How to solve NumberLogic questions

Find the next number. Check differences, second differences, ratios, alternating patterns and Fibonacci-style sums.

How to solve them

Arith

1. Cue: neighbours differ by the same amount.
2. Method: subtract neighbours. If every difference is equal, add it to the last term.
3. Mental math: find the step from the last two terms only (like 47 - 38 = 9), then add it.
4. Check order when stuck: differences, then second differences, then ratios, then odd/even positions, then sum of the previous two, then (x m) + c. Skip after about 45 seconds.

Quad

1. Cue: differences change, but they change by the same amount each time.
2. Method: write the first differences, then the second differences. If the second differences are constant, extend the differences row, then add back.
3. Mental math: next difference = last difference + second difference. Next term = last term + next difference.
4. Check order when stuck: differences, then second differences, then ratios, then odd/even positions, then sum of the previous two, then (x m) + c. Skip after about 45 seconds.

Geo

1. Cue: terms grow fast, like doubling or tripling.
2. Method: divide neighbours. If the ratio is constant, multiply the last term by it.
3. Mental math: powers of 2: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. Powers of 3: 3, 9, 27, 81, 243, 729. Times 3 is the number doubled, plus itself.
4. Check order when stuck: differences, then second differences, then ratios, then odd/even positions, then sum of the previous two, then (x m) + c. Skip after about 45 seconds.

Alt

1. Cue: differences zigzag, like a big step then a small step.
2. Method: split into odd and even positions and treat each as its own sequence. The next term belongs to the sequence whose turn it is.
3. Mental math: for the 7th term use positions 1, 3, 5, then add that step.
4. Check order when stuck: differences, then second differences, then ratios, then odd/even positions, then sum of the previous two, then (x m) + c. Skip after about 45 seconds.

Fib

1. Cue: each term is about 1.6 times the last, and the differences look like earlier terms.
2. Method: test whether each term is the sum of the previous two on the first three terms.
3. Mental math: just add the last two terms.
4. Check order when stuck: differences, then second differences, then ratios, then odd/even positions, then sum of the previous two, then (x m) + c. Skip after about 45 seconds.

Affine

1. Cue: neither the differences nor the ratios are constant, but the ratio hovers near 2 or 3.
2. Method: guess the multiplier m from the ratio of the first two terms. Find c = term2 - m x term1. Test the rule on the next pair.
3. Mental math: the leftover c is small, so try m = 2 or 3 first.
4. Check order when stuck: differences, then second differences, then ratios, then odd/even positions, then sum of the previous two, then (x m) + c. Skip after about 45 seconds.

Square

1. Cue: second differences are constant and equal to 2.
2. Method: the terms are near perfect squares. Subtract the nearest square from each term. If the leftover is constant, that is the offset.
3. Mental math: squares 1 to 20: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.
4. Check order when stuck: differences, then second differences, then ratios, then odd/even positions, then sum of the previous two, then (x m) + c. Skip after about 45 seconds.

AltOps

1. Cue: the terms alternate between small jumps and big jumps.
2. Method: look at consecutive pairs. One step adds, the next step multiplies (or similar). Find both operations, then see which one comes next.
3. Mental math: the multiply step is often x2, and the add step is a small constant.
4. Check order when stuck: differences, then second differences, then ratios, then odd/even positions, then sum of the previous two, then (x m) + c. Skip after about 45 seconds.

Worked examples

• What comes next? 70, 64, 58, 52, 46, 40, ?

  Answer: 34

  1. First differences: -6, -6, -6, -6, -6. All equal -6.
  2. Next = 40 - 6 = 34.

• What comes next? 19, 28, 39, 52, 67, 84, ?

  Answer: 103

  1. Remove 3: 16, 25, 36, 49, 64, 81. These are perfect squares.
  2. They are 4^2, 5^2, ..., 9^2.
  3. Next = 10^2 + 3 = 103.

• What comes next? 4, 12, 36, 108, 324, 972, ?

  Answer: 2916

  1. Ratios between neighbours: 12/4, 36/12, 108/36, 324/108, 972/324. All equal 3.
  2. Next = 972 x 3 = 2916.

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