  The jigsaw percolation process, introduced by Brummitt, Chatterjee, Dey and
Sivakoff, was inspired by a group of people collectively solving a puzzle. It
can also be seen as a measure of whether two graphs on a common vertex set are
"jointly connected". In this paper we consider the natural generalisation of
this process to an arbitrary number of graphs on the same vertex set. We prove
that if these graphs are random, then the jigsaw percolation process exhibits a
phase transition in terms of the product of the edge probabilities. This
generalises a result of Bollob\'as, Riordan, Slivken and Smith.
