boxes (and eggs?) of hanoi
Why not make Towers of Hanoi a bit more fun? Put three tables in a row. You have 3 objects in a stack on a table, and you need to move them from one table to another. One of the objects is a heavy box (at the bottom!), one is an smaller and lighter one (that would be squashed by the heavy box) and the other is a really small/light squashable object (an egg?). If you can't find an two boxes and an egg (!) what else could you use? Maybe 3 different sized boxes that fit inside each other, like Russian dolls? You can only move one object from table to table at a time. How can you get the stack from one table to another without crushing anything! What is the minimum number of moves? You could play this with cards, say J,Q,K  and stacks must be in the order top to bottom J, Q, K as shown in the gallery below 

Do you notice anything about how each card moves when you do this in the best possible way? Just watch the Jack only, then the Queen, then the King. What do you notice?
Interestingly the best sequence of moves is related to binary, and can also be thought of as moves around a cube. This is related to the Icosian game, invented by Hamilton, which is related to the idea of traversability.
Playing with 4 cards is related to moves around a hypercube; for more details see the Martin Gardner book 'Hexaflexagons and other mathematical diversions'. Can you find the sequence of moves for n cards?
This reminds me of a nice simple card trick; lay out 4 piles of cards: A,K,Q,J in each. Then collect the cards up and ask volunteers to cut them. Then deal them out as you would when dealing cards to 4 players. What do you notice? Why?
Interestingly the best sequence of moves is related to binary, and can also be thought of as moves around a cube. This is related to the Icosian game, invented by Hamilton, which is related to the idea of traversability.
Playing with 4 cards is related to moves around a hypercube; for more details see the Martin Gardner book 'Hexaflexagons and other mathematical diversions'. Can you find the sequence of moves for n cards?
This reminds me of a nice simple card trick; lay out 4 piles of cards: A,K,Q,J in each. Then collect the cards up and ask volunteers to cut them. Then deal them out as you would when dealing cards to 4 players. What do you notice? Why?