AMC 8 Formula Cheat-Sheet — Printable PDF

ARITHMETIC & OPERATIONS

Arithmetic & Operations

FORMULAS TO KNOW COLD

PEMDAS: Parentheses, Exponents, Mul/Div, Add/Sub — in tiers.
Arithmetic series: sum = (first + last) ÷ 2 × count.
Sum 1+2+…+n = n(n+1)÷2. (1+2+…+100 = 5050.)
Sum of first n odds 1+3+5+… = n².
Inclusive count: from a to b is b − a + 1 (the +1!).
Distributive: a×b + a×c = a×(b + c).
Minus over parens: a − (b + c) = a − b − c.
Average ↔ total: total = average × count.
Deviations from the mean sum to zero.
Shift / scale a list: add n to every value → mean (median, mode) +n; multiply every value by k → mean ×k.
Swap one value: the total changes by the difference; new mean = (old total + change) ÷ count.
Powers of 10: 10³ = 1,000; 10⁶ = million; 10⁹ = billion.
Exponent laws (same base): \(a^m \cdot a^n = a^{m+n}\); \(a^m \div a^n = a^{m-n}\); \((a^m)^n = a^{mn}\).
Zero / negative power: \(a^0 = 1\); \(a^{-n} = 1/a^n\).
Scientific notation: \((a \times 10^m)(b \times 10^n) = (a b) \times 10^{m+n}\) — multiply the fronts, add the exponents.
Square of a sum: (a + b)² = a² + 2ab + b² — never just a² + b².
Outliers: drag the mean, leave the median alone.

Common traps

Reading left to right instead of in PEMDAS tiers.
Dropping parentheses after a minus sign. a − (b + c) ≠ a − b + c.
Forgetting the +1 when counting inclusively (fence-post error).
Averaging two averages. Combine totals first, divide once.
Trusting size instincts with negatives or fractions. Test the cases.
Rounding the leftover the wrong way — check whether a partial group counts.
Misreading a graph axis. A ‘50’ tick means different things in different graphs.
Multiplying exponents when you should add. \(a^m \cdot a^n = a^{m+n}\) (add); only a power of a power multiplies.
Splitting a power across a plus sign. (a + b)² = a² + 2ab + b², never a² + b² — the 2ab strip is the forgotten piece.
Trusting the mean when one value is an outlier. Check the median — it is robust.

→ Full Arithmetic & Operations lesson

NUMBER THEORY

Number Theory

FACTS TO KNOW COLD

Primes ≤ 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
The only even prime is 2. 1 is NOT prime.
To test if n is prime: trial-divide by primes ≤ √n. Beyond that, no factors can hide.
Divisibility by 3 or 9: digit sum.
Divisibility by 4: last 2 digits.
Divisibility by 8: last 3 digits.
Divisibility by 11: alternating digit sum.
Divisibility by 7: chop last digit, double it, subtract from what's left.
Divisibility by 13: chop last digit, multiply by 4, add to what's left.
1001 = 7 · 11 · 13 — split into 3-digit groups, alternating sum tests all three at once.
Difference of squares: a² − b² = (a − b)(a + b).
Divisor count formula: for n = p^a · q^b · …, count = (a+1)(b+1)·…
Divisor SUM formula: (1 + p + … + p^a)(1 + q + … + q^b)·… — one bracket per prime, then multiply.
Odd divisors: cover the 2's, count the rest. Even divisors: total − odd.
Perfect-square ↔ odd number of divisors. Every other number has an even count.
Perfect-square divisors of n: count divisors using only EVEN exponents.
Divisors pair up: d ↔ n/d, product = n.
GCD(a, b) × LCM(a, b) = a × b (two positive integers — fails for three or more).
Scaling: gcd(na, nb) = n·gcd(a, b) and lcm[na, nb] = n·lcm[a, b]. Coprime ⟹ lcm = product: if gcd(a,b)=1 then lcm[a,b] = ab.
Composite test: split into RELATIVELY PRIME pieces (36 = 4·9 ✓, not 3·12); the split is valid only when the pieces' lcm equals the original.
Remainder shortcut: if every remainder is (divisor − 1), answer = LCM − 1; if every remainder is 1, answer = LCM + 1.
Euclidean algorithm: gcd(a, b) = gcd(b, a mod b). Replace the bigger with the leftover until you hit 0.
Units-digit cycles: 2/3/7/8 → period 4; 4/9 → period 2; 0/1/5/6 → stuck.
Digit sum ≡ N (mod 9).

