This is my favourite investigation to help pupils understand the basic processes involved in carrying out an investigation, EXPERIMENT, HYPOTHESISE, SIMPLIFY, STRUCTURE and FEEDBACK are all important elements of this.
I use it mainly with year 7's but even some 6th form pupils have shown an interest in the past.
The Power Point can be found here and the only other things you will need are mini whiteboards and some pens or pencils and paper for recording results.
I usually introduce the investigation as something I was working on and I don't really want to tell them about it as there is a big reward from my uncle if I can solve the problem. "Reluctantly" I show them it and let them play around with the objective for a while, you will find it becomes very competitive very quickly and within minutes someone will have solved it. Until you point out that they have drawn the plan incorrectly.
Soon someone will ask "Is it possible" or just state outright "It's impossible" in either case my response is "I'm glad you asked/say that" as it just so happens that my uncle will still give me the millions if I can prove why it's impossible.
From here the investigation can take on many forms, pupils can develop a systematic approach of listing the number of walls and doors, trying easier shapes, try to find any other shapes that appear to be impossible, you are really only bound by your imagination at this point.
I may add some examples of pupil work in the future, but for now see how far you can take it.