Conjugacy problem
Okay, so let's imagine you have two different shapes made out of blocks, like Legos. One shape is a tower made up of 4 blocks in a straight line, and the other shape is a pyramid made up of 3 blocks with one block on top.
Now, let's say you want to know if you can move the blocks around to make the pyramid shape look like the tower shape. This is called the "conjugacy problem" - basically trying to figure out if two shapes are the same but just arranged differently.
To figure out if you can do this, you need to try moving the blocks around in different ways - maybe you can move the top block of the pyramid down to make it a straight line, or maybe you can move the two bottom blocks of the tower closer together to make it more like a pyramid.
Sometimes it is easy to tell if two shapes are conjugate (the same but arranged differently), but other times it can be very tricky and take a lot of experimenting with moving the pieces around. Mathematicians study this problem to try and find ways to make it easier to tell if two shapes are conjugate or not.
Now, let's say you want to know if you can move the blocks around to make the pyramid shape look like the tower shape. This is called the "conjugacy problem" - basically trying to figure out if two shapes are the same but just arranged differently.
To figure out if you can do this, you need to try moving the blocks around in different ways - maybe you can move the top block of the pyramid down to make it a straight line, or maybe you can move the two bottom blocks of the tower closer together to make it more like a pyramid.
Sometimes it is easy to tell if two shapes are conjugate (the same but arranged differently), but other times it can be very tricky and take a lot of experimenting with moving the pieces around. Mathematicians study this problem to try and find ways to make it easier to tell if two shapes are conjugate or not.