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Counting Studio
Build one example on the figure, then pick the tool that turns it into a count. Nothing gets counted one by one. Boards marked “Worked example” walk you through a new tool step by step.
Foundations
9Multiply, line up, and count what you don't want.
Ice-cream stand
≈ 400Worked exampleMultiply the choices
Five books
≈ 450Worked exampleLine them all upEarns Line them all up
Podium
≈ 500Multiply the choices
Giant bowling display
≈ 550Bowling pins
Palindromes
≈ 600Split into cases
A PIN with a 7
≈ 650Count the complement
Even numbers
≈ 750Pickiest first
Side by side
≈ 800Glue a block
Not side by side
≈ 850Count the complementGlue a block
Choosing
11Order that doesn't matter: divide it out, then choose.
Handshakes
≈ 700Worked exampleDivide the overcount
Three sodas
≈ 850Worked exampleDivide the overcountEarns Choose r of n
Three toppings
≈ 900Choose r of n
BANANA
≈ 900Divide the overcount
Starting seven
≈ 950Choose who stays out
Diagonals
≈ 950Divide the overcountCount the complement
Triangles in a circle
≈ 1000Divide the overcountChoose r of n
Round table
≈ 1000Divide the overcount
Walk to the park
≈ 1000A route is a word
Rectangles in a grid
≈ 1050Two lines each way
BOOKKEEPER
≈ 1100Divide the overcount
Sharing
5Identical things, gaps and grids.
Mixed
6No hints on the figure: decide which tools the count needs.
Competition
9Overlaps, recursion and counting twice.
Chess and band
≈ 1300Worked exampleInclusion–exclusion
Digits that add to 9
≈ 1300Give one each first
Multiples of 3 or 5
≈ 1350Inclusion–exclusion
Paving a path
≈ 1350Worked exampleBuild from smaller
A committee and its chair
≈ 1400Choose r of nCount it two ways
How many committees?
≈ 1400Count it two ways
Cats, dogs and fish
≈ 1450Inclusion–exclusion
Presents for everyone
≈ 1450Worked exampleMultiply the choicesInclusion–exclusion
The hopping frog
≈ 1500Build from smaller
Olympiad
7Bijections, lanes, sieves and symmetry: the ideas behind contest counting.
No two heads
≈ 1500Worked exampleTrack the ending
No neighbours
≈ 1600Worked exampleShift the gaps
Nobody gets more than 4
≈ 1650Stars and barsInclusion–exclusion
Never behind
≈ 1700Worked exampleReflection
Every card in the wrong envelope
≈ 1700Line them all upInclusion–exclusion
Balanced brackets
≈ 1800Reflection
Necklaces
≈ 1900Worked exampleCount the fixed ones
Final round
6Olympiad-style counts: name the tools, stack them, and watch for the step that breaks another.
Three dice make 12
≈ 1800Give one each firstInclusion–exclusion
Nobody idle
≈ 1800Multiply the choicesInclusion–exclusion
A spread-out draw
≈ 1850Shift the gaps
Nobody marks their own
≈ 1950Line them all upInclusion–exclusion
Three in a row
≈ 2000Track the endingCount the complement
Three-colour badge
≈ 2050Count the fixed ones