Common traps

1 is not prime. The smallest prime is 2.
Confusing “multiple of” with “divisor of”. Multiples of 6: 6, 12, 18, 24, … Divisors of 6: 1, 2, 3, 6.
Mod-cycle off-by-one. If a cycle has period 4 starting at n=1, then n=4 hits the 4th entry (the last), n=5 wraps to the 1st.
Forgetting primes with exponent 1. 60 = 2²·3·5 has (2+1)(1+1)(1+1) = 12 divisors. The exponent-1 primes still contribute a (+1) factor.
Computing huge powers directly. Always reduce mod the relevant modulus first.
Splitting a composite the wrong way. To test divisibility by 36, use 4×9 (coprime), NOT 3×12 — 3 and 12 share a factor, so “divisible by 3 and 12” only forces divisibility by 12. The pieces must be relatively prime (their lcm equals the original).
GCD × LCM = product is for TWO numbers only. It fails for three or more.

→ Full Number Theory lesson

FRACTIONS, DECIMALS & PERCENTS

Fractions, Decimals & Percents

CONVERSIONS TO MEMORIZE

1/2 = 0.5 = 50%; 1/3 ≈ 0.333 = 33⅓%; 2/3 ≈ 0.667 = 66⅔%
1/4 = 0.25 = 25%; 3/4 = 0.75 = 75%
1/5 = 0.2 = 20%; 2/5 = 0.4 = 40%; 3/5 = 0.6 = 60%; 4/5 = 0.8 = 80%
1/6 ≈ 0.167; 5/6 ≈ 0.833
1/8 = 0.125 = 12.5%; 3/8 = 0.375; 5/8 = 0.625; 7/8 = 0.875
1/9 ≈ 0.111; 1/11 ≈ 0.0909; 1/12 ≈ 0.0833
Scale to 100: if the bottom divides 100 (2,4,5,10,20,25,50), multiply top and bottom up to /100 — the top is then the percent (2/25 = 8/100 = 8% = 0.08). No long division.

FRACTION SKILLS

Mixed ↔ improper: improper → mixed = divide top by bottom (quotient = whole, remainder = new top, same bottom): 23/5 = 4⅗. Mixed → improper = (whole × bottom + top) over the bottom: 4⅗ = (4×5+3)/5 = 23/5.
Add/subtract mixed numbers: wholes with wholes, fractions with fractions; carry a whole when the fraction part tops 1, borrow a whole when it is too small. (Or flip both to improper and skip carrying.)
KCF for division: Keep, Change, Flip — keep the first, change ÷ to ×, flip the second. ÷(1/n) = ×n, so dividing by a number under 1 makes the result BIGGER.
Compare fractions: a/b vs c/d → cross-multiply, a×d vs c×b, bigger product wins (both bottoms positive). Compare to ½ by checking whether 2a > b.
Terminating vs repeating: simplify first; the decimal terminates exactly when the bottom's only primes are 2 and 5, else it repeats. A one-digit repeating block over 9, two digits over 99: 0.3̄ = 1/3, 0.2̄7̄ = 27/99.

