Is this a difficult math problem for ninth grade?

Depends on whether they’re going to punish people for getting it wrong, or if it’s to provoke thinking. I think it’s an actual interesting problem, and fine for 9th graders.

I can see how most ninth-graders might have issues with this problem, though sadly I can see how many adults would to.

That said, it seems about the level where a ninth-grader was expected to be when I went to school years ago as well, probably still should be, and something I personally could’ve done before then.

These days it is probably asking a little. But it looks like a typical problem solving exercise that should be doable but kids nowadays seem to have little patience to do problem solving. They want everything to be easy. Is the “rush” of solving something gone, replaced with instant gratification?

I don’t think it is too difficult for ninth grade. If it becomes frustrating, the student can try taking the brute force approach to solving the problem by drawing out various possible arrangements. In an effort to shorten that task, the student will probably find patterns. And out of that, the glimmer of a mathematical principle might start to emerge. It might just be tantalizing enough to get the student to pay attention the next day.

I think few will get it who can’t see how to solve it in the first minute. It’s more of an intelligence test than a learning exercise. (40).

Let me explain how I solved the problem and why I think the reasoning is fairly sophisticated for ninth grade. The student I worked with had facility with combinatorial concepts, but I still had difficulty explaining what I was doing.

The first step is a little tricky, but I can see how some ninth graders may see it. Choose one of the beads, say the green one, and consider the possible locations. Due to symmetry considerations, there are only 3 positions that need be considered: a corner, a center of an edge and center of array. Due to the adjacency constraint, the center can be eliminated, leaving two positions to consider. The student readily understood this part.

Now consider the green bead in the center of an edge.
xgx
xxx
xxx

The adjacency restriction allows the purple bead only to be in the last row. The difficult thing to see (for a ninth grader) is that we only need to work with one of the bottom corner positions, since vertical reflection exchanges corners. So we have 2×7! positions. The student was able to see this, but it took some convincing.

Things broke down when considering the corner position.
gxx
xxx
xxx

Here the adjacency restriction allows the bottom row and right column. for placing the purple bead. What I was unable to convince the student of was that diagonal reflection exchanges the bottom row with the right column, so we need only work with one of them. This yields 3×7! additional positions, for a total of 5×7! possibilities.

I would expect a good high school junior or senior math student to be able to solve this, but ninth grade seems to early to me. At that age, an extra two or three years can make a big difference.

I agree with @Zaku. In any case, it’s a lovely little problem. I might try it out with some of my students.

The answer I gave is not correct. When the two beads are opposite each other vertically or diagonally, you have to divide 7! in half to account for reflection. The answer then is 4×7!.

One last comment. I like the idea of @Zaku and @finkelitis of turning the problem into a class project. There is a way of checking the answer. After working out the problem, compute the number of ways of arranging the beads where the green and purple are adjacent. The sum of adjacent plus non-adjacent possibilities has to work out to the total number of ways of arranging the beads. The total is easy to calculate. The total is 9! divided by the 8 symmetry operations, giving 9×7!. The 8 symmetry operations are 4 rotations plus reflection followed by one of four rotations.

I also agree that using it to provoke thinking rather than as one more thing to grade would be better. Because while I think it’s fine for ninth grade, it is certainly one of the harder problems a ninth grader will face. Moreover, the process of figuring it out is just so much more valuable than getting it right—especially at this stage of education. So a group project, or an individual assignment that is then turned into a group project the next day, seems optimal.