Section 32.1

Algebraic Tricks

Factorisations, symmetric functions, nested radicals, recovering values from pairwise sums, Diophantine word puzzles (hundred fowls, dollar-cent swaps).

Toolkit

Algebraic tricks turn word problems and identities into manageable equations by substitution, factoring, and exploiting symmetric structure rather than brute computation.

Key tools:

Symmetric functions: $x^2+y^2=(x+y)^2-2xy$ , $x^4+y^4=(x^2+y^2)^2-2(xy)^2$ .
Telescoping differences of squares: $n^2-(n-1)^2=2n-1$ .
$(x-1)(y-1)=xy-x-y+1$ turns $xy=x+y$ -type equations into factor pairs.
Infinite self-similar expressions (nested radicals, continued fractions): set the whole equal to $x$ and solve the resulting finite equation, checking convergence.
Linear Diophantine equations: reduce modulo a coefficient to find all integer solutions; impose range constraints (e.g. digits $0$ – $99$ , calendar years) to pin down the unique answer.
Pairwise-sum/product recovery: summing all $\binom{k}{2}$ pair sums counts each element $(k-1)$ times; complementary products of pairs from four numbers each equal $abcd$ .
Working backwards: invert each step of a sequential process (fractions taken, amounts added) from the final known state.
$e^{i\theta}$ form for complex exponents: $i=e^{i\pi/2}$ .

Problems