Fluid Reasoning curriculum

Most pattern items use one of a short list of rules: a constant gap, a constant ratio, powers, a rule built from earlier terms, or two series woven together. Check them in roughly that order. A rule that needs special cases to fit is almost never the intended one, so if your explanation keeps growing, look again for a plainer rule.

Two terms fit many rules: 7 then 14 could be +7 or ×2. A rule earns trust only when it carries through every term given. In 7, 14, 28, 56 the +7 reading breaks at the third term, while ×2 holds all the way. Check the whole row before you extend it.

Once you have a candidate rule, compute the next term yourself and then look for it among the options. If it is missing, the rule is wrong, not the choices. Working backward from the options tempts you to accept the first one that fits a single step, and wrong options are often built to fit exactly one step.

Patterns are easier to see on paper than in your head. Write the differences beneath the terms, or the alphabet positions beneath letters, and the rule often shows itself in that second row. Under 4, 11, 25, 53 the gaps are 7, 14, 28, which double each time. That is hard to notice by staring at the original numbers.

Subtract each term from the next. If every gap is the same, add that gap once more. In 23, 34, 45, 56 every gap is 11, so the next term is 67. A falling series works the same way with a negative gap: 130, 123, 116, 109 drops by 7, so 102 comes next. Check the sign, because a subtraction slip turns a falling series into a rising one.

If the first differences are not constant, take differences again. In 2, 7, 23, 50, 88 the gaps are 5, 16, 27, 38, and those rise by 11 each time. So the next gap is 49 and the next term is 137. A constant second difference means the series grows like the squares, even when its terms do not look like squares.

When the gaps run 1, 2, 3, 4, the series adds the next counting number at each step. Starting from 10, that gives 11, 13, 16, 20, 25, and the step after that is +6. The same structure starting from 0 gives the triangular numbers, the sums of the counting numbers. Recognizing the growing gaps saves testing ratios or powers that will not fit.

Sometimes the row of gaps is itself a familiar sequence. Consecutive odd gaps always build the perfect squares: 81, 100, 121, 144 has gaps of 19, 21 and 23, so the next gap is 25 and the next term is 169. Gaps that are squares or cubes continue the same way, with the next square or cube.

A series can alternate between two steps. In 12, 16, 25, 29, 38 the gaps run +4, +9, +4, +9, so the next step is +4, giving 42. Find where you are in the cycle before you add. The usual slip is repeating the step you just used instead of taking the next one. Cycles of three steps work the same way.

When the gaps grow quickly, divide each term by the one before it. If the result is constant, keep multiplying. In 7, 28, 112, 448 each term is 4 times the last, so the next is 1,792. Halving works the same way: 1,000, 500, 250 halves each step, so 125 comes next.

If each term is roughly double or triple the last but not exactly, try "multiply, then add or subtract". In 4, 10, 22, 46 the terms about double: 4 × 2 + 2 = 10, 10 × 2 + 2 = 22 and 22 × 2 + 2 = 46, so the next is 94. Read the multiplier from the growth, take the constant from the first step, and confirm both on the other steps.

If the ratios between terms change, check whether they rise steadily. In 5, 10, 30, 120 the ratios are 2, 3 and 4, so the next term is 120 × 5 = 600. Starting the same rule from 1 gives the factorials. A growing multiplier outruns any fixed ratio, so very fast growth is itself the hint.

Terms that grow by about the same amount each step suggest a gap rule; terms that grow by about the same proportion suggest a ratio. 40, 52, 64, 76 grows by 12 each time, while 7, 21, 63, 189 grows threefold. Checking the shape of the growth first tells you which second row to write, the differences or the quotients.

In a Fibonacci-style series every term adds together the two terms just before it. In 4, 9, 13, 22, 35 every term after the second adds the previous two: 4 + 9 = 13, 9 + 13 = 22 and 13 + 22 = 35. So the next is 22 + 35 = 57. The gaps in such a series look irregular, which is the clue to stop testing single steps.

Harder series blend the previous two terms with weights. In 5, 7, 19, 45, 109 each term is twice the previous one plus the one before: 2 × 7 + 5 = 19, 2 × 19 + 7 = 45 and 2 × 45 + 19 = 109. The next is 2 × 109 + 45 = 263. Try the plain sum first; if it falls short by a growing amount, test a weight on one of the terms.

When numbers grow faster than any ratio explains, test multiplying the two previous terms. In 4, 5, 20, 100 each term after the second is the product of the two before it, so the next is 20 × 100 = 2,000. Series like this become huge within a few terms, which is itself the hint.

If a row jumps up and down, read every other term. In 21, 200, 33, 191, 45, 182 the odd positions run 21, 33, 45, rising by 12, and the even positions run 200, 191, 182, falling by 9. The next term is the seventh, so it continues the first strand: 57. Decide which strand the missing position belongs to before you compute.

Squares hide in many series, and recognizing them is faster than computing gaps. Learn them through 256; 169, 196, 225 and 256 are the ones people miss. A row of 169, 196, 225 is 13², 14², 15², so 16² = 256 comes next. The gaps between consecutive squares are the odd numbers, which gives a quick confirmation.