TELESCOPING & PERCENT

Telescoping product: (1−1/2)(1−1/3)…(1−1/N) = 1/N.
Telescoping sum: 1/(n(n+1)) = 1/n − 1/(n+1).
Percent is a multiplier: up p% → ×(1+p/100); down p% → ×(1−p/100); p% OF → ×(p/100).
+25% then −20% returns to start (because 1.25 × 0.8 = 1).
+1/n undoes with −1/(n+1). +25% (=+1/4) undoes with −20% (=−1/5).

Common traps

Adding percents from successive applications. 50% off then 20% off ≠ 70% off (it's 60% off). Multiply the multipliers.
+25% then −25% returns to less than the start. +25% then −20% returns exactly (because ×1.25 × ×0.8 = 1).
Averaging two averages without weighting. Use total ÷ count, not (a+b)/2 when groups differ.
Confusing 'A is x% of B' with 'A is x% more than B'. The first is A = x%·B; the second is A = (1+x%)·B. 'Percent OF the whole' can never exceed 100% — divide by the total, not the other group.
Forgetting to KCF when dividing fractions. Dividing by a fraction is multiplying by the flipped fraction; the answer grows when you divide by something under 1.
Bigger bottom = bigger fraction. No — with the same top, more pieces means each is smaller: 2/7 < 2/5.
Subtracting mixed numbers without borrowing. 5¼ − 2¾ is not 3 (½). Borrow a whole (5¼ = 4 and 5/4) or flip both to improper first.
Mis-aligning a partly-repeating decimal. For 0.2̄8̄ = 0.2888…, line the repeating tails up before subtracting (10x − x), or the 8s won't cancel.
Adding tops and bottoms. 1/2 + 1/2 ≠ 2/4. Match bottoms first, add only the tops (that across-rule is for multiplying).

→ Full Fractions, Decimals & Percents lesson

RATIOS, RATES & PROPORTIONS

Ratios, Rates & Proportions

FORMULAS / FACTS TO KNOW COLD

D = S × T (and its two cousins). Same shape as Work = Rate × Time.
Ratio parts: a : b has a + b parts; one part = total ÷ total parts. Each side is also a fraction of the whole: a/(a+b) and b/(a+b) — multiply by the total in one step.
Join two ratios on a shared term: scale each so the shared quantity becomes their LCM, then read the chain (blue:white 2:3 and white:red 4:5 → white = 12 → blue : red = 8 : 15).
Average speed = total distance ÷ total time. NEVER the average of the speeds (unless the times are equal). This is the marquee trap.
Factor-label conversion: multiply by fractions equal to 1 (5280 ft / 1 mi); unwanted units cancel, the wanted unit survives.
1 hour = 3600 seconds = 60 minutes.
1 mile = 5280 feet. 1 km = 1000 m. 1 yard = 3 feet. 1 foot = 12 inches.
1 mph ≈ 1.467 ft/s (so 60 mph = 88 ft/s).
Equal-distance round trip avg speed: 2ab / (a+b) (harmonic mean).
Equal-time legs avg speed: (a+b)/2 (simple average).
Work-rates ADD; times don't. A alone a, B alone b → together T = ab / (a+b); for 3+ workers add all rates 1/a + 1/b + … and flip. k identical workers each taking t: t/k.
Same-direction closing speed: faster − slower.
Opposite-direction closing speed: sum of speeds.
Exponential growth: V(n) = V₀ · rⁿ. Compound interest: V₀ · (1 + p/100)ⁿ.
Percent = ratio out of 100. Percent of change = change ÷ original. Chain percent changes by MULTIPLYING multipliers (down 20% = ×0.80), never by adding.

