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Grades 4–8
7 sections · 7 questions

Count without listing

Seven questions where writing them all out is not an option — and seven different ways of getting the count wrong.

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What's on this test

Section 1 is handshakes: everyone meets everyone, so each person's count is easy and the total is exactly twice the answer, because a handshake belongs to two people. Section 2 is a polygon's diagonals, which is the same count with the sides taken off — and the halving is still there to forget. Section 3 is socks in a dark drawer, and it is not a count of cases at all: it is a count of the unluckiest run of bad luck, plus one. Section 4 is a corridor of lockers, opened and shut by every number that divides them, where only the perfect squares end up open. Section 5 is a painted block cut into little cubes, where the paint on each one is decided entirely by where it sat. Section 6 is triangles with whole-number sides and a fixed perimeter, where most ways of splitting the number are not triangles at all. Section 7 is a rod pushed corner to corner through a block of unit cubes, counted by the walls it crosses rather than cube by cube.

How the paper is put together

The order is deliberate. Each of the first three sections' classic mistake is the previous section's lesson: a child who has just learned to halve meets the question where the subtraction takes their attention and the halving slips, and a child who has learned both meets a question where neither applies and the method is to imagine the worst instead. The later sections change the trap entirely — the lockers surprise is that almost every one ends shut, the painted cube's is that stripping the outer shell leaves a smaller cube whose size does the counting, the triangles' is that a whole list of splits has to be thrown away for failing to close up, and the rod's is that a coordinated box like 2 × 2 × 2 lines its walls up so the naive six becomes two.

Count without listing

Name: Date:

1. Challenge puzzlesCount the pairs without listing them. Show your working.1 question

1A league has 9 teams. Every team plays every other team twice — once at home and once away.How many matches are played?Why

Count without listing

2. Challenge puzzlesCount them without drawing them. Show your working.1 question

27 towns are arranged in a ring, and a road already joins each town to the two beside it. New roads are built so that every pair of towns is joined directly.How many new roads are built?Why

Count without listing

3. Challenge puzzlesSay how you know your number is always enough, however the picking goes.1 question

3A bag holds 12 red beads and 11 blue beads. You reach in without looking.How many beads must you take to be sure two of them match?Why

Count without listing

4. Challenge puzzlesA row of lockers, and every student flips a share of them. Say how many end open, and why.1 question

4A long hallway has 26 doors, numbered 1 to 26, all closed. Person 1 opens every door. Person 2 changes every second door, person 3 every third, and so on to person 26 — person k changes every door whose number k divides.In the end, how many doors are left open?Why

Count without listing

5. Challenge puzzlesA block is painted all over, then cut into little cubes. Count the ones the question asks for, and show why.1 question

5A big cube of sugar, 3 small cubes wide each way, is dipped so every outer face is coated, then broken into 3 × 3 × 3 small cubes.How many of the small cubes have no paint on them at all?Why

Count without listing

6. Challenge puzzlesCount the triangles with whole-number sides and the given perimeter, and show why the impossible ones are out.1 question

6A triangle is to be made with three whole-number side lengths adding up to 13.How many different triangles have three whole-number sides and a perimeter of 13? (A triangle and the same one turned around or flipped over count as one.)Why

Count without listing

7. Challenge puzzlesA rod is pushed corner to corner through a block of unit cubes. Count the ones it passes through, and show why a + b + c is too many.1 question

7A 2 × 2 × 6 box is packed exactly with unit cubes. A laser is aimed from one corner to the corner diagonally opposite, passing straight through the block.How many of the unit cubes does it pass through?Why

Think about it

Every count here was too big to write out. What did you count instead of the thing itself?

TeacherMath Picnic

Answer key

Challenge puzzles

If they are stuck: Pick one of them and count who they meet. Is that true of every one of them?

1. Each of the 9 meets the other 8, and this time the two directions are different events. So 9 × 8 = 72 matches, with no halving.

Challenge puzzles

If they are stuck: Join every pair first, including the ones already joined. What do you then take off?

2. Every pair of the 7 could be joined. Each town reaches 6 others, so 7 × 6 = 42 — and each pair is counted from both ends, so 42 ÷ 2 = 21 lines in all. But 7 of those roads already exist, so take them off: 21 − 7 = 14 new roads. More than one way works here and it never counts the roads already there: each town joins all but itself and its two neighbors, so 7 × 4 = 28, and 28 ÷ 2 = 14.

Challenge puzzles

If they are stuck: Do not count the cases — ask what the unluckiest possible run looks like.

3. There are 2 colors here (12 red and 11 blue). The worst that can happen is taking one of each — 2 beads and still no pair. The next one has to match something, so 2 + 1 = 3. Notice the counts never came into it.

Challenge puzzles

If they are stuck: How many times does each one get changed? Ask when that count is ODD.

4. Each door numbered L is changed once for each divisor of L. Divisors come in pairs (d with L ÷ d), an even count, so a door ends closed unless a divisor pairs with itself — that is, unless L is a perfect square. The doors left open are the perfect squares up to 26: 1, 4, 9, 16, 25. The largest is 5 × 5 = 25, so there are 5 of them and 5 doors stay open.

TeacherMath Picnic

Answer key

Challenge puzzles

If they are stuck: Where must a little cube sit to touch that many faces? Peel the painted shell off and look at what is left.

5. No paint at all means a cube deep inside, never on the surface. Peel one layer off all six sides and the cube left inside is 1 on each edge: 1 × 1 = 1, then 1 × 1 = 1 cube hidden in the middle.

Challenge puzzles

If they are stuck: Fix the longest side first, then ask which shorter pairs can actually reach across it.

6. Sides a ≤ b ≤ c with a + b + c = 13 only make a triangle when a + b > c — the two shorter sides have to out-reach the longest, or they never close up. So a split like (1, 1, 11) is out. Walking the shortest side up from 1 and keeping the splits that obey it: (1, 6, 6), (2, 5, 6), (3, 4, 6), (3, 5, 5), (4, 4, 5). That is 5 different triangles.

Challenge puzzles

If they are stuck: Count a + b + c first. Then ask where the rod crosses two walls at once — those got counted twice.

7. Corner to corner, the rod enters a new unit cube at every wall it crosses — 10 of them if none lined up, which is the count to beat. But two walls meeting on one line are crossed together: that happens gcd(2,2) = 2, gcd(2,6) = 2 and gcd(6,2) = 2 times, 6 in all, each an over-count of one. Taking them off: 10 − 6 = 4. All three walls meet together gcd(2,2,6) = 2 times, and that was subtracted once too often, so add it back: 4 + 2 = 6.

Count without listing

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