Argthtjtr

I have a disk with 6 equally spaced dents around the edge. The disk balances on the center point. I want to place marbles around the edge so that it stays balanced. There are four ways that this can be done:

One with two balls, one with three, one with four, and one with six. It can't be balanced with five or one.

Now the question is:

How many ways (all combinations excluding rotations and reflections) are there to balance the disk if it has 12 dents


(Imagine that my drawing skills were better and the 12 dents are all equally spaced!)

Now that there's a valid answer, I'm going to add a source with the inspiration and an interesting fact about it:

As some of the answerers noted, each answer has a complement. If it can be balanced in one configuration, then you can swap every marble for a hole and every hole for a marble and it will still be balanced.

It turns out that with $n$ holes and $k$ marbles it can be balanced if and only if both $k$ and $n-k$ can be written as the sum of prime factors of $n$.

Here's a link to the numberphile video that inspired this: Numberphile video

There are more possibilities than one expects because

perhaps surprisingly, neither the number of balls nor the number of holes need be a factor of 12! (You can put together a solution with 2 and a solution with 3 to get one with 5, for instance.)

Here is the full list, after removing symmetries; I have marked some of the more surprising ones:

............
oo..oo..oo..
o.o.o.o.o.o.
o.....o.....
oooo..oooo..
o..o..o..o..
oo....oo....
ooo..oo.oo.. ** (seven balls!)
ooo.ooo.ooo.
ooooo.ooooo.
o...o...o...
o.o...o.o...
ooo.o.ooo.o.
oo.oo.oo.oo.
oooooooooooo
oo..o..oo... ** (five balls!)
ooo...ooo...
oo.o..oo.o.. * (no mirror symmetry)

So there are

18 ways to do it. This includes the "empty" solution; OP implies that this one isn't acceptable, so the answer OP is looking for is 17.

A visual representation:

Another answer has missed one permutation of

9 balls

Note that each solution has its complement too.

The answer is:

$15$ (counting $0$ marbles as an option)

because

For a board to balance, it must be symmetric about two distinct axes. $k$ must therefore be a divisor of $12$ greater than $1$ to achieve a balance, hence $2,3,4,6,12$, but leaving these as holes also balances the board, and so we also have $10,9,8,6,0$, which makes for $9$ cases in total. There are $3$ versions with $4$ marbles (and therefore also with $8$ marbles), and $3$ versions with $6$ marbles, giving $15$ in total.


Using $4$ marbles:



Using $6$ marbles:

I will start solving the questions by mentioning the two required solution techniques.

Technique 1:

The marbles can be arranged such that every pair of marbles are facing each other (used with an even number of marbles). The total number of ways is given by $^{N/2}C_{n/2}$, where $N$ is the total number of dents, and $n$ is the number of marbles to be used.

Technique 2:

The marbles can be arranged such that they are separated by an equal number of dents. (used with an odd number of marbles).

Using these two techniques, the number of possible combinations for each number of marbles can be calculated.

2 marbles:

They can be arranged in $^6C_1 = 6$ ways.

3 marbles:

They can be arranged in $4$ ways.

4 marbles:

They can be arranged in $^6C_2 = 15$ ways.

6 marbles:

They can be arranged in $^6C_3 = 20$ ways.

8 marbles:

They can be arranged in $^6C_4 = 15$ ways.

9 marbles:

They can be arranged in 2 ways.

10 marbles:

They can be arranges in $^6C_5 = 6$ ways.

12 marbles:

There is only one way of arranging them ($^6C_1 = 1$).

One more thing...

The disk will also balance without any marbles.

Hence, the total number of arrangements that can be used to balance the disk is

$6 + 4 + 15 + 20 + 15 + 2 + 6 + 1 + 1 = 70$ ways.