Common traps

Averaging speeds for equal-distance trips — THE marquee trap. The slow leg eats more time, so the average is pulled toward the slow speed. Always use total distance ÷ total time (60 & 30 mph round trip averages 40, not 45).
Averaging speeds to close a gap. Two cars approaching each other close the gap at the SUM of their speeds, not the average — both eat into it at once.
Lining up two ratios without matching the shared term. blue:white 2:3 and white:red 4:5 can't chain until you scale both so white agrees (=12).
Forgetting to convert minutes to hours before applying D = S × T with mph. (Wrong final units = a flipped conversion fraction.)
Inverse vs. direct proportion. 'More workers, less time' is inverse (product fixed); 'more time, more distance' is direct (ratio fixed).
Averaging times for work-rate problems. Rates add, times don't — two helpers must finish FASTER than the quicker one alone.
Cumulative vs. per-period graphs. Cumulative answers are differences of two heights; per-period answers are direct reads.
Linear vs. exponential. 'Adds the same each year' is linear; 'multiplies each year' is exponential — a 10% raise four times compounds to 46.4%, not 40%.
Doubling-jar 'halfway in time' error. If something doubles each step and is full at step 12, it was half full at step 11, not at the time-midpoint — count backward from the finish.
Counting only the vehicles that depart during your trip. You also pass those already en route when you started (the train-passing trap).
Dropping a stop or a head start from total time in an average-speed problem.
Assuming −p% then +p% cancels. It never does: −20% then +20% lands at ×0.96, a 4% loss, because the rise is taken on a smaller amount.
Reading 'percent of the whole' as 'percent of the other part'. A 2 : 3 split makes boys 2/5 (40%) of the class, not 2/3.

→ Full Ratios, Rates & Proportions lesson

ALGEBRA & PATTERNS

Algebra & Patterns

SOLVING MOVES

Balance rule: do the SAME thing to both sides — peel weights off both pans (subtract), or split both pans evenly (divide) — until the letter stands alone.
Peel order: when the letter has an add/subtract AND a multiply/divide on it, undo the add/subtract FIRST, then the multiply/divide (order of operations, run backward).
Tidy first: combine like terms on each side (2x + 5x = 7x; but 2x and 5 never merge) before you peel.
Inequality flip: multiply or divide BOTH sides by a NEGATIVE → reverse the sign (\(<\) becomes \(>\)). Positive or +/− → no flip.
Word-to-symbol: '5 less than n' → n − 5 (not 5 − n); '5 less than twice n' → 2n − 5; 'three times the sum of n and 7' → 3(n + 7).

FORMULAS TO KNOW COLD

Linear: y = mx + b. Slope m = rise/run.
Arithmetic sequence: a_n = a_1 + (n−1)d. Sum = n(a_1 + a_n)/2.
Geometric sequence: a_n = a_1 · r^(n−1).
Difference of squares: a² − b² = (a+b)(a−b). (Mental math: 51·49 = 50²−1 = 2499.)
Square of binomial: (a±b)² = a² ± 2ab + b².
Sum & product bridge: a² + b² = (a+b)² − 2ab; and 1/a + 1/b = (a+b)/(ab). (Answer symmetric questions without finding a and b.)
Sum 1+2+…+n = n(n+1)/2 (the triangular numbers 1, 3, 6, 10, 15, …).
Sum 1+3+5+…+(2n−1) = n².
Sum of squares 1²+2²+…+n² = n(n+1)(2n+1)/6.
Sum of cubes 1³+2³+…+n³ = [n(n+1)/2]².
Sum of n consecutive integers = n × middle.
Average of an arithmetic sequence = (first + last) / 2.
Exponent rules: x^a·x^b = x^a+b; x^a/x^b = x^a−b; (x^a)^b = x^ab; x^−a = 1/x^a; x⁰ = 1.
Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … (each = sum of previous two).
Digit-counts for pages: 1–9 → 9, 10–99 → 180, 100–999 → 2700.