A term near a cube is worth testing as n³. 343 is 7³, 512 is 8³ and 729 is 9³, so a row of 343, 512, 729 continues with 10³ = 1,000. Some numbers are both a square and a cube, such as 64 and 729. When you meet one, check its neighbors to see which pattern the row follows.

If the terms are close to squares, subtract the nearest square from each. In 41, 54, 69, 86 the terms are 36 + 5, 49 + 5, 64 + 5 and 81 + 5, so the next is 100 + 5 = 105. The same check finds a series that sits a fixed amount below the squares. Compare each term with the squares before you try anything more complicated.

The numbers 42, 56, 72, 90 are 6 × 7, 7 × 8, 8 × 9 and 9 × 10, so the next is 10 × 11 = 110. Each is a little more than a square (42 is 36 + 6), which is the tell. Their gaps rise by 2 each time, so a second-difference check confirms the pattern too.

A prime is divisible only by 1 and itself. Rows of primes, and sets with one composite among primes, are common in odd-one-out items, so test small divisors before you trust a number that looks prime. 91 = 7 × 13, 87 = 3 × 29 and 119 = 7 × 17 are the usual impostors. Checking 2, 3, 5 and 7 settles any number below 121.

Run through a short checklist on all four numbers: 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 and 171 is odd, so 171 is the outlier. Testing one property at a time beats staring at the numbers and hoping something stands out.

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, while 160 sums to 7, so 160 is the one that is not a multiple of 9.

When the numbers look familiar, check each against the squares. In 121, 169, 196, 210 the first three are 11², 13² and 14², while 210 falls between 196 and 225. Near-misses are chosen to look plausible, so confirm each square rather than trusting that it feels like one.

A property is the answer only if exactly one number lacks it. In 105, 114, 123, 130 parity splits two to two, so it is not the rule. Every number but 130 has a digit sum of 6, which makes it a multiple of 3, and 130 is not. When a property splits two to two, set it aside and test the next one on your list.

For words or pictures, name candidate categories and test each against all four items: what it is, what it does, how many parts it has, whether it is flat or solid. In oak, maple, tulip, birch, three are trees and the tulip is a flower. If a category fits only two items, drop it and try a more specific one.

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 step is ×3, so the answer is 42. Apply exactly the same operation to the third number. Switching operations partway, such as adding 22 to 14, is the standard trap.

One pair rarely pins a rule: 11 is to 121 fits both ×11 and squaring. A second pair settles it. If the item adds "as 13 is to 169", squaring holds, because 13² = 169, while ×11 fails, because 13 × 11 = 143. Test your rule on every example the item gives before you use it.

If no single operation fits, look for a power followed by a small adjustment. In 10 is to 104 as 12 is to 148, 10² + 4 = 104 and 12² + 4 = 148, so the rule is "square, then add 4". Peel the steps one at a time: find the power that lands close, then name what is left over.

Analogy rules escalate in a predictable order. If neither a fixed addition nor a fixed multiplier fits both pairs, test squaring, then cubing, then n × (n + 1). 10 is to 1,000 as 11 is to 1,331 is cubing. 10 is to 110 as 12 is to 156 is n × (n + 1). Escalate one step at a time rather than guessing.

Turn the first pair into a bridge sentence, then drop each option into the same sentence. Caterpillar is to butterfly as acorn is to what? The bridge is "a caterpillar grows into a butterfly", so look for what an acorn grows into: an oak. If several options fit a loose sentence, make the sentence more specific until only one survives.

Write each letter's position, A = 1 through Z = 26, and treat the row as a number series. C, J, Q is 3, 10, 17, a gap of 7, so the next position is 24, which is X. A few anchor letters help: E = 5, J = 10, O = 15, T = 20, Y = 25. Convert the answer back to a letter before you choose.

If the letters come from the end of the alphabet, check for a falling series. Z, U, P, K is 26, 21, 16, 11, a step of 5 downward, so the next position is 6, which is F. Read the direction from the first two letters, then confirm it on the rest.

Some series continue past the end of the alphabet. Count on from Z to A as if the letters sat in a circle: four steps on from X lands on B (Y, Z, A, B). In position terms, subtract 26 from any total over 26, or add 26 to any total below 1. Check whether a step would carry past either end.

When letters seem to jump at random, split them into odd and even positions. B, Y, D, W, F, U holds B, D, F rising by two and Y, W, U falling by two. The next term is the seventh, so it continues the first strand: H. This mirrors an interleaved number series and is solved the same way.

For terms like K12, M24, O36, split each into its letter and its number and solve each column alone. The letters skip one each time (K, M, O, so Q next) and the numbers rise by 12 (12, 24, 36, so 48 next), giving Q48. Join the columns back together only at the end.

Find the shortest block that repeats, such as red, red, blue, a unit of three. To find the 20th item, divide 20 by 3: the remainder is 2, so the 20th item matches the second item of the unit, red. A remainder of 0 means the last item of the unit. This avoids counting term by term.

Count what changes from stage to stage and write the counts as a series. A tile pattern with 31, 43, 55 and 67 tiles in its first four stages adds 12 each time, so stage 10 uses 31 + 9 × 12 = 139 tiles. Jumping to a far stage with a formula avoids drawing every stage in between.