Number Series and Patterns
The difference ladder — one method that cracks almost every series
Free trail · Analytical · ages 11–12 · 6 stations
The Difference Ladder
Guessing at a series is a waste of time. Use the difference ladder: write the gaps underneath the terms, then the gaps of those gaps, until a row goes constant.
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Row 1 is the series. Row 2 is the gap between neighbours. Row 3 is the gap between those gaps. Keep going until one row is constant — that row names the rule.
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The series is arithmetic: it adds a fixed amount. Term n = first term + (n − 1) × gap, so you can jump to term 100 without listing anything.
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The rule involves n². Squares, triangular numbers and expressions like 2n² + 1 all have a constant third row. Two rows down means squared.
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If the gaps grow faster than the terms themselves, stop laddering — the rule is multiplication. Divide neighbouring terms instead and look for a constant ratio.
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The Difference Ladder
Flip cards
Guessing at a series is a waste of time. Use the difference ladder: write the gaps underneath the terms, then the gaps of those gaps, until a row goes constant.
One reliable method beats a hundred guesses. Flip every card to learn it.
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Run the Ladder
Try it
Work the ladder on paper before you look at the options — the options are designed to tempt anyone who guessed.
Write the gaps first. Only predict once a row has gone constant.
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Watch the Ladder Settle
Slider explorer
This series follows 2n² + 1. Slide to add terms and watch how the gaps themselves grow by a fixed amount — the fingerprint of a squared rule.
Slide to add terms and watch the second row of gaps go constant.
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Name the Rule Family
Sorting baskets
Half the battle is naming the family. Sort each series by the kind of rule running underneath it.
Before solving, decide which family the series belongs to.
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Find the Broken Term
Try it
Here the rule holds for every term but one. Build the rule from the terms that agree, then hunt for the one that does not fit.
One term breaks the rule. Establish the rule from the majority, then test each term.
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Practice Zone
Practice Zone
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More Analytical for ages 11–12
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