Common traps

Off-by-one in nth term. The 100th term is 99 steps past the first, not 100.
Forgetting the inequality flip. Dividing both sides by a negative reverses the sign: −8y ≥ −24 gives y ≤ 3, not y ≥ 3.
Peeling in the wrong order. In 8t + 9 = 65, undo the +9 first, then the ×8 — not the other way.
Solving for the wrong variable. Re-read what the question asks before reporting.
Custom operations. Re-read the definition each time — symbols mean different things in different problems.
Forgetting to verify all conditions. Check your solution against EVERY constraint, not just the ones you used.
Sign errors in difference of squares. a² − b² = (a+b)(a−b), not (a−b)².
Dropping the middle term. (a+b)² = a² + 2ab + b², NOT a² + b² — the 2ab is the one everyone forgets.

→ Full Algebra & Patterns lesson

GEOMETRY & MEASUREMENT

Geometry & Measurement

FORMULAS TO KNOW COLD

Rectangle: A = lw, P = 2(l + w).
Square: A = s², P = 4s, diagonal = s√2.
Triangle: A = ½ b h. Pythagorean: a² + b² = c².
Equilateral triangle (side s): A = (√3/4) s². Height = (√3/2) s.
Parallelogram: A = b h (h ⊥ base).
Trapezoid: A = ½ (b₁ + b₂) h. Midsegment = average of the two parallel sides = ½(b₁ + b₂).
Regular hexagon (side s): A = (3√3/2) s² (= 6 equilateral triangles of side s).
Circle: A = π r², C = 2π r. Sector: π r² · θ/360. Arc: 2π r · θ/360.
Cube: SA = 6 s², V = s³. Space diagonal = s√3.
Box (a × b × c): SA = 2(ab + bc + ca), V = abc. Space diagonal = √(a² + b² + c²).
Cylinder: V = π r² h, SA = 2π r² + 2π r h.
Cone (radius r, height h): V = ⅓ π r² h.
Sphere: V = (4/3) π r³, SA = 4π r².
Distance between two points: √((x₂−x₁)² + (y₂−y₁)²).
Equation of a circle: (x−a)² + (y−b)² = r² (center (a,b), radius r).
Pick's Theorem (lattice polygons): A = I + B/2 − 1.
Shoelace (coordinates): A = ½ |Σ(xₙyₙ₊₁ − xₙ₊₁yₙ)|.
13-14-15 triangle: area = 84 (splits into 5-12-13 + 9-12-15).
Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, 20-21-29.
Special right triangles: 45-45-90 sides 1:1:√2. 30-60-90 sides 1:√3:2.
Simplifying roots: pull out the largest perfect square — √50 = 5√2, √12 = 2√3, √75 = 5√3, √18 = 3√2. Multiply across: √a · √b = √(ab). Add only like roots.
Polygon interior angle sum: (n−2) · 180°.
Regular polygon angles: interior = (n−2)·180/n; exterior = 360/n. Pentagon 108, hexagon 120, octagon 135, 9-gon 140, 10-gon 144.
Exterior angles sum to 360° for ANY convex polygon (one full lap of turning).
Exterior angle of a triangle = sum of the two remote (non-adjacent) interior angles.
Parallel lines + transversal: Z (alternate interior) equal; F (corresponding) equal; C (co-interior / same-side) sum to 180°.
Triangle inequality: each side < sum of the other two (the two shorter sides must beat the longest).
Same height → area ratio = base ratio. A median (corner to midpoint of opposite side) splits a triangle into two equal areas.
Inscribed right triangle: a side equal to the diameter forces a 90° opposite angle.
Tangent ⊥ radius: a tangent meets the radius at the touch-point at 90°.

QUADRILATERAL AREAS

square s² · rectangle / parallelogram b·h
rhombus ½·d₁·d₂ (the diagonals — not the sides!)
trapezoid ½(b₁+b₂)·h (average the parallel sides, times height)
Hierarchy: square ⊂ rectangle & rhombus ⊂ parallelogram ⊂ quadrilateral. A rhombus or square has perpendicular diagonals — solve through the diagonals, not the sides.

