Hunting the high seas

by Jason Davison, Mathematical Magician

Victorian Computer Scientists, Ada Lovelace and Charles Babbage worked to try to automatically calculate nautical charts. In this pirate treasure themed magic effect you are able to mysteriously guide your spectators freely chosen moves over a map to find the hidden booty.

Hunting the high seas - What your spectator sees on the sea

You present to your spectator a gridded treasure map showing sea, sand, more sea, islands, monsters ... in fact whatever you fancy drawing. The word 'start' is marked on a central square. Place a counter on the start. Explain to the spectator that they are about to take a random wander around the map to try and find some hidden treasure, it is up to them to make the right choices to track the treasure down.

Much more than one direction

Show them a stack of cards with a variety of compass directions on them (North, North East, East, and so on). Cut the pack a few times and then ask the spectator to cut the cards anywhere so they can start with a completely random first move. Let them take the top card and move the counter in the direction given by one grid square, e.g., one step North. Mention it would have been impossible to predict what that first step would be. Deal them eight more cards from the top and have them mix them up. One by one they should pick a direction card at random from their set that leads to the counter moving in a random path around the board.

The digging done did the distant dithering dance deliver?

Once they have finished their cards take off the counter and stab a pencil through the final square they landed on, this is the dig to find the treasure. Hold the pencil with the map skewered on it and show them the map from the other side. They will see the pencil has skewered a cartoon of a treasure chest perfectly. They have wandered but have still found the treasure! Unfathomable?

The method making the map

Once you know the secret you can create your own custom treasure maps. Here we give a simple example. On your gridded treasure map draw a compass with NSEW, choose a start square somewhere near the middle, then find the square that is say 1 square south and 3 squares east of the start. This is where you should draw your treasure chest on the opposite side of the paper.

The mathematical process that moves to the magic

Now create a set of cards with compass directions on them. The stack of cards should be a multiple of 9. (9, 18, 27, ...) The cards should cycle through this set of 9 however many times you like: E. E. N. NE. SE. W. S. W. SE.

The trick is now entirely self working! The first useful fact is that cutting the deck will not change which cards are in each stack of 9. The second and most important fact is that vector addition is commutative. What does that mean? A vector is just a distance combined with a direction, like "move West one step". Adding vectors just means following one such instruction then another. It gives you a new vector that just takes you a different distance in some new direction. Adding moving a step North to a step East gives you a new vector of taking you North East.

What about 'commutative'? That is about it not having an effect if you swap the order of things round. So, all that it means for vector addition to be commutative is that it doesn't matter what order you add a set of vectors in (ie moves in given directions can be done in any order). You will always end up at the same overall vector - ie the sequence of moves will always take you to the same place. Going North a step and then East a step takes you to the same place as first going East and then North.

So for the example set of directions we used here - E. E. N. NE. SE. W. S. W. SE. These vectors combine to give a resulting vector displacement of 1 square south and 3 squares east of the start, as required to find the treasure on the back of the map, and the mathematical principle of commutativity means it doesn't matter what order these 9 instructions are taken in.

You are welcome to create an alternative set of cards and end point to experiment with this crucial and amazing maths property yourself. You could use a different presentation using a local street map and a start point and end point that have some meaning, eg moving between your house and a friend's, or escaping from their house only to find they end up drawn unswervingly to the local school!

The power of commuting

The property of a set of entities being commutative or not being commutative is fundamental to forming and developing many theories underlying computer science. Truth tables and mathematical proofs often require that a substitution is made and that substitution is only possible because the entities commute, so it is a useful mathematical trick used in many an adventure on the high seas of logic.

We are always looking for clever new ideas, so If you create your own magic mathematical or computer tricks why not share them with us at cs4fn. We will publish the best.

Author author

This original trick was created by Jason Davison. Jason is a creator and lecturer of mathematical magic. He is featured on the YouTube channel Numberphile and has filmed video resources for the Manual of Mathematical Magic, and has worked with us on the Math Made Magic and CS4FN books. He is keen on using his passions in mathematics and magic to find new ways to guide people into mathematical explorations