Discrete math question :(

[nwicole]

We have several piles of koala bears. In an attempt to disperse them, we remove exactly one koala bear from each pile and place all of those koalas into one new pile. For example, if we started with koala piles of sizes 1, 4, and 4, we would then end up with koala piles of sizes 3, 3, and 3; or, if we started with piles of size 3 and 4, we would end up with piles of size 2 and 3 and 2. It is possible that we do this operation exactly once and end up with the exact same pile sizes as we started with? (the order of them is irrelevant; only the sizes matter).

Identify all the collections of piles that have this property and explain why they are the only ones.

thank you so much

 

[Ishuda]

The way I would do it:
To help get you started: Assume there are m piles.
(1)Since there are m piles and you are going to pull one from each pile, how many are there in the new pile for the second set of piles?
(2)Take the pile(s) with the most bears, call the number of bears n. Considering the fact that there must be a pile with n bears in the new set of piles, what does that tell you about the relationship between n and m.
(3)How many bears will be left after pulling 1 from a pile with n bears? What does that tell you about the existence of a pile with n-1 bears. Continue that to its logical end.
(4)Can there be more than one pile with n bears? Consider the answer to that, can there be more than one pile with n-1 bears? Continue that to its logical end.
...

 

[Ishuda]

Can't follow what you're doing/showing Ishuda...
If we end up with what we started with, then one pile must have only one bear; no?
And I can only see 1 pile of 1 as possible: from [1] to [0][1]

I'm obviously missing something...

As an example we have 3 piles of 1, 2, and 3 bears each. What do we have for the next set?

Old set = the one we start with.
New set = the one after we have reduced each of the old set by 1 and added a new pile.
Old set = New set:

(1)Since there are m piles and you are going to pull one from each pile, how many are there in the new pile for the second set of piles?
Answer: the new pile has m bears.

(2)Take the pile(s) with the most bears, call the number of bears n. Considering the fact that there must be a pile with n bears in the new set of piles, what does that tell you about the relationship between n and m.
Answer: All the piles in the new set have less than n bears except possible the new one which has m bears. Therefore that new pile must have n bears, so n=m.

(3)How many bears will be left after pulling 1 from a pile with n bears? What does that tell you about the existence of a pile with n-1 bears. Continue that to its logical end.
Answer: Since n = m, we can use use m instead of m.
When we take 1 each from m piles of the old set, we will have a new pile in the new set with m bears. If the new set is going to be the same as the old set, then
- there was a pile with m bears in the old set, which means there is a pile with m-1 in the new set which means
- there was a pile with m-1 in the old set, which means there is a pile with m-2 in the new set which means
- there was a pile with m-2 in the old set, which mean there is a pile with m-3 in the new set which means
-...
-there was a pile with 2 in the old set, which means there is pile with 1 in the new set which means
-there was a pile with 1 in the old set and it became non existent when we took one away from it.

(4)Can there be more than one pile with n bears? Consider the answer to that, can there be more than one pile with n-1 bears? Continue that to its logical end.
Answer:
Can there be more than one pile with m bears? Suppose there were two or more. Then there would two or more in the new set of piles. But, in the new set, all of the piles have less than the maximum number of bears, i.e. less than m (see (2) above), except the one new pile with m bears.
-Since there is only one pile with m bears in the old set, there is only one pile in the new set with m-1 bears which means
-there is only one pile with m-1 in the old set, so there is only one pile in the new set with m-2 and
-...
-there is only one pile with 2 in the old set, so there is only one pile with 1 in the new set and
-there is only one pile with 1 in the old set

...Can we have a pile with more than m bears.
Answer: No. Let n be the pile(s) with the maximum number of bears. Then the new set has pile(s) of n-1 bears
and fewer and, since it is the same as the old set, a pile of n bears. But all of the piles in the new set, except the new pile which has m bears, has less than n bears. Therefore n=m.

So, there are m piles with 1, 2, 3, ..., m bears in each pile.

Hopefully I haven't made a type somewhere and what I said was clear enough.

 

[nwicole]

oh i see, finally i got what you mean, thank you for helping , your explanation is much better to my professor

As an example we have 3 piles of 1, 2, and 3 bears each. What do we have for the next set?

Old set = the one we start with.
New set = the one after we have reduced each of the old set by 1 and added a new pile.
Old set = New set:

(1)Since there are m piles and you are going to pull one from each pile, how many are there in the new pile for the second set of piles?
Answer: the new pile has m bears.

(2)Take the pile(s) with the most bears, call the number of bears n. Considering the fact that there must be a pile with n bears in the new set of piles, what does that tell you about the relationship between n and m.
Answer: All the piles in the new set have less than n bears except possible the new one which has m bears. Therefore that new pile must have n bears, so n=m.

(3)How many bears will be left after pulling 1 from a pile with n bears? What does that tell you about the existence of a pile with n-1 bears. Continue that to its logical end.
Answer: Since n = m, we can use use m instead of m.
When we take 1 each from m piles of the old set, we will have a new pile in the new set with m bears. If the new set is going to be the same as the old set, then
- there was a pile with m bears in the old set, which means there is a pile with m-1 in the new set which means
- there was a pile with m-1 in the old set, which means there is a pile with m-2 in the new set which means
- there was a pile with m-2 in the old set, which mean there is a pile with m-3 in the new set which means
-...
-there was a pile with 2 in the old set, which means there is pile with 1 in the new set which means
-there was a pile with 1 in the old set and it became non existent when we took one away from it.

(4)Can there be more than one pile with n bears? Consider the answer to that, can there be more than one pile with n-1 bears? Continue that to its logical end.
Answer:
Can there be more than one pile with m bears? Suppose there were two or more. Then there would two or more in the new set of piles. But, in the new set, all of the piles have less than the maximum number of bears, i.e. less than m (see (2) above), except the one new pile with m bears.
-Since there is only one pile with m bears in the old set, there is only one pile in the new set with m-1 bears which means
-there is only one pile with m-1 in the old set, so there is only one pile in the new set with m-2 and
-...
-there is only one pile with 2 in the old set, so there is only one pile with 1 in the new set and
-there is only one pile with 1 in the old set

...Can we have a pile with more than m bears.
Answer: No. Let n be the pile(s) with the maximum number of bears. Then the new set has pile(s) of n-1 bears
and fewer and, since it is the same as the old set, a pile of n bears. But all of the piles in the new set, except the new pile which has m bears, has less than n bears. Therefore n=m.

So, there are m piles with 1, 2, 3, ..., m bears in each pile.

Hopefully I haven't made a type somewhere and what I said was clear enough.