Common traps

Confusing slanted side with height in parallelogram / triangle area formulas. Height is always perpendicular.
Forgetting symmetry. Look for fold axes or rotational symmetry before computing.
Misreading the figure. Always redraw it on your own paper and write your finds on it.
Area vs perimeter mix-up. Fence is a length; dirt is squares. Keep them apart.
Off-by-one on grid counts. A 5×5 grid of squares has 6 horizontal lines and 6 vertical lines.
Forgetting overlap in two-region area sums. Use A + B − (A ∩ B).
Doubling the area ratio instead of squaring it. Twice the size = four times the area (k²), not twice.
Splitting a root across a sum. √(a + b) is NEVER √a + √b, and √(a² + b²) is NEVER a + b. Add inside the radical first, then take the root.
Radius vs diameter. The distance straight across a circle is the diameter; every formula uses the radius r = d/2. A “fits inside” or “width across” gives the diameter — halve it.
Using Pythagoras without a right angle. a² + b² = c² (and the triples) hold only when the angle between the two legs is 90°.
Midpoint square is HALF, not a quarter. Joining the midpoints of a square’s sides gives an inner square with half the area (the four corner triangles fold in to fill it).
Stopping at the first case. “An isosceles triangle has a 70° angle” can mean three different triangles — “sum of all possible values” means more than one case survives. List every case, check each is valid, then combine.
Forgetting a triangle can’t exist. Before computing, check the triangle inequality — sides 2, 3, 8 form no triangle (2 + 3 < 8).
Counting only one source of rotation when a shape rolls. Total spin = rolling along edges (distance ÷ perimeter × 360°) PLUS pivoting at each corner (the exterior angles).

→ Full Geometry & Measurement lesson

COUNTING & PROBABILITY

Counting & Probability

FORMULAS TO KNOW COLD

Count from a to b (both ends): b − a + 1. (Keep both fenceposts — that is the +1.)
Multiplication principle: independent steps multiply. Multiply WITHIN a case (AND), add ACROSS cases (OR).
Factorial: n! = n·(n−1)·…·2·1. 5! = 120. 6! = 720. 7! = 5040. (0! = 1 by convention.)
Permutations (order matters): P(n, k) = n! / (n−k)!.
Combinations (order doesn't matter): C(n, k) = n! / (k! · (n−k)!).
Symmetry: C(n, k) = C(n, n−k).
Pairs / handshakes: C(n, 2) = n(n−1) / 2. (5 → 10, 10 → 45, 20 → 190.)
Polygon diagonals: n(n−3) / 2.
Gauss sum: 1 + 2 + … + n = n(n+1) / 2. (1 + … + 100 = 5050.)
Subsets of an n-element set: 2ⁿ.
Circular arrangements: (n−1)! (rotations of the same arrangement count once).
Letters with repeats: n! / (d₁! · d₂! · …).
Stars and bars: identical N items among k distinct recipients (zeros allowed) = C(N+k−1, k−1).
Probability: favorable / total (only when outcomes are equally likely).
Independent events: P(A and B) = P(A) · P(B).
Mutually exclusive: P(A or B) = P(A) + P(B).
Complement: P(not A) = 1 − P(A).
Inclusion-exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|.
Lattice paths (right/up): C(m+n, m).
Pigeonhole (guarantee N from k): k(N−1) + 1 items.
Degree-sum parity: sum of everyone's handshake counts = 2 × (handshakes), so it is always even — the number of people who shook an odd number of hands is even.

Common traps

Fencepost / off-by-one. Counting a to b is b − a + 1, not b − a; “span ÷ step” misses the endpoints. List a tiny case to check both ends.
Order matters or not? Permutation vs. combination — read the problem carefully.
With or without replacement? Without = the pool shrinks for the second draw.
Forgetting to subtract overlap in “A or B” with both possible. If a total exceeds the whole group, you forgot to subtract the overlap.
“At least one” trap. Use complementary counting; count the “none” case and subtract.
Outcomes not equally likely. Favorable / total only works when every counted outcome is equally likely — the 11 dice-sums (2…12) are NOT equal; count the 36 ordered pairs instead.
Dividing by 2 when you didn't double-count. Halve a handshake/pair count only if EVERY item was counted twice; mixed cases (e.g. some shakes one-way) must be split first.
Over-counting symmetric arrangements. If items can be rotated/reflected to the same arrangement, divide by the symmetry count.

