MindProofMindProof · Puzzle guides
How to solve number series puzzles
Learn a practical method for number series: differences, ratios, alternating patterns and checked examples you can play.
A number series puzzle shows a short list and asks you to choose a plausible next term. Begin with a simple rule that explains every displayed step. Checking the entire sequence matters more than spotting a relationship between the last two numbers.
A finite sequence can support many different rules. These examples explain the intended pattern; they do not prove that no other continuation exists. The answer choices and a clear explanation help establish the puzzle’s convention.
A repeatable solving method
- Write the difference between each pair of neighbors. Equal gaps suggest addition or subtraction.
- If the gaps vary, compare the gaps themselves. They may grow steadily or form squares.
- Check multiplication and division, especially when values grow or shrink quickly.
- Look for familiar number families, such as squares and cubes.
- For alternating patterns, separate odd and even positions or check for a repeating cycle of operations.
- Calculate the proposed next term, then go back and test your rule against every earlier term.
Worked examples
1. A constant difference
1 → 3 → 5 → 7 → 9 → ?
The differences are all 2: 3 − 1 = 2, 5 − 3 = 2, 7 − 5 = 2 and 9 − 7 = 2. Continue the same step: 9 + 2 = 11.
2. Growing differences
3 → 8 → 14 → 21 → ?
The gaps are 5, 6 and 7. Increase the gap once more to 8: 21 + 8 = 29. Adding the last gap again would give 28, but that would stop the established progression.
3. A constant ratio
162 → 54 → 18 → 6 → ?
Divide each term by 3: 162 ÷ 3 = 54, 54 ÷ 3 = 18, and 18 ÷ 3 = 6. Next comes 6 ÷ 3 = 2. The differences are changing, so subtraction is a less direct description here.
4. Add the previous two terms
1 → 2 → 3 → 5 → 8 → 13 → ?
From the third term onward, each term is the sum of the previous two: 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8 and 5 + 8 = 13. Next: 8 + 13 = 21.
5. Alternate two operations
5 → 15 → 18 → 54 → 57 → 171 → ?
Alternate ×3 and +3: 5 × 3 = 15; 15 + 3 = 18; 18 × 3 = 54; 54 + 3 = 57; 57 × 3 = 171. The next operation is +3, giving 174.
6. Interleave two sequences
1 → 60 → 5 → 100 → 9 → 140 → ?
Read positions 1, 3 and 5: 1, 5, 9 increase by 4. Positions 2, 4 and 6 are 60, 100, 140 and increase by 40. Position 7 belongs to the first sequence, so 9 + 4 = 13.
Common mistakes
- Using only the final gap. Write all gaps before extending one.
- Ignoring operation order. “Multiply by 2, then add 2” differs from “add 2, then multiply by 2.”
- Forcing one operation across every step. Alternating sequences may need two operations or two separate position tracks.
- Choosing a complicated rule too soon. Check addition, multiplication and standard number families first. A rule should explain the observations, not just produce an available answer.
Practice with explanations
Open Number Series, choose an answer, and read the worked steps. For a new challenge, try doubling gaps or adding three previous terms. If you prefer building an expression, read the Target Number guide.