NEED HELP!

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Days of Frusturation have lead us here :(

A teacher has 247 students. To reward them for a good week he is going to buy them ice-cream. There are only two flavors of ice-cream, Chocolate, and Vanilla. Each cone must be made different.

So the question is how many scoops of ice-cream will each cone need to be different. Not only do we need the answer...but how we got it. Have tried for days...come up with 17 but there is always a remainder. Would like to say one child is milk intollerant :lol:
If anyone can help...Thanks
 
Please reply with the definition of "different". Does this mean "a different combination of vanilla and chocolate", or does this mean, in a Saturday-morning-cartoons sort of way, "a different stacking of scoops rising out of the cone", or does this mean something else?

Thank you.

Eliz.
 
The way I understand your problem:

scoops:flavors:students:students so far
1: c v : 2 : 2
2: cc vv cv : 3 : 5
3: ccc vvv cvv ccv : 4 : 9
4: cccc vvvv cvvv cccv ccvv : 5 : 14
5: ccccc vvvvv cvvvv ccccv ccvvv cccvv : 6 : 20
...
20: (ain't typing 'em!) :**21 : 230
21: remaining 17 students each get indigestion from 21-scoops cones :shock:

n(n+1) / 2 - 1 = 247
n^2 + n - 248 = 0
**floor(n) = 21

21(22) / 2 - 1 = 230

So 247-230=17 students get 21 scoops.

Your problem should state something like: what is minimum number of scoops?
If not, you can have everybody getting chocolate, from 1 to 247 scoops :shock:
 
Hello, AVHear2!

A teacher has 247 students. To reward them for a good week he is going to buy them ice-cream.
There are only two flavors of ice-cream, Chocolate, and Vanilla. Each cone must be made different.

How many scoops of ice-cream will each cone need to be different?
Click to expand...

I will assume that the scoops will be stacked and the order of the flavors is important.
\(\displaystyle \;\;\)For example: \(\displaystyle \,\begin{Bmatrix}c \\ v \\ v\end{Bmatrix} \:\neq\:\begin{Bmatrix}v \\ c \\ v\end{Bmatrix}\)

The first (bottom) scoop has 2 choices of flavors.
The second scoop has 2 choices of flavors.
The third scoop has 2 choices of flavors . . . etc.
\(\displaystyle \;\;\)For \(\displaystyle n\) scoops, there will be: \(\displaystyle 2^n\) possible stacks of flavors.

To have at least 247 different arrangements, \(\displaystyle n\,=\,8\)
\(\displaystyle \;\;\)There will be be: \(\displaystyle \,2^8\,=\,256\) possible 8-scoop cones.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Edit: \(\displaystyle \;\)Ha . . . Denis had a different approach.
\(\displaystyle \;\;\;\;\;\;\)Nice going, Denis!

And I thought a stack of eight scoops was pretty silly . . .
 
You guys are all awesome...!!!! I think I didn't write the question right. Lets try again.

All 247 students have to have the same amount of scoops on thier cone. The difference comes in how they are stacked. No two cones can have the same combination, but they must have the same number of ice cream scoops. Ok...I am leaving this to the brains. Thanks again...you are amazing!
 
AVHear2 said:
You guys are all awesome...!!!! I think I didn't write the question right. Lets try again.
All 247 students have to have the same amount of scoops on thier cone. The difference comes in how they are stacked. No two cones can have the same combination, but they must have the same number of ice cream scoops. Ok...I am leaving this to the brains. Thanks again...you are amazing!
Click to expand...

Hmmm.....3 scoops example:
C V V
V C V
V V C

Do you consider that ONE or THREE combinations?
 
“Do you consider that ONE or THREE combinations?”
From the first, I thought that it would be THREE combination(S).
I had prepared almost exactly what Soroban wrote.
I did not post it because of the confusion about interpretation.
I now think that we are right, i.e. Soroban’s solution is correct.
 
I think that is Three different combinations. So....how many scoops would there need to be for all 247 to have a different combination?
 
Each gets an 8scoop cone (see soroban's, the ice cream man!).

247 * 8 = 1976 total scoops (teacher has expense account?)

Maximizing the chocolaters
1: c c c c c c c c [8]
2: c c c c c c c v [16]
3: c c c c c c v c [24]
4: c c c c c c v v [32]
...
247: v v v v c v v c [1976]
(1011 chocolate scoops, 965 vanilla scoops :shock: )
...
256: v v v v v v v v [2048]
 
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