→ Full Counting & Probability lesson

LOGIC & WORD PROBLEMS

Logic & Word Problems

FACTS TO KNOW

Contrapositive: ‘If A then B’ = ‘If not B then not A’.
Converse (not equivalent): ‘If B then A’.
Inverse (not equivalent): ‘If not A then not B’.
De Morgan: NOT(A and B) = (NOT A) or (NOT B). NOT(A or B) = (NOT A) and (NOT B).
Counterexample disproves a ‘for all’. One case with A true and B false kills ‘A → B’.
One example doesn’t prove a ‘for all’. You’d need a general argument.
For ‘there exists’: one example proves it; disproving it means checking every case.
Posts vs gaps: n posts make n−1 gaps. Decide which you’re counting before you add.
Honest speakers agree: two truth-tellers asked the same fact give the same answer.
Reduce and expand: shrink a scary number to 1, 2, 3, 4, tabulate, read the jumps to find the rule, then grow it back.
Lines / handshakes: \(n\) things joined in every pair make \(\dfrac{n(n-1)}{2}\) connections (each of \(n\) joins the other \(n-1\), then halve for the two ends). 6 people shake hands \(\dfrac{6\times5}{2}=15\) times.
Think about the LAST move: ways to reach step \(n\) often equal ways to reach the earlier steps it could come from — 1-or-2 hops give \(f(n)=f(n-1)+f(n-2)\) (Fibonacci 1, 2, 3, 5, 8, …).
Single-elimination: each game knocks out one entrant, so \(N\) entrants need \(N-1\) games.
Cycles & remainder: if a list repeats with cycle length L, the position’s slot is decided by (position mod L). Remainder 0 lands on the LAST slot, not the first.

Common traps

Confusing the converse with the original. ‘All cats are mammals’ does NOT mean ‘all mammals are cats’.
Negating ‘all’ to ‘none’. The negation of ‘all X are Y’ is ‘some X is not Y’, not ‘no X is Y’.
Dropping the ‘or equal’ when you negate > or <.
‘Or’ includes ‘both’ (inclusive).
Skipping casework when the clues leave more than one possibility.
The fencepost slip: adding two arc-counts around a circle and double-counting the endpoints.
Forgetting to recheck ALL constraints on your final answer — not only the ones you used to find it.
Trusting a confident claim. In liar puzzles, a stated number proves nothing — test it; honest speakers must agree.
Applying a steady rate past the finish. A net-per-day rate stops the moment you cross the line — the last move can overshoot, so check it by hand.
Remainder-0 off-by-one in a cycle. A position that is an exact multiple of the cycle length lands on the LAST slot of the block, not the first.
Trusting an unproven pattern. A loop you only eyeballed can break on the next term — know WHY it repeats before extending it.
Stopping the small-case table too early. The staircase counts 1, 2, 3 tempt you to guess 6 next, but the real rule adds the two before (5, then 8). Expand a few more cases before you trust the jump.

