Computation: fluency is the foundation
Computation comes first, but the goal is not raw speed - it is the idea of rounding structures and rewriting: building 100s like 25x4, extracting common factors, converting between fractions and decimals. Classic types: clever arithmetic, invented operations, series sums (Gauss). Start in grades 2-3 once the four operations are fluent. Tools: the math drill, the 24 game (the best gamified rounding practice), fractions, times tables.
Number theory: divisibility and remainders
Number theory studies properties of integers; at primary level the core is divisibility rules, parity analysis, prime factorisation and remainders. Classic types: tests for 2/3/4/5/9/11, GCD and LCM applications, parity arguments, page-number puzzles. Systematic study from grade 4. Tools: the primes tool, base conversion, Base encoding.
Word problems: translating text into relations
Word problems form the main body of olympiad maths; the core skill is finding the invariant - the age gap, the fixed sum or difference, the fixed total. Classics: chickens and rabbits (assumption), ages, sum-difference-multiple, surplus-and-shortage, averages. What is trained is the habit of sorting out relations before naming unknowns; attack one type at a time from grades 3-4. Tools: chickens & rabbits, age problems, sum-diff-multiple, travel, work, concentration.
Geometry: shapes and area transformations
Primary olympiad geometry is mostly rectilinear area: from rectangles to triangles and parallelograms, with the core idea of cut-and-paste and equal-area transforms (same base, same height). Classic types: counting rectangles and squares, square arrays, lattice polygons. Start with shape counting in grade 3, area ratios in grade 5. Tools: square arrays, magic squares, patterns.
Counting: order matters or it does not
Counting teaches completeness without duplication: categories must be exhaustive and mutually exclusive. From enumeration to the addition and multiplication principles and the first look at permutations and combinations. Classic types: counting segments, pairings, routes, and the pigeonhole principle (the worst-case principle is counting next of kin). From grade 4. Tools: pigeonhole, grouping, tree diagrams (enumeration made visible).
Strategy: extremes, logic and games
Max-min problems train the extreme principle (fixed sum favours equal numbers, split mostly into 3s, fixed perimeter favours the square); logic puzzles train elimination and assumption; strategy games teach working backwards. What these share is the absence of a formula - the skill is recognising which known structure the problem hides. Suits grade 5 onwards. Tools: max-min problems, the mission game, towers of Hanoi, sudoku.
A sensible order
Suggested order: grade 3 - computation plus simple word problems (sum-diff-multiple, chickens and rabbits); grade 4 - number theory, travel, enumeration; grade 5 - areas, work and concentration, max-min and pigeonhole; grade 6 - synthesis and past papers. More important than the order is how each type is trained: understand one worked idea, do five same-structure questions, revisit after three days. Every solver on this site shows step-by-step reasoning - when stuck, read the first step, not the last answer.
想马上练：从 数学与奥数专题 挑当前模块的工具，每天 20 分钟；口算基础用 口算出题 保持。家长怎么安排练习节奏，看 三年级口算怎么练。Practise now: pick the tool for the current module in the maths and olympiad collection, twenty minutes a day; keep arithmetic sharp with the math drill. For scheduling, see the mental maths guide.
FAQ
Which grade should olympiad maths start?
Clever arithmetic and shape counting can start in grades 2-3; systematic number theory and travel problems suit grade 4 onwards. The test is schoolwork: expand only when in-school accuracy is stable above ninety percent. Starting a year or two late costs nothing - maturity matters more than a head start.
Does olympiad maths conflict with school maths?
Not if positioned right: it extends school maths rather than pre-teaching the textbook. The same ideas - rounding, invariants, classification - deepen understanding of school content. Conflict usually means the pace or difficulty is too high; step down half a level and it smooths out.
Do solver tools stop children thinking?
It depends on usage. The healthy loop: attempt it yourself, and only after ten stuck minutes open the step-by-step reasoning; compare where your path diverged; test with a similar question the next day. Copying the final answer is the least useful use - the value is in clarifying the step you were stuck on.