Systematic listing strategies

Options evening. Two subjects to pick from four, and you are already writing out the pairs. Then the head of year says one line. Geography and History are timetabled at the same time. You cannot take both. Your counting was fine. One sentence you nearly skipped has just removed a pair. So there are two jobs here. List every possibility in an order that proves you missed nothing, then check each one against the rule you were given.

This is video four of five in our chapter on factors, multiples and counting.

Start with no rule attached. Using the digits three, five and eight, list all the different two digit numbers you can make. Each digit used at most once.

The tempting way is to write whatever comes into your head. Thirty five. Eighty three. Fifty eight. Quick, until you lose track. Did you already do thirty eight? That is how one gets counted twice, or missed.

So never vary two things at once. Fix the first digit. Cycle the second. Three at the front gives thirty five, then thirty eight.

Your turn, and it is a small one. Fix five at the front. Which pair comes next? A, fifty three and fifty eight. B, fifty three and eighty five. C, thirty five and fifty eight.

The answer is A. Fifty three and fifty eight. First digit stays put, second one changes.

Here is why a fixed order matters, rather than just looking tidy. With the first digit locked, there is only one thing left to track. The order remembers for you. And you know you have finished when every digit has had its turn at the front. Nothing to trust, nothing to recall.

Full list. Thirty five, thirty eight. Fifty three, fifty eight. Eighty three, eighty five. Six numbers, none repeated, none hiding.

That was the easy half. Almost every listing question you meet has a rule attached, and reading that rule is the harder job.

A cafe offers three sandwich fillings. Ham, cheese and egg. On either white or brown bread. But ham is never available on brown bread, because the brown loaves run out before the ham delivery arrives. List all the different sandwiches that could actually be ordered.

Before a single sandwich goes on the list, write the rule as one plain sentence. Ham can never go with brown. It is on your paper now, so it cannot fall out of your head halfway through.

Then draw the grid. Fillings down the side in a fixed order. Ham, cheese, egg. Breads across the top. White, brown. Every box is one sandwich, and the grid is your proof you looked at every one.

Now spend the rule. Ham with brown. Closed. Cross it out before you list anything, not after.

Read the survivors off in order. Ham on white. Cheese on white, cheese on brown. Egg on white, egg on brown.

So the answer is five sandwiches. Not six. The grid held six boxes and one was never on offer.

Then the last move. Walk back along the list with the rule in your hand. Ham on white, allowed. Cheese on brown, allowed. Egg on brown, allowed. Nothing breaks the sentence you wrote down.

One piece of exam craft. Examiners have written about this exact skill. The report says: A poorly answered question where the majority of students misinterpreted the given conditions in the systematic listing question in the context of timetabled subjects. Hear what that does not say. Not listed them in a muddle. Misinterpreted the given conditions. The listing was fine. The rule was misread.

So, a check on exactly that. Different cafe, different rule. Egg is never served on white. Someone hands in this list. Ham white, ham brown, cheese white, cheese brown, egg white. One item breaks the rule. Which one?

The answer is egg on white. It should never have reached the paper. That is the job of the final pass. One item at a time, against one sentence.

Three steps to take in with you. Rule first. Fixed order. Check back.

So, let us pull it together. Rule first: one plain sentence before you list anything. Fixed order: fix one thing, cycle the other, and the grid proves you looked everywhere. Check back: walk the finished list against your rule.

That was video four of five. That works while the list is short enough to write out. When a question has hundreds of possibilities, there is a way to count them without listing them. That is the next video.

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