From the Start hexagon, alternate jumping 1 hexagon, then 2 hexagons (in a straight line). You cannot jump across solid black bars (walls) between hexagons. You can land on the same hexagon multiple times.
When you land on a green hexagon, add the number given on that hexagon to your total. When you land on an orange hexagon, subtract the number given on that hexagon from your total. Starting with zero, the Start hexagon immediately adds 9 to the total.
At no time when traveling through the puzzle can your total be greater than 9, nor can it be less than negative 9.
Can you find the shortest path to land on the Finish hexagon that returns you to your starting total of zero?
To solve this puzzle (in other words, to figure out what the appropriate instructions should be), you need to carefully observe some consistencies between the solution and the incorrect solution examples, as well as deduce why the incorrect solution attempts are wrong.
First, the solution connects the hexagons marked S and F, presumably the Start and Finish, and all but the first two of the incorrect solution examples do not. The solution is also labeled "shortest". So one rule is: we need to go from the Start to the Finish in the shortest way possible. All but two of the examples stop before reaching the Finish hexagon, and there must be problems with the two that do reach the finish.
The list of numbers that accompanies each diagram must also mean something. Comparing the first example to the solution, notice that both number lists start with a green 9, and both finish on a grey 0. But the first example crosses a solid black bar between hexagons, twice actually. The solution does not do this (nor do any of the other examples). Thus, we can conclude another rule: you cannot jump across solid black bars between hexagons.
The other example that goes from start to finish avoids crossing any solid black bars. However, the list of numbers ends with an orange 1, and not a grey 0. This must make a difference, although right now it isn't clear what that difference is. So another (partially understood) rule is: when you get to the Finish hexagon, your list of numbers must end in a grey 0. We'll still need to deduce how these numbers relate to the numbers on the hexagons.
The next example illustrates another rule about how we must move about the hexagons. In all of the examples so far, you start at the green 9 hexagon, move one hexagon in some direction, then move two hexagons in a straight line, then move one hexagon, then move two hexagons in a straight line, and so on. Note that you never turn while jumping two hexagons, nor do any of the examples start by jumping two hexagons from the green 9.
In this example, once you get to the orange 2, you cannot jump two hexagons, in a straight line, in any direction. In three options moving up/right, you can only jump one hexagon before running out of hexagons. In the other options moving left/down, you are blocked by a black bar before you can get to the second hexagon. All of these observations lead to the rule: from the Start hexagon, alternate jumping one hexagon, then two hexagons (in a straight line).
Going back to the solution, given this rule about alternatively jumping 1 or 2 hexagons, you would need to move around the loop (either clockwise or counter-clockwise), return to the green 9 hexagon, and then go up the path towards the top (9 – 4 – 8 ...), since you couldn't immediately jump two hexagons from the green 9 to the orange 4. But which way should you move around the loop?
Now is a good time to figure out what the string of numbers means. You could move around the loop of hexagons in one of two ways: 9 – 7 – 5 – 6 – 4 – 9... or 9 – 4 – 6 – 5 – 7 – 9.... The string of numbers starts 9, 2, .... Notice that the difference between 9 and 2 is 7. This suggests that the appropriate route is the clockwise one (9 – 7 – 5 – 6 – 4 – 9...) and that the string of numbers relates to the numbers in the hexagons through differences.
But the difference between the next numbers in the string, 2 and 3, is just 1, and the next hexagon is a 5. However, the 2 is green and the 3 is orange. Those numbers are 5 apart if one color is considered positive and the other is negative. This suggests that orange hexagon numbers turn the list numbers more orange (let's say, negative) and green hexagon numbers turn the list numbers more green (positive). This also explains why 0 is grey, and not green nor orange, since 0 is neither positive nor negative. And it explains why every list starts with a green 9, since that's the value of the starting hexagon.
Thus, we can summarize the above observations as the following set of rules: When you land on a green hexagon, add the number given on that hexagon to your total. When you land on an orange hexagon, subtract the number given on that hexagon from your total. Starting with zero, the Start hexagon immediately adds 9 to the total.
Just to be sure, if we check the solution, we can match up the hexagon numbers with the list numbers following the above operations:
Returning to the second example, now our earlier rule makes a little more sense. When you get to the Finish hexagon, your list of numbers must end in a grey 0. In other words: you must finish with a total of zero. Now that we know what the list of numbers means, we can see that the running total for the second example is not zero when we arrive at the finish.
So what is wrong with the remaining attempted solution examples?
Both of these end in red 10's. None of the previous examples (nor the solution) have had totals in the double-digits, and none of the hexagons have numbers in the double-digits either. Perhaps there is a limit to how large (or how small) the total can be.
Before examining the totals in each of the last examples, there is one more thing we need to pay attention to, which is that the paths appear to overlap or double-back on themselves. As we saw in the solution where the Start hexagon is used twice: you can land on the same hexagon multiple times. We will need to be careful when examining the sequence of moves in the remaining examples, since the order isn't completely obvious.
In the fourth example, we jump 9 – 7 – 5, and then we could jump 6 – 7 or 7 – 6. To match the string of numbers given, you need to jump in the order 6 – 7. Notice that the black line connecting the hexagon jumps is bolder between the 5 and the 6. This must mean that path is used twice: once to jump down from the 5 to the 6, then again on the way up to the 7. This indication of when paths are used twice will be helpful when tracing the jumps in the remaining examples.
On the last jump up to 7 in the fourth example, we obtain a total of 10. Apparently this is too big, as 10 is in red and the solution attempt ends. Similarly, in the fifth example, we obtain a total of negative 10, and this is apparently too small, as negative 10 is in red and this solution attempt also ends.
How big or small is still acceptable? We've seen poitive 9 in every example (and the solution), since that number is on the starting hexagon. 9 is ok, but 10 is too big. In the sixth example, we obtain negative 9 in the middle of the solution example, and the example continues (only to end when obtaining negative 13). This must mean that negative 9 is ok, but negative 10 is too small. Thus, the final rule is: at no time when traveling through the puzzle can your total be greater than 9, nor can it be less than negative 9.
Finally, these rules were all combined in a concise way to form the instructions for the puzzle.