Problem Solving Heuristics

Stage 22 of 23 Strand 4 of 4 4 lessons

4 illustrated lessons, each teaching the why before the how.

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Working Backwards from the Goal #

Ask what would produce the answer you want.

Starting at the goal and asking what would produce it turns an open search into a short chain of steps

Instead of pouring at random, ask which state would give you the answer straight away.

Four is 5 − 1, so the question becomes how to take exactly one unit out of a full jug.

That state is reachable, so the chain is complete and can be read forward as the solution.

Working backward narrows the search, because only some states could produce the goal.

Now you

Why undo the operations in reverse order?

A number is doubled, then 7 is added, giving 23. What was it?

Shrink it, find the method, then scale up.

Shrinking a problem until it can be solved by hand often exposes the method the full problem needs

A 20-sided polygon has far too many diagonals to draw and count reliably.

Draw the small cases instead. Four, five and six sides can all be counted by hand.

The small cases show the reason: each corner has n − 3 diagonals, and each is counted from both ends, so n(n − 3)/2.

Check the rule against a case you counted, then use it on the case you could not.

Now you

What is the point of solving a smaller version first?

You find a rule from three small cases. What must you still do?

A pattern noticed, then a pattern earned.

Checking special cases tests a result, and only a general argument proves it

Add the odd numbers in order and the totals keep coming out as squares.

That is a conjecture. The cases make it worth believing, but they do not prove it.

An n by n square is built from L-shaped layers, and each layer holds an odd number of dots.

That argument covers every n, so the pattern is now proved for all of them.

Now you

What does a special case of a proved result tell you?

You have checked a claim for n = 1 to 10. What do you have?

A false ending means one step was not earned.

A false conclusion means some step is unjustified, and reading an argument means finding which one

Every line so far follows from the one above, and each move is stated plainly.

The conclusion is false, so some step is unjustified, and only one of them is.

Since a = b, the factor a − b is 0, and dividing by 0 is the one move not allowed.

A few moves account for most flaws, so check those first when a conclusion is absurd.

Now you

An argument uses the statement it is trying to prove. What is that called?

From x² = 9 an argument concludes x = 3. What is missing?