→ Full Logic & Word Problems lesson

SPATIAL & VISUAL REASONING

Spatial & Visual Reasoning

SPATIAL FACTS

A cube has 6 faces, 12 edges, 8 vertices; opposite faces never touch.
A cube net is 6 squares — exactly 11 different ones exist. Six squares alone do not guarantee a net; fold-check. Opposite faces skip one along a straight row (1–3, 2–4); two squares that touch are neighbours, never opposite.
Count 3D by layers: an n×n×n cube is n³ small cubes; remember the hidden ones. Each flat view is the tallest cube along your line of sight.
Die: opposite faces sum to 7, all six sum to 21, neighbours are never opposite. Opposite corners of a cube share the same corner-sum.
Tiles needed ≥ region area ÷ tile area — then check it is achievable (checkerboard colouring).
Back view = front view flipped left-right. Reflection flips across a line; rotation turns about a point. To shorten a bent touch-the-line path, reflect the far point across the line and draw it straight.
Cuts on a line: pieces = cuts + 1 (so 3 pieces need 2 cuts). Slicing a corner creates a new face with its own corners — count what a cut adds, not only what it removes.
Count figures inside a figure by size, then by orientation (up vs down); the down-pointing ones are the ones beginners miss.
Overcount, then subtract the double-counts (shared edges, tunnel crossings). When a count comes out, redo it a second way — two roads to the same number means you can trust it.

Common traps

COMMON TRAPS

Counting only visible cubes and forgetting the hidden ones.
Assuming any 6-square shape folds into a cube.
On a net, reading two squares that point opposite ways on the paper as opposite faces — once folded they often share an edge, so they are neighbours.
Confusing a reflection (mirror flip) with a rotation (turn).
Trusting the area ceiling without checking the tiles actually fit.
Forgetting the faces or corners a cut creates, or trusting an impossible die.
Reading ‘3 pieces’ as 3 cuts — pieces are always one more than cuts.
Forgetting to subtract overlaps when counting — or subtracting them twice; missing the down-pointing figures in a count-the-triangles problem.

→ Full Spatial & Visual Reasoning lesson

TEST-DAY HABITS

Test-day Habits

THE SEVEN HABITS

Read the question twice. Underline the question word. What does it actually ask?
Circle the tiny words. “inclusive,” “except,” “positive,” “non-negative,” “at most,” “different,” “distinct.”
Match the units. Cents vs dollars. Minutes vs hours. Inches vs feet.
Plug your answer back in. Does it satisfy every condition the problem stated?
Don’t trust mental math past 3 digits. Write it on scratch.
Re-read your final letter. The bubble has to match your work.
Two-minute rule. If a problem’s taking too long, mark it and move on.

USE THE ANSWERS

The answer choices are part of the problem. Ask: what do they have in common? What’s DIFFERENT about each?

All whole numbers → the answer is a whole number.
Three choices obviously too big or too small → cross them off without solving.
Choices differ by units digit → find the units digit; you’re done.
Choices differ by parity → find the parity first.

STUCK? RUN THE TOOLBOX

Work backward when the finish is clearer than the start — undo each step (add→subtract, times→divide).
Try the extreme case when sizes are free — push a quantity to its smallest or simplest value.
Make a table when facts pile up — let the grid do the remembering.
Split into cases on “how many ways” — tidy, non-overlapping cases that miss nothing.

THE THREE-PART CHECK

Conditions: does your answer satisfy every “and” and “each” in the problem?
Ballpark: is it roughly the size a quick estimate predicts?
Units & question: right units, and the thing actually asked (total not part, count not value)?

Common traps

Stopping at step 1. The question may have asked for X+Y, but you found just X.
Wrong unit. Answer is right in seconds but the question wanted minutes.
Off-by-one inclusive vs exclusive. “Pages 5 to 12” is 8 pages, not 7.
Bubbling the wrong letter. You wrote (C) but bubbled (B). A 2-second re-read prevents this.
Burning time on a hard problem. Every minute past 2 on one problem is two easy problems you didn’t reach.
Casework that overlaps or leaves a gap. Cases must miss nothing and double-count nothing — organize them around one thing (e.g. the number of dimes) and march through it in order.
Forgetting to divide by 2. When you count handshakes or pairs by “each of \(n\) does \(n-1\),” every pair is counted twice — divide by 2.

→ Full Test-day Habits lesson

→ Reaching for a perfect score (advanced techniques)