Finding the Rule

One pair is never enough to find a rule. Here one goes to four, which could be add three — but two goes to seven, and two plus three is five, so add three is out. Try multiply then add instead: one times three plus one is four, two times three plus one is seven, three times three plus one is ten. It fits all three, so now you can trust it, and five times three plus one is sixteen. The first guess that fits a single line is almost never the rule, and the wrong options on this paper are built from exactly those near-misses.

When the numbers are laid out in a grid, check whether the rule runs down the columns as well as across the rows. In this square each row multiplies — two twos are four, seven fives are thirty-five — and so does each column: two sevens are fourteen, and four times thirty-five is a hundred and forty. That gives two different routes to the bottom-right cell, and they have to agree. That agreement is a free check on your work, so use it: if the two routes give different answers, the rule you found is not the right one.

With a partly complete grid, do not start at the cell you have been asked for. Start wherever you have the most information and work towards it. Row one has both of its first numbers, so the top right is three times four, which is twelve. Row two already has its answer, so five times something is ten and the middle cell is two. Now the columns fill the bottom row: three fives are fifteen, and four twos are eight. That makes the star fifteen times eight, which is a hundred and twenty — and twelve times ten also gives a hundred and twenty, so it checks.

When the gaps are all different, write them down and look at the gaps between them. Two, three, six, eleven, eighteen gives gaps of one, three, five and seven — the odd numbers, each two more than the last. So the numbers are not doubling; the gap is growing by a fixed amount. The next gap is nine, and eighteen plus nine is twenty-seven. Three things to try, in this order: is the gap the same, is each number double the one before, and are the gaps growing steadily. Always write the gaps above the numbers, because staring at the sequence does not work.

Sometimes the options are sentences rather than numbers, and then the fastest method is to test each sentence against the second step rather than the first. Here add one each time fits the jump from two to three and then dies. Double it fails immediately. Multiply by two and subtract one survives two to three, and even survives three to five — except the sequence goes to six, so it fails there. Add the odd numbers in turn fits every gap, so that is the rule. Every wrong option is designed to survive the first step, which means the first step tells you nothing at all. Quiz and exercises now.