Fluid reasoning, oriented

6 chapters

about 9 min read

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Fluid reasoning questions show you a pattern and ask you to continue it, spot the member that breaks it, or carry a relationship from one pair to another. Nearly all of them use a short list of rules, and the fastest solvers test those rules in a fixed order rather than staring until something jumps out. The chapters cover that method, then gap and ratio series, series built from earlier terms, the odd one out, analogies, and letter series.

Each chapter opens with the short version. Tap one to read the detail.

A method for any pattern

~2 min

Test the simplest rules first, confirm a rule on every term before trusting it, and compute the answer yourself before looking at the options. Repeating patterns and picture patterns become number problems once you count or divide.

Most pattern items use one of a handful of rules: a constant gap, a constant ratio, powers, a rule that combines earlier terms, or two series woven together. Test them in roughly that order. A rule that needs special cases to fit is almost never the intended one.

Two terms fit many rules, so a rule earns trust only when it holds for every term given. In 7, 14, 28, 56 the step looks like +7 at first, but the third term breaks that reading while ×2 holds throughout. Writing a second row under the terms, the differences or the quotients, makes these checks quick and visible.

Compute the next term before you look at the choices. Wrong options are often built to fit one step of a rule, so working backward from the options invites you to accept a partial fit. If your answer is not among the options, the rule is wrong, not the list.

Some items are patterns in another form. A repeating sequence such as red, red, blue has a unit of three, so the 20th item is found by division: 20 leaves a remainder of 2 when divided by 3, so it matches the second item, red. A picture pattern becomes a series once you count each stage: 31, 43, 55, 67 tiles grows by 12, so stage 10 uses 31 + 9 × 12 = 139 tiles.

Rule: test simple rules first, confirm on every term, and compute before you choose.

Gap series and ratio series

~2 min

Write the gaps between terms; a constant gap, a cycle of gaps, or gaps that change steadily each has its own next step. If the terms grow by a proportion rather than an amount, divide instead and look for a constant or growing ratio.

Start with the differences. If every gap is the same, add it once more: 23, 34, 45, 56 adds 11 each time, so 67 is next. If the gaps repeat in a cycle, such as +4, +9, +4, +9 in 12, 16, 25, 29, 38, continue the cycle from where it stopped, which here means +4 for 42.

When the gaps change, look at the gaps between the gaps. In 2, 7, 23, 50, 88 the gaps are 5, 16, 27, 38, rising by 11 each time, so the next gap is 49 and the next term 137. Sometimes the gaps form a sequence you know. Consecutive odd gaps build the perfect squares: 81, 100, 121, 144 has gaps of 19, 21 and 23.

If the terms grow by a proportion rather than an amount, divide each term by the one before. A constant ratio means keep multiplying: 7, 28, 112, 448 is ×4 throughout. Growth that is close to a whole multiple but not exact often means "multiply, then adjust": 4, 10, 22, 46 is ×2 + 2 each step. And the multiplier itself can grow, as in 5, 10, 30, 120, where the ratios are 2, 3 and 4.

The shape of the growth tells you which row to write. 40, 52, 64, 76 grows by a steady amount; 7, 21, 63, 189 grows by a steady proportion.

Rule: write the differences first; if they grow in proportion, switch to the ratios.

Rules built from earlier terms, and special numbers

~2 min

When single steps do not fit, test whether each term combines the two before it, by adding, weighting or multiplying, or whether two strands alternate. Know the squares, cubes and primes well enough to spot them on sight.

Irregular gaps are a clue that each term depends on the two before it. In a Fibonacci-style series each term is their sum: 4, 9, 13, 22, 35 continues with 57. A weighted version doubles one term before adding: in 5, 7, 19, 45, 109 each term doubles the one before it and adds the term before that, so 263 is next. When terms explode in size, test the product of the previous two: 4, 5, 20, 100 continues with 2,000.

A row that jumps up and down may be two series interleaved. In 21, 200, 33, 191, 45, 182 the odd positions rise by 12 and the even positions fall by 9. The missing term is the seventh, so it belongs to the first strand: 57. Always decide which strand the gap belongs to.

