11 plus Number Series questions

What a Number Series question looks like

Number Series is one of the sequence puzzles that 11 plus papers use to test reasoning with numbers rather than arithmetic on its own. A short run of numbers is printed, one of them is replaced by a bracket, and five possible answers are offered. The rule behind the run is always something a child can do in their head or on the edge of the page: add the same amount each time, take the same amount away, double, halve, multiply, or something a little cleverer built out of those.

The missing number is usually at the end, but not always. Some runs hide a number in the middle and a few hide the very first one, so “what comes next” is not quite the whole story. When the bracket is not at the end, the number your child picks has to fit the numbers on both sides of it. Here are three worked through in full.

The question. 7, 14, 21, 28, (?) with the options 35, 36, 34, 42 and 28.

The method. Write the gaps underneath. From 7 to 14 is 7, from 14 to 21 is 7, from 21 to 28 is 7. Every gap is the same, so the rule is add 7, and one more step takes 28 to 35.

The answer. 35.

Why a wrong option tempts. 42 is the trap. It is in the same times table and it is the very next number after the right one, so a child reciting “7, 14, 21, 28, 35, 42” under their breath runs one step too far and lands on it.

The question. 3, 6, 12, 24, (?) with the options 48, 42, 36, 60 and 30.

The method. The gaps here are 3, then 6, then 12, so they are not equal and the rule is not adding. Ask what each number does to become the next one instead. 3 doubled is 6, 6 doubled is 12, 12 doubled is 24. So double 24 as well.

The answer. 48.

Why a wrong option tempts. 36 is 24 plus 12, the last gap used one more time. It is a reasonable thing to try, and it is wrong for a reason worth saying out loud: when the gaps are getting bigger, the gap you just measured is never the one you need next.

The question. 10, 17, 13, 20, 16, (?) with the options 24, 22, 23, 12 and 30.

The method. The numbers go up, then down, then up, then down, so write the direction with each gap. From 10 to 17 is up 7. From 17 to 13 is down 4. From 13 to 20 is up 7. From 20 to 16 is down 4. Two rules are taking turns. The last move was a down 4, so this one is an up 7, and 16 plus 7 is 23.

The answer. 23.

Why a wrong option tempts. 12 is 16 take away 4. A child who has spotted both rules and simply lost track of whose turn it is lands there, which is why it is worth marking the ups and downs on the page rather than holding them in your head.

The harder questions are built from the same parts. In 1, 3, 6, 10, 15 the gaps grow steadily as 2, 3, 4, 5, so the next number is 21. In 4, 5, 7, 11, 19 the gaps double as 1, 2, 4, 8, so the next number is 35. And 7, 34, 9, 30, 11 is two sequences woven together, 7, 9, 11 going up 2 and 34, 30 going down 4, so the next number is 26.

How to answer it

1. Write the gap between each pair of numbers underneath the sequence, with an arrow or a plus and minus sign to show the direction. Most of the thinking is done once those numbers are on the page.
2. If every gap is the same, you are finished. Carry it on one more step, and check the direction: sequences that count down are as common as sequences that count up.
3. If the gaps are not the same, ask whether each number is being multiplied or divided instead. Doubling, halving, times 3 and divide by 3 are easy to miss if you only ever look at gaps.
4. If it is neither, look at what the gaps themselves are doing. They may be growing by the same amount each time, doubling, or following 1, 4, 9, 16, 25. If the numbers jump up and down, check whether two rules are taking turns, or whether every other number makes a sequence of its own.
5. Test the rule on the whole run before choosing. If the bracket sits in the middle or at the start, the number you pick has to work on both sides of it, which sometimes means running the rule backwards.

Common mistakes

• Adding the last gap when the numbers are being multiplied. In 3, 6, 12, 24 the last gap was 12, so 36 feels right and is offered. The numbers are doubling, and the answer is 48.
• Reciting a times table one step too far. In 7, 14, 21, 28 both 35 and 42 are offered, and a child counting quickly in sevens says one number too many.
• Losing the turn in an up and down run. In 10, 17, 13, 20, 16 the next move is up 7, not down 4, but 12 is there for anyone who applies the wrong half of the pattern.
• Letting a growing gap stop growing. When the gaps run 2, 3, 4, 5, the next one is 6. Using 5 again gives an answer that is one short, and one short is usually an option.
• Assuming the number is always the last one. A run printed as (?), 17, 24, 31, 38 is asking for the number before 17. The rule is add 7, so the answer is 17 take away 7, which is 10.

How to practise 11 plus Number Series questions

A little every day beats a long session at the weekend here, because the whole skill is a habit: gaps written underneath, direction marked, rule tested against the whole run before an answer is chosen. A child who has met doubling, halving, growing gaps and woven sequences a few times each stops being surprised by them. 11+ Daily has a Number Series project among its verbal reasoning skills, with questions original to 11+ Daily rather than taken from past papers and a written explanation of the rule after every answer, so a wrong one turns into the next thing your child learns. The Today screen decides when the skill comes round again, which keeps it in rotation with the other eleven plus reasoning work instead of being drilled on its own. You can see where it sits on the 11 plus verbal reasoning page, and there is more on session length in our guide to how much 11 plus practice a child needs each day.

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