Classification (Odd One Out)

high importance~3 Q in Tier 19 formulas⚡ 9 shortcuts6 subtopics

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Number odd one out

Divisibility by 11

|S_{odd} - S_{even}| \equiv 0 \pmod{11}

S = sums of alternate digits; e.g. 2728: (2+2)-(7+8) = -11 → divisible.

Perfect square endings

n^2 \in \{0,1,4,5,6,9\}

A square never ends in 2, 3, 7 or 8 — instant elimination.

Digit sum 9 rule

9 \mid n \iff S(n) \equiv 0 \pmod 9

Numbers with digit sum exactly 9: 45, 54, 63, 72, 81.

Step of a cluster

s_i = Q(L_{i+1}) - Q(L_i) \pmod{26}

Compute steps for every option; three must match as tuples.

Opposite-pair sum

Q(X) + Q(Y) = 27

Letter pairs whose positions add to 27 (AZ, BY, ..., MN).

Pair rule

b = f(a),\; f(a) \in \{a^2,\ a^3,\ a^2 \pm k,\ ka \pm r\}

Find f from one pair, check the other three.

Chain triad

(a,\ f(a),\ f(f(a)))

Same operation applied twice, e.g. ×3: (3, 9, 27).

Outer-to-middle triad

(a,\ g(a, c),\ c),\; g = ac \text{ or } a^2 + c^2

Middle number made from the two outer numbers.

First-two-to-third triad

(a,\ b,\ h(a, b)),\; h = k(a+b) \text{ or } ab \pm r

Third number made from the first two.