These are the advanced ideas that turn an 18 into a 23 — identities, theorems, and meta-tools that cover the “#22–25 territory” on each topic. They’re also embedded in each lesson’s appendix (under the “Want to climb higher?” expander), so you can study one topic at a time too.
Heads up: most of this material is harder than the AMC 8 main band needs. Don’t feel obligated to learn all of it. Use it as a reference when you’ve already mastered a lesson’s main content and want the next layer.
CLIMB · ARITHMETIC & OPERATIONS
Arithmetic & Operations — Look for structure before you calculate.
Want a quick check? — casting out nines
A 19th-century shortcut for verifying arithmetic without redoing it. It catches MOST mistakes in seconds.
The fact: a number and its digit sum leave the SAME remainder when divided by 9. (Why? Because 10, 100, 1000, … all leave remainder 1 mod 9; so a digit in any place contributes itself.)
How to use it. Compute your answer, then check:
- Compute the digit sum of each input. Reduce mod 9 (or keep adding digits until single).
- Combine those single-digit remainders the same way the problem combines the inputs (+ if you added, × if you multiplied).
- Compute the digit sum of your computed answer.
- Both single digits should match. If they don’t — you made an arithmetic mistake.
Try it. “Is 472 + 351 = 823 correct?” Digit sums: 4+7+2 = 13 → 1+3 = 4. 3+5+1 = 9 → 0 (since 9 ≡ 0 mod 9). Sum of remainders: 4 + 0 = 4. Digit sum of 823: 8+2+3 = 13 → 1+3 = 4 ✓. The answer is consistent.
If your answer’s digit sum HAD differed, you’d know to redo the math. (Casting nines doesn’t catch every error — e.g., swapping two digits leaves the digit sum unchanged — but it catches most.)
CLIMB · NUMBER THEORY
Number Theory — What whole numbers are made of.
Want to climb higher? — advanced number-theory tools (#22–#25 territory)
- Perfect-square filter (mod 3, mod 4). A perfect square can ONLY have these remainders:
· mod 3: 0 or 1 (never 2). So if N ≡ 2 (mod 3), N is NOT a perfect square.
· mod 4: 0 or 1 (never 2 or 3). So if N ≡ 2 or 3 (mod 4), N is NOT a perfect square.
Great for “is X a perfect square?” questions — rules out most numbers in one glance. - Sum of divisors of
n = pa · qb:σ(n) = (1 + p + p² + … + pa) · (1 + q + q² + … + qb). So σ(112) = σ(2⁴ · 7) = (1+2+4+8+16) · (1+7) = 31 · 8 = 248. - Product of divisors of n:
nd/2where d is the number of divisors. (Pair each divisor with its partner; the product of every pair is n.) - Chicken McNugget Theorem. If
aandbare coprime positive integers, the largest amount you CANNOT make as a non-negative combination of a’s and b’s isab − a − b. (For 4 and 7: the largest unmakeable amount is 4·7 − 4 − 7 = 17.) - Legendre’s formula (exponent of prime p in n!):
vp(n!) = ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + …— useful for “how many trailing zeros does 100! have?” (answer: v5(100!) = 20 + 4 = 24).
CLIMB · FRACTIONS, DECIMALS & PERCENTS
Fractions, Decimals & Percents — Three names for the same number.
Want to climb higher? — telescoping with bigger gaps
- Two-apart split:
1 / (n · (n+2)) = ½[1/n − 1/(n+2)]. So 1/(1·3) + 1/(3·5) + 1/(5·7) + … telescopes too, just with a 1/2 out front. - Product telescoping with (1 + 1/k):
(1 + 1/1)(1 + 1/2)(1 + 1/3) … (1 + 1/n) = (2/1)(3/2)(4/3) … ((n+1)/n) = n + 1. Each numerator kills the next denominator. The whole product collapses to just n + 1. - The harmonic series. 1 + 1/2 + 1/3 + … does NOT telescope — it grows without bound (slowly). Don’t try.
