Classification (Odd One Out)
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Number odd one out
Divisibility by 11
|S_{odd} - S_{even}| \equiv 0 \pmod{11}
S = sums of alternate digits from the left; 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 rule for 9
9 \mid n \iff S(n) \equiv 0 \pmod 9
S(n) is the sum of the digits of n.
Letter-cluster odd one out
Step of a cluster
s_i = Q(L_{i+1}) - Q(L_i) \pmod{26}
Q is the alphabet position. 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.
Number pairs and sets odd one out
Pair rule
b = f(a),\; f(a) \in \{a^2,\ a^3,\ a^2 \pm k,\ ka \pm r\}
Find f from one pair, then check the other three pairs.
Chain triad
(a,\ f(a),\ f(f(a)))
The same operation applied twice, e.g. ×3 gives (3, 9, 27).
Outer-to-middle triad
(a,\ g(a, c),\ c),\; g = ac \text{ or } a^2 + c^2
The middle number is 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
The third number is made from the first two.