Alex Kidd in Miracle World

The telepathy ball can be found in Stage 7 and is portrayed as a blue ball with a red star on it. You'll see it at a lower level, next to some lava. To get it, stand on the ledge to the top-left of it, and hold right. You'll drop down, but miss the lava. Then move further right and the wall next to the second patch of lava will disappear, revealing an exit.

You only need to lose one round of rock-paper-scissors while using the telepathy ball, not the entire game.

If you're having issues with the Janken matches, for this trophy or otherwise, the solutions can be found below (thanks to jimmylad):

1) Rock, Scissors
2) Scissors, Paper
3) Rock, Scissors
4) Paper, Paper
5) Rock, Rock
6) Rock, Scissors
7) Paper, Paper