Proofs To Make You Go WOW

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For those of us who love maths, it's sad to see how many people hate it, or have a phobia of it, or proudly announce "I was never very good at maths at school." The question we often counter with is "Would you be so proud of being unable to read?"

But what's true is that despite the dedication and best efforts of maths teachers in schools all over the world, many people are left with no real appreciation of what the subject is, or of the beauty that's there to be experienced.

So what would be great would be to have a collection of activities that have, at their heart, proofs. Proofs to make you go "Wow!"

Wouldn't it be great if just for one lesson a year a teacher had a resource that let them present an age-appropriate, curriculum appropriate proof that was accessible, engaging, and genuinely beautiful.

Maybe that's not possible, but we can try. Let's gather together a collection of candidate proofs, then bundle them to be accessible to teachers so they can let the students see beyond the sometimes motiveless manipulations and tedious tinkerings. We know that teachers want to instill a sense of awe, wonder, and enthusiasm in their students, so let's try to provide the resources they need to do exactly that.

The proofs

So we need a collection of proofs to be bundled and presented, to be made accessible and useable for time-pressed teachers. At this stage we don't know exactly what will fly, or what will best engage the students, but we can start by collecting candidates, and then working to attach them to the curriculum and thereby determine the best age group for them.

We need candidates.

Some Suggestions ...

So here are some suggestions:

  • The proof that a 10x10 board cannot be covered by 1x4 strips
    • Possibly grade 5
  • Euclid's Proof of the Infinitude of Primes
    • Possibly grade 6
  • Pick's Theorem
    • Possibly grade 8
  • A proof without words to Nicomachus theorem
    • Possibly grade 9

Others, currently ungraded:

  • A chessboard with opposite corners removed cannot be covered exactly by dominoes of size 2x1
    • But any chessboard with one black and one white removed can always be so covered
  • For every number N, (N-1)!+1 is a multiple of N if and only if N is prime. (Wilson's Theorem).
  • If you take two copies of any map, scrumple one, and place it on top of (and inside the boundary of) the other, then there is at least one point that "matches". (Brouwer's Fixed Point Theorem).
  • Any "doodle" that can be drawn in a single pen-stroke can be drawn without crossing over a line already drawn. More, if you return to where you started, the resulting division of the plane can be coloured with two colours.
  • Any prime of the form 4k+1 can always be written as the sum of two squares, but any prime of the form 4k+3 cannot.
  • Given any six people, either there are (at least) three who have all shaken hands, or there are "at least) three none of whom have shaken hands.
  • The number e is irrational
  • There exist transcendental numbers
    • Not via diagonalisation
  • Any traversable network has all nodes of even degree (except perhaps 2), and any connected network with all nodes of even degree is traversable (has an Euler Cycle)
  • Construction of a regular pentagon.

A challenge for you ...

So, what mathematical proof could you present to a six-year-old? Or an eight year-old? What is there in the school curriculum that you think has a lovely proof that the students really ought to get to experience?

Let me know.


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