[Prof. Templeton]

Long before Professor Templeton started his career at Redrum University he had a job as a stockboy at the Manhattan Fruit Exchange. One day the Prof. had placed nine baskets of apples into a three by three square and being of a puzzling mind even back then he arranged the apples so that the number of fruit in each basket would form a magic square with no two baskets containing the same amount. Just as he finished, his manager was leading a tour of school children past and the manager instructed the Prof. to distribute one of the baskets fully to the group of students so that each student received an equal amount of apples to take home. The Professor having just made a magic square out of those very baskets told his manager that the task was quite impossible.

So now I ask you, what were the quantities of apples in those baskets?

In order to help you out in discovering how many each basket contained, I’ll tell you a little bit about the baskets or bushels. They were very much similar in size to those that you might put laundry into. If they were capable of retaining water they might hold 10 gallons or so. Some of the baskets were quite full and others held considerably fewer apple, but none of them was nearly empty as far as baskets of apples go.

[Guest]

Long before Professor Templeton started his career at Redrum University he had a job as a stockboy at the Manhattan Fruit Exchange. One day the Prof. had placed nine baskets of apples into a three by three square and being of a puzzling mind even back then he arranged the apples so that the number of fruit in each basket would form a magic square with no two baskets containing the same amount. Just as he finished, his manager was leading a tour of school children past and the manager instructed the Prof. to distribute one of the baskets fully to the group of students so that each student received an equal amount of apples to take home. The Professor having just made a magic square out of those very baskets told his manager that the task was quite impossible.

So now I ask you, what were the quantities of apples in those baskets?

In order to help you out in discovering how many each basket contained, I’ll tell you a little bit about the baskets or bushels. They were very much similar in size to those that you might put laundry into. If they were capable of retaining water they might hold 10 gallons or so. Some of the baskets were quite full and others held considerably fewer apple, but none of them was nearly empty as far as baskets of apples go.

Perhaps I'm missing something. Did you intentionally omit the number of children in the tour?

[Prof. Templeton]

Did you intentionally omit the number of children in the tour?

Yes. The number of children is irrelevant.

[Guest]

Long before Professor Templeton started his career at Redrum University he had a job as a stockboy at the Manhattan Fruit Exchange. One day the Prof. had placed nine baskets of apples into a three by three square and being of a puzzling mind even back then he arranged the apples so that the number of fruit in each basket would form a magic square with no two baskets containing the same amount. Just as he finished, his manager was leading a tour of school children past and the manager instructed the Prof. to distribute one of the baskets fully to the group of students so that each student received an equal amount of apples to take home. The Professor having just made a magic square out of those very baskets told his manager that the task was quite impossible.

So now I ask you, what were the quantities of apples in those baskets?

In order to help you out in discovering how many each basket contained, I’ll tell you a little bit about the baskets or bushels. They were very much similar in size to those that you might put laundry into. If they were capable of retaining water they might hold 10 gallons or so. Some of the baskets were quite full and others held considerably fewer apple, but none of them was nearly empty as far as baskets of apples go.

There are 17 students in the class. The bushels contain 16,32,48,64,80,96,112,128,144...none of which are evenly divisible by 17.

[Prof. Templeton]

There are 17 students in the class. The bushels contain 16,32,48,64,80,96,112,128,144...none of which are evenly divisible by 17.

I'll tell you this...

The manager wanted each student to take home some "apples", so every basket must contain more apples then there are students.

[Guest]

Yes. The number of children is irrelevant.

If that's the case, then the problem seems extremely simple: Find a magic square whose numbers range from 10 to 200, such that there exists n where 5 <= n <= 30 and none of the square's numbers are evenly divisible by n. In addition to tpaxatb's solution there are lots of solutions that meet these criteria.

So what am I missing?

[CaptainEd]

If that's the case, then the problem seems extremely simple: Find a magic square whose numbers range from 10 to 200, such that there exists n where 5 <= n <= 30 and none of the square's numbers are evenly divisible by n. In addition to tpaxatb's solution there are lots of solutions that meet these criteria.

So what am I missing?

You've seen more than I have--I don't get the range 10,200 or 5,30.

It looks to me like the Prof is saying we need a magic square composed entirely of primes (so that the number of students will always fail to divide any basket).

I'm concerned about the (apparently new) constraint that the number of apples in EVERY basket must be greater than the number of students. Maybe I've misread it. It's possible that when the teacher said to distribute "one of the baskets", the choice of basket was left to the Prof, and the Prof found no baskets with more apples than students. I'm still not stating it comfortably. I think the Prof got to choose a basket, found no baskets could be so divided. But I don't think that WE get to choose the size of the tour.

