Analogy

high importance~3 Q in Tier 112 formulas⚡ 11 shortcuts6 subtopics

Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.

Letter / letter-cluster analogy

Shift rule

\text{new position} = \text{old position} + k

k is the shift. Above 26, subtract 26; below 1, add 26.

Opposite letter

\text{opposite} = 27 - p

p is the position. A and Z, or M and N, always add to 27.

Rising shift

\text{shift of slot } i = i

Slot 1 moves +1, slot 2 moves +2 and so on, e.g. ABCD → BDFH.

Multiply-and-add rule

b = k a + r

a is the first number, b the second. Find k and r from the model pair, then check.

Square family

b = a^2 \pm r,\ a(a+1),\ (a+1)^2

Try these when b is close to the square of a.

Digit rules

b = \text{digit sum},\ \text{digit product},\ (\text{digit sum})^2

Not allowed when the whole-number note is given.

Chain rule

b = ka + r,\quad c = kb + r

a, b, c are the three numbers of a set. One step is used twice.

Third from first two

c = ab,\ a^2 + b^2,\ k(a+b),\ k(a-b)

Test on every model set.

Power set

(a,\ a^2 \pm r,\ a^3 \pm r)

Near-squares and near-cubes of the first number.

Word to number

\text{code} = k \times (\text{sum of positions}) + r

Often k = 1 and r = 0: the plain sum of letter positions.

Reverse position

\text{reverse value} = 27 - p

A = 26, B = 25 ... Z = 1.

Reverse sum shortcut

\text{reverse sum} = 27n - \text{plain sum}

n is the number of letters in the word.