CLIMB · RATIOS, RATES & PROPORTIONS
Ratios, Rates & Proportions — Same ratio in two places.
Want to climb higher? — advanced rate ideas (#22–#25 territory)
- Three-worker together-time (A, B, C alone take a, b, c hours):
T = 1 / (1/a + 1/b + 1/c). Same pattern, three terms. - Filling AND draining at once. If pipe A fills at rate 1/a and drain D empties at rate 1/d, the net rate is
1/a − 1/d. The drain SUBTRACTS from the filling. - Mixture problems. Mixing x liters of A% solution with y liters of B% solution gives a (x+y)-liter mix with concentration
(xA + yB) / (x + y). Same as a weighted average. - “Train passes a person” vs “train passes a station.” To pass a person (a point): train travels its own length. To pass a station (also has length): train travels train length + station length. Time = (relevant length) / train speed.
- Continuous compound interest (rare on AMC 8): ert grows faster than discrete (1+r)t for the same r, t.
CLIMB · ALGEBRA & PATTERNS
Algebra & Patterns — Turn the story into a letter.
Want to climb higher? — advanced algebra tricks (#22–#25 territory)
- Symmetric-sum trick. When several equations share a symmetric structure (e.g.,
2x + y + z = a,x + 2y + z = b,x + y + 2z = c), ADD them all. The left side simplifies to a clean multiple of (x + y + z). Solve for the symmetric sum first; then individual variables. - Sum & product of two numbers. If x + y = S and xy = P, then x and y are the roots of
t² − St + P = 0. So x² + y² = S² − 2P. Useful when the problem gives you S and P without giving x and y individually. - Simon’s Favorite Factoring Trick (SFFT). An equation like
xy + 3x + 2y = 17can be tamed by adding a constant to both sides to factor:xy + 3x + 2y + 6 = 23 ⇒ (x+2)(y+3) = 23. Now hunt integer factor pairs of 23. - Sum/difference of cubes: a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²).
CLIMB · GEOMETRY & MEASUREMENT
Geometry & Measurement — Draw the picture. Then look for symmetry.
Want to climb higher? — three advanced geometry facts (#22–#25 territory)
- British Flag Theorem. For any point P inside a rectangle ABCD: PA² + PC² = PB² + PD². The sum of squared distances to opposite corners is equal — doesn't matter where P sits inside.
- Euler’s polyhedron formula. For any solid with flat faces: V − E + F = 2 (vertices − edges + faces). Cube: 8 − 12 + 6 = 2 ✓. Tetrahedron: 4 − 6 + 4 = 2 ✓.
- Inradius of a right triangle (legs a, b, hypotenuse c): r = (a + b − c) / 2. The circle inscribed in a 3-4-5 right triangle has radius (3 + 4 − 5)/2 = 1.
- Heron’s formula for area from sides only: with semi-perimeter
s = (a+b+c)/2, area = √(s(s−a)(s−b)(s−c)). Avoid unless you must — usually decomposing or finding a height is cleaner.
CLIMB · COUNTING & PROBABILITY
Counting & Probability — Count carefully. Don't compute.
Want to climb higher? — advanced counting tools (#22–#25 territory)
- PIE for 3 sets: |A ∪ B ∪ C| = |A| + |B| + |C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|.
- Pascal’s identity: C(n, k) + C(n, k+1) = C(n+1, k+1). (Useful for building Pascal’s triangle row-by-row.)
- Binomial sum: C(n,0) + C(n,1) + … + C(n,n) = 2n. (Same fact as “number of subsets” — two views.)
- Linearity of expectation: E[X + Y] = E[X] + E[Y], even when X and Y are NOT independent. Lets you compute the expected total without disentangling cases.
- Recursion-as-counting: when a problem builds on smaller cases, count f(1), f(2), f(3) and look for a relation like f(n) = f(n−1) + f(n−2) (Fibonacci). Then chase the recurrence.