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**Ansette****Member**- Registered: 2006-02-19
- Posts: 21

Hey,we've been set an assignment in pure maths and have no clue how to go about it..

"A student on a year abroad buys a number of presents and wants to send them home. The goods are already wrapped up as individual packets: by a curious chance there is an integer n such that there is precisely one packet of depth a, width b and length c, for each set of integers (a,b,c) with 1<=a<=b<=c<=n

The student could send each packet home separately, but it is cheaper (and mathematically more interesting) to make the packets up into parcels. To avoid breakages, the parcels must have no empty spaces (the country is experiencing a shortage of bubble wrap). In other words, a parcel of size pxqxr will have volume pqr that is equal to the sum of the packets it contains. Assuming that it does not matter how large the parcels are, how many parcels does the student need?"

Seriously we're so unsure what to do :s

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,562

I have two questions? Is n different for each package? How many packages are there?

I don't really get what you are trying to say. But it seems interesting.

**igloo** **myrtilles** **fourmis**

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**Ansette****Member**- Registered: 2006-02-19
- Posts: 21

Thats the thing.. we really do not understand! That is literally what we've been given..

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,562

Since your professor has given such an erroneous question, I would tell him the answer is that the packages cannot be combined into parcels, and therefore he must send them out separately!

**igloo** **myrtilles** **fourmis**

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**Ansette****Member**- Registered: 2006-02-19
- Posts: 21

looking at it again.. (and again, etc) i'm thinking it suggests the number of parcels is the number of combinations of dimensions (a,b,c) with the bound that is set as n.. I'm going to email him now to see if that's correct.. any suggestions if that is the case?

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**Ansette****Member**- Registered: 2006-02-19
- Posts: 21

In the case that there are x many packets with the dimensions bounded as suggested above...

I've been looking at the number of packets there are.. e.g. each x for n.

n=2, x=4

n=3, x=10

n=4, x=20

n=5, x=35

This sequence by the pyramidal numbers...

Another thing I noticed, the number of occurences of each size per set.

In other words, for instance with n=2,

out of all the combinations, the number 1 appears 6 times, the number 2 also 6 times.

for each n, the number of occurences is the same (for that n).

the number of occurences (y) is as follows:

n=2, y=6

n=3, y=10

n=4, y=15

n=5, y=21

This sequence being the triangular numbers..

Can anyone give me a reason as to why this is the case?

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,562

So for n = 2, we have dimensions 1,1,1 or 2,1,1 or 1,2,1 or 1,1,2. Is that what you mean?

**igloo** **myrtilles** **fourmis**

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**Ansette****Member**- Registered: 2006-02-19
- Posts: 21

no, see the dimensions must be 1,1,1 or 1,1,2 or 1,2,2 or 2,2,2

because 1≤a≤b≤c≤n

that's 4 different combinations. however what im now querying of the lecturer is that for each n it's not one of the combinations, there are indeed all of those packages (so for n=2, 4 packages.)

any idea on why the combinations take those patterns? i mean i can see it but unless i can prove it.. it's no use to us :s

so for combinations, forn any n.

Number of packages = (n)(n+1)(n+2)/6 or alternative (n+2)C3 [(n+2)Choose 3]

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