 19, 23.

(The number of marbles in each bag must be at least 3 more than the total number of bags.)

If are to interpret that inside each bag the number of marbles of one color is not equal to the number of marbles of the other color. Then also taking into account that if we knew the total number of marbles, we'd know the number of bags, it would have to be a product of two prime numbers such that only one of them could taken for the number of marbles in the bag. The only two prime numbers that can mean the number of bags, but not the number of marbles inside each bag are 2 and 3. Then the possibilities are:

BAGS -- Marbles

2 ------- 101, 103, 107, 109, 113, 127, 131, 137, 139, 143, 149.

3 ------- 67, 71, 73, 79, 83, 89, 97.

Or are we are missing some other possible interpretation here?

[bonanova]

No bag has the same number of one color that is has of the other color

[Guest]

there were 17 bags. If by telling the total number of marbles someone could determine the number of bags then surely the number of bags must have been equal to the number of marbles. And the number of marbles and the number of bags need to be prime too otherwise there would be more than one way to factor them.

[bonanova]

there were 17 bags. If by telling the total number of marbles someone could determine the number of bags then surely the number of bags must have been equal to the number of marbles. And the number of marbles and the number of bags need to be prime too otherwise there would be more than one way to factor them.

Yup.