Hello Ferk, This is a good question. "Take on" and "take up" really mean quite different things, but in this case you can use either. I would define them as: "Take on" means either 1. To confront ("I am going to take this bully on") 2. To accept some long-term piece of work ("I've just taken on a new project"). You could think of this as "taking on board", as a ship might take an extra piece of cargo on board. "Take up" means to accept something that is offered to you - 90% of the time this is indeed some kind of offer. Eg. "You kindly offered me a lift home earlier. May I take you up on that?" You can probably use "take ON the challenge" in all cases. You could "take UP" the challenge if you wanted to acknowledge that someone had offered the challenge to you.