Edited June 26, 2009 by CaptainEd

[Guest]

I'll tell you this...

The manager wanted each student to take home some "apples", so every basket must contain more apples then there are students.

19,38,57,76,95,114,133,152,171 with 17 students in the class.

Since it is a magic square, there are nine bushels. for some x, each bushel b_n contains count_n = x*n apples. The above is a possibility, since none of the containers are now evenly divisible by 17... I don't know that i have time (or feel like) a generality to find some relation between the number of students and the value of x beyond:

students < x < 2*number of students

since if x = number of students, bushel b₁ is divisible by the number of students. And if x == 2 * number of students then b₂ is distributed. It may be that that is the final relation...but someone else can maybe go from there...

[Guest]

You've seen more than I have--I don't get the range 10,200 or 5,30.

It looks to me like the Prof is saying we need a magic square composed entirely of primes (so that the number of students will always fail to divide any basket).

I'm concerned about the (apparently new) constraint that the number of apples in EVERY basket must be greater than the number of students. Maybe I've misread it.

Those numbers were just what i thought to be reasonable estimations of the constraints given the description - and they illustrate my point of the existence of several solutions.

Only using primes would make it more interesting (although I still don't gather that from the OP), but I still don't see how to ensure that all baskets have more apples than there are children if the number of children is irrelevant.

[CaptainEd]

Professor, I'm not quite getting the constraints properly. Please correct my understanding, and forgive me for appearing to overlook some of your responses so far, as I can't get my understanding of all of them to jibe.

We seek a magic square with 9 baskets, all different positive numbers of apples. (Magic square: all rows, all columns, and both major diagonals add to the same number of apples)

1) No matter what size tour came through, the Professor could not find a basket that could be shared equally by the number of students

2) Tour size >1

3) share size > 1

[Guest]

181 | 43 | 151

97 | 127 | 157

103 | 211 | 73

Is this along the lines of what we're looking for?

Edited June 26, 2009 by hookemhorns

[Prof. Templeton]

Professor, I'm not quite getting the constraints properly. Please correct my understanding, and forgive me for appearing to overlook some of your responses so far, as I can't get my understanding of all of them to jibe.

We seek a magic square with 9 baskets, all different positive numbers of apples. (Magic square: all rows, all columns, and both major diagonals add to the same number of apples)

1) No matter what size tour came through, the Professor could not find a basket that could be shared equally by the number of students

2) Tour size >1

3) share size > 1

Yes that's it. Also what I've told you about the baskets should eliminate some solutions. Someone said 10-200 and that would be an acceptable range.

Is this along the lines of what we're looking for?

181 | 43 | 151

97 | 127 | 157

103 | 211 | 73

Yes, except thats not a magic square.

[bushindo]

Yes that's it. Also what I've told you about the baskets should eliminate some solutions. Someone said 10-200 and that would be an acceptable range.

Yes, except thats not a magic square.

I cheated on this problem...

checking all possible 3x3 table constructed from prime numbers between 2-200 (46 primes ), by hand. Just kidding. It was a fun algorithm optimization exercise. The magic squares are

	 [,1] [,2] [,3]

[1,]  101	5   71

[2,]   29   59   89

[3,]   47  113   17


	 [,1] [,2] [,3]

[1,]  109	7  103

[2,]   67   73   79

[3,]   43  139   37


	 [,1] [,2] [,3]

[1,]  101   29   83

[2,]   53   71   89

[3,]   59  113   41


	 [,1] [,2] [,3]

[1,]  149   11  107

[2,]   47   89  131

[3,]   71  167   29


	 [,1] [,2] [,3]

[1,]  163	7  139

[2,]   79  103  127

[3,]   67  199   43

[Prof. Templeton]

I cheated on this problem...

checking all possible 3x3 table constructed from prime numbers between 2-200 (46 primes ), by hand. Just kidding. It was a fun algorithm optimization exercise. The magic squares are

	 [,1] [,2] [,3]

[1,]  101	5   71

[2,]   29   59   89

[3,]   47  113   17


	 [,1] [,2] [,3]

[1,]  109	7  103

[2,]   67   73   79

[3,]   43  139   37


	 [,1] [,2] [,3]

[1,]  101   29   83

[2,]   53   71   89

[3,]   59  113   41


	 [,1] [,2] [,3]

[1,]  149   11  107

[2,]   47   89  131

[3,]   71  167   29


	 [,1] [,2] [,3]

[1,]  163	7  139

[2,]   79  103  127

[3,]   67  199   43

No. 3 was the one I was going for, although No. 4 could also work. I was trying to exclude solutions with lower numbers like 7.