Special numbers save time. Know the squares through 16², including the ones people miss: 169, 196, 225 and 256. Know the cubes through 10³: 343, 512 and 729 are 7³, 8³ and 9³. A series can also sit a fixed distance from the squares, as 41, 54, 69, 86 sits 5 above 36, 49, 64 and 81. Products of neighbors, such as 42, 56, 72 and 90, sit just above the squares. Finally, a few numbers look prime but are not: 91 = 7 × 13 and 119 = 7 × 17.

Rule: when no single step fits, look back two terms, look for two strands, or look for a power.

Finding the odd one out

~2 min

Run a fixed checklist of properties across all four numbers and stop at the first that splits them exactly three to one. Digit-sum tests decide divisibility fast, and near-miss squares are the classic trap.

For a set of four numbers, test one property at a time on all of them: even or odd, a multiple of 3, a multiple of 5, a perfect square, a prime. Stop at the first property that splits the set three to one. In 112, 136, 154, 171 three are even, so 171 is the outlier.

A property only counts if it gives that exact split. In 105, 114, 123, 130 parity splits two to two, so it is not the rule. Moving on through the list, three of them are multiples of 3 and 130 is not, so 130 is the answer.

Divisibility tests make the checklist fast. A number is a multiple of 3 when its digit sum is, and a multiple of 9 when its digit sum is a multiple of 9. A multiple of 5 ends in 0 or 5. In 117, 135, 153, 160 the first three have digit sums of 9 and 160 does not, so it is the one that is not a multiple of 9.

Sets of squares often include a near-miss chosen to look familiar: in 121, 169, 196, 210, the last number falls between 196 and 225. The same method works for words and shapes. Name a category and test it on all four, as in oak, maple, tulip, birch, where three are trees and one is a flower.

Rule: test one property at a time and accept only a clean three-to-one split.

Number and word analogies

~2 min

Name the step that changes the first number into the second, confirm it on every example pair, and apply exactly that step to the last term. For words, state the relationship as a sentence and test each option in it.

Read "a is to b as c is to ?" as "what turns a into b?" Try adding, multiplying and squaring. For 11 is to 33 as 14 is to ?, the operation is ×3, so the answer is 42. The usual trap is to switch operations midway, adding 22 to 14 because 33 is 22 more than 11.

One pair rarely settles the rule. 11 is to 121 fits both ×11 and squaring, so check the second example pair if the item gives one. If it says 13 is to 169, squaring holds and ×11 does not, because 13 × 11 is 143.

When neither a fixed addition nor a fixed multiplier fits every pair, escalate one step at a time: squaring, then cubing, then n × (n + 1). 10 is to 1,000 as 11 is to 1,331 is cubing, and 10 is to 110 as 12 is to 156 is n × (n + 1). Some rules take two steps, such as squaring and then adding a small number: 10 is to 104 and 12 is to 148 both follow "square, then add 4".

Word analogies work the same way with a sentence in place of an operation. For caterpillar is to butterfly as acorn is to what, the bridge sentence is "a caterpillar grows into a butterfly", so the answer is what an acorn grows into, an oak. If several options fit, make the sentence more specific.

Rule: name the operation, confirm it on every pair, then apply that operation and nothing else.

Letter series

~2 min

Turn letters into alphabet positions and solve the number series underneath, watching for backward steps, wraparound past Z, and two strands woven together. For letter-and-number tokens, solve each column separately.

Letter series are number series in disguise. Write each letter's position, A = 1 through Z = 26, and solve the numbers. C, J, Q is 3, 10, 17, so the next position is 24, which is X. A few anchors make conversion quick: E is 5, J is 10, O is 15, T is 20 and Y is 25.

Series can run backward. Z, U, P, K is 26, 21, 16, 11, a step of 5 downward, so F comes next. They can also wrap around: count past Z to A as if the letters sat in a circle, so four steps on from X lands on B. In number terms, subtract 26 from any total over 26.

When letters seem to jump at random, try two interleaved strands. B, Y, D, W, F, U holds B, D, F rising and Y, W, U falling, so the seventh term continues the first strand with H.

Tokens that pair a letter with a number usually follow one rule per column. If the tokens are K with 12, M with 24 and O with 36, the letters skip one each time and the numbers rise by 12, so the next token is Q with 48. Solve the columns separately and join them only at the end.

The common slip is converting the answer back incorrectly, so count the final letter carefully, or use the nearest anchor and step from it.

Rule: convert to positions, solve the numbers, and convert back with care.

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