On a clearing, ladybugs gathered. If a ladybug has six spots on its back, it always tells the truth, and if it has four spots, it always lies, and there were no other ladybugs on the clearing. The first ladybug said: "Each of us has the same number of spots on our backs." The second said: "All of us together have 30 spots on our backs." - "No, all of us together have 26 spots on our backs," - the third one objected. "Exactly one of these three told the truth," - each of the other ladybugs declared. How many ladybugs gathered on the clearing?
If the first ladybug tells the truth, then the second and third ones should also tell the truth, as they should have the same number of spots on their backs as the first one. But the second and third ladybugs contradict each other, so at least one of them is lying. Therefore, the first ladybug is also lying.

Suppose each of the first three ladybugs lied, then all the others lied as well, since none of these three told the truth. This means all the ladybugs are liars. But in this case, the first ladybug did tell the truth, which cannot be. Therefore, the first three ladybugs cannot lie simultaneously, so either the second or the third told the truth, while the other two are liars. Thus, each of the other ladybugs told the truth.

Therefore, there are two ladybugs with four spots on their backs, and several ladybugs with six spots on their backs, and in total, the spots on the backs of all the ladybugs are either 30 or 26. But \(30 - 2 \cdot 4 = 22\), which is not divisible by 6, so the total number of spots is 26, and the number of honest ladybugs is \((26 - 2 \cdot 4) : 6 = 3\). Therefore, there are five ladybugs on the meadow.

## Answer

5 ladybugs.