A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:
There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?
As shown in the image, the nine Dogs are square fenced. By constructing just two square fences can you make sure that two Dogs cannot meet each other without crossing the fence?
The king of Octopuses has servants who have six, seven or eight legs. The distinguishing characteristics of the servants is that the one with seven legs always lie but the one with either six or eight legs speak the truth always.
One day, four servants meet and converse:
The black one says, 'We have 28 legs altogether.'
The green one says, 'We have 27 legs altogether.'
The yellow one says, 'We have 26 legs altogether.'
The red one says, 'We have 25 legs altogether.'
Can you identify the colour of the servant who is speaking the truth?
How many triangles can you count in the figure given?
