Identifiying the Impossible

A lesson that I like to introduce the idea of thinking problems through carefully is one that I call Identifying the Impossible.  It takes a little setting up and a few bits that aren’t in most classrooms but it isn’t expensive to sort out.  You may recognise the puzzles as two are old parlour tricks that have been popular for years.  It works well for those aged around 10.  You may want to have an alternative puzzle as backup in case one of the students has seen one of the solutions before.

It is an effective lesson as the impossible problem is easy to explain why it is impossible.  Most students grasp it and realise that thinking about the problem rather than prolonged trial and error would have helped them. 

Tell the students that they have three problems to look at.  They will have 10 minutes to attempt each one.  One of the problems is impossible the other two they will see completed by the end of the session.

I talk through and demonstrate each puzzle with the children and take a limited amount of feedback from the children.  At the end of the demonstrations I ask each child which one they think is the impossible one and make a note.  I repeat this at each changeover of problems to record how their opinion of each problem changes.  After the y have tried all three I ask for their final decision.

Problem One.

Shrink or Grow. This is the old walking through a postcard trick.  I ask them to walk though a sheet of A5 paper (half of U.S. letter size).  The group have a pile of suitably sized paper and scissors.

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Here’s a video of the solution if you’ve not enountered it before:

Problem Two.

Balancing Nails.  Balance 10 nails on the head of one other nail. 

The following video gives you the best explanation of the solution here.  This fits in very well if you are likely to do any science work on centre of gravity as you can use that to explain why it works.

Problem Three.

Mutilated Chessboard.

First introduced by Gammow and Stern in their Puzzle Math Book in 1957 this is a useful way of explaining mathematical proof and was used to do this in Fermat’s Last Theorem: The story of a riddle that confounded the world’s greatest minds for 358 years
an excellent book giving a very readable and entertaining  insight to non-mathematicians of the world of professional and talented mathematicians.

Show a chessboard and ask the students how many squares it has.  Bring in square numbers, 8 X 8 etc and get the answer of 64.  Next show them a domino and explain that it covers two squares of the chessboard when placed either horizontally or vertically.  Diagonally does not cover the squares fully so is not allowed.  The students should see that it will take 32 dominos to cover the chess board.  Show this.

mutilated chessboard

Next take one domino away so there are two squares not covered.  Now tell them that they must rearrange the dominos to cover the chessboard but as one domino is missing they now no longer need to cover 2 diametrically opposite corners.  The ones marked with an X above.  If you use a real chessboard put a sticker over the corners.

This is the impossible one which is surprising to the children as it is the one that seems most likely to be possible. 

Why is it impossible?

Each domino must cover a black and a white square yet as 2 white squares have been removed that means one domino must cover 2 black squares.  Clearly impossible and is why they end up with two uncovered squares always apart from each other.

I’ve always found this a fun lesson that really gets the children talking and drives home the point that they need to think carefully about problems.  At the end talk them through how to do the paper trick and the nail balancing and let them have a go.

Mutilated Chessboard. Includes printable chessboard and dominos.  Best printed on to card.

If you buy nails ensure that the heads are flat on both sides.  Some nails are angled from the stem to the head.

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