Find the missing relation

Start with comparisons

A sequence offers ordered observations. The task is usually to propose a compact rule that explains them and then continue that rule. This is different from proving that no other continuation exists. Keeping those two tasks separate makes number puzzles more honest and more interesting.

Begin with inexpensive checks. Are the terms increasing? Are the gaps constant? Is each term a fixed multiple of the previous one? Does odd or even position matter? Write these observations down rather than cycling through guesses in your head. A short table often reveals more than a long stare.

Consider 4, 9, 16, 25. The gaps are 5, 7 and 9. The terms are also consecutive squares, starting with 2². Both descriptions suggest 36 next. They are not rival explanations here: they describe the same continuation in two different ways.

Separate observation from extension

1. Observed: the supplied gaps are 5, 7 and 9.

2. Hypothesised: each later gap increases by 2.

3. Derived: under that hypothesis, the fifth term is 35.

The first statement is directly checkable from the list. The second extends beyond the list. The third follows from the second. If your explanation collapses these stages into “obviously 35,” it hides the most important part of the reasoning.

This distinction is not pedantry. For any finite collection of terms, another rule can agree with all the supplied values and differ later. For example, adding a multiple of (n − 1)(n − 2)(n − 3)(n − 4) to a formula changes nothing at positions 1 through 4, because one factor is zero at each position. At position 5, that extra term need not vanish.

Consequently, a good open sequence question asks for a simple plausible rule. A strictly unique question needs additional restrictions, such as “the terms are consecutive squares” or “the sequence has constant second differences.”

Use remainders when size distracts

Not every number puzzle is about a sequence. Sometimes the useful relationship is a remainder. If an integer leaves remainder 1 when divided by 3, it can be written as 3k + 1 for some integer k. That description reduces a large search to a specific family of candidates.

Suppose a number between 10 and 20 leaves remainder 1 on division by 3 and is even. The first condition leaves 10, 13, 16 and 19. The parity condition leaves 10 and 16. There are still two answers. If the puzzle also says the number exceeds 12, only 16 remains. Notice how each clue earns its place.

Keep a useful checking habit

When you reach an answer, substitute it back into every condition. This is particularly helpful after mental arithmetic, where an attractive pattern can conceal a small subtraction error. Check the starting index too: n² + 1 beginning at n = 0 produces a different opening term from the same formula beginning at n = 1.

For further context, the On-Line Encyclopedia of Integer Sequences collects many documented integer sequences. A matching entry can suggest a connection, but it does not prove that its rule was intended by a puzzle author. Treat a reference as evidence to inspect, not an authority that removes the need to explain.

You have finished this route when you can distinguish the observed data, your proposed rule and the conclusion that follows. Finding the number is only one part of that achievement.