How to spot the rule in a number sequence

What comes next after 3, 6, 9, 12? Most people say 15. Each step adds three, so the answer follows the pattern. The useful part is not the number 15 alone. It is the sentence “add three each time.” That sentence gives you a rule you can check against every number in the row.

Number sequences are small reasoning puzzles. You look for a rule, test it, and explain why it works. The University of Cambridge's NRICH maths project recommends going beyond spotting a pattern to asking why it occurs. That question helps children see the mathematical structure rather than only guess the next item (NRICH, 2014).

Start with the gaps

Write the difference between each pair of neighbors. For 4, 7, 10, 13, the gaps are +3, +3, +3. For 2, 5, 10, 17, the gaps are +3, +5, +7. That second row does not add the same number each time, but the gaps themselves follow a pattern: they go up by two. The next gap would be +9, making the next number 26.

Try another: 20, 18, 16, 14. The gaps are -2 each time. You can say “subtract two,” rather than guessing 12 because it looks smaller. Now check the rule against the whole row: 20 - 2 = 18, 18 - 2 = 16, and 16 - 2 = 14. If one step fails, look again.

A child who is learning this can draw arrows between the numbers and write the change above each arrow. The Institute of Education Sciences recommends visual representations and asking students to monitor their problem-solving steps. Both are useful here: the arrows make a possible rule visible, and checking every arrow tests it (IES, 2012).

If the gaps do not help, try another view

Some sequences multiply instead of add: 2, 4, 8, 16 doubles each time. Other rows switch between two actions. For example, 2, 5, 4, 7, 6, 9 alternates between “add three” and “subtract one.” If you only inspect the first two numbers, you miss the second action. Look at enough steps to see whether the idea repeats.

Another useful move is to split odd and even positions. Consider 1, 10, 2, 20, 3, 30. The first, third and fifth entries count 1, 2, 3. The second, fourth and sixth count 10, 20, 30. The next entry would be 4. This row has two smaller patterns woven together.

No single trick works every time. Start with differences because they are easy to check. Then try multiplication, alternation or two interleaved rows. If nothing fits, ask whether you made a copying error. It is fine to pause instead of picking an answer that you cannot explain.

More than one rule can fit

A short row does not always have one certain continuation. Take 1, 2, 4. “Double each time” gives 8 next. “Add one, then add two, then add three” gives 7. Both fit the three numbers shown. A puzzle needs enough information, or a set of answer choices, to make its intended rule clear.

This is why explaining a rule matters more than blurting out a number. Ask, “Does your rule fit every step?” Then ask, “Could another rule also fit?” These are fair questions, not traps. NRICH encourages children to create and explain patterns, and the IES guide recommends exposing students to multiple strategies for a problem (NRICH, 2014; IES, 2012).

Make your own sequence

Choose a rule first. It might be “add four,” “double,” or “add two, then add five.” Write five numbers from it. Give the row to someone else without the rule. If they find a different answer, see whether their rule fits all five entries. You may need a sixth number to make your idea clearer.

This small exercise changes the child from a guesser into a puzzle maker. They must work forward to build a fair row and backward to check another person's explanation. There is no score to chase. A pencil and scrap paper are enough.

Try a round

What Comes Next asks you to find a rule and pick the next number. Use the same routine: inspect the gaps, test a possible rule across the row, and then choose. If you practise, your scores in this game may improve. That shows progress on its number-sequence puzzles; it does not prove a broad change in reasoning outside the game.