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Numbers with a trickGrades 1–6

Parity and alternation

Can 15 children stand in a ring, boy, girl, boy, girl, all the way round? In these tasks even and odd answer the question without counting everything or trying every way: which bank a ferry is at, whether a score is possible, who makes the last move. For many children parity is the first thing they meet that stays the same however the moves go.

The main move
Watch what each move does to even and odd
What it teaches
To answer “can it be?” without trying every way
●  ○  ●  ○  ●  ○
1  2  3  4  5  6

The rules of even and odd

Count only the odd numbers

To tell whether a sum is even, there is no need to add it up. Count how many odd numbers are in it.

The sumOdd numbers in itThe sum is
4 + 6 + 8noneeven
3 + 52even
3 + 5 + 73odd
1 + 2 + … + 105odd

How to solve: follow even and odd

Find a number in the task that is even or odd: a count, a sum, a place in a row. See what each move or each step does to it, whether it changes or stays, and compare the start with the goal. The examples below go from the simplest to an olympiad's.

  1. Find what is even or odd
  2. See what each step does to it
  3. Compare the start with the goal

Example 1 · Grades 1–2

The ferry

A ferry carries cars across a river and back, again and again. It starts at the left bank. At which bank is it after crossing the river 7 times?

  1. After 1 crossing the ferry is at the right bank, and after 2 back at the left.
  2. Every two crossings bring it back to where it started. So after an even number of crossings it is at the left bank, and after an odd number at the right.
  3. 7 is odd, so the ferry is at the right bank.
 left    |  right
0 2 4 6  |  1 3 5

Answer At the right bank.

Example 2 · Grades 3–4

Four darts

Ben throws 4 darts at a target whose rings are worth 1, 3, 5 and 7 points, and every dart hits it. Ben says he scored 15 points. Can that be true?

  1. Every ring is worth an odd number of points.
  2. Two odd numbers add up to an even one. 4 darts are two pairs, and each pair scores an even number, so the 4 darts together score an even number.
  3. 15 is odd, so Ben cannot have scored it.

Answer No.

Example 3 · Grades 5–6

A spider on a cube

A spider sits at a corner of a cube made of wire. Every minute it crawls along one edge to the next corner. Can it be back at the corner it started from after exactly 7 minutes?

  1. Colour the corners: the start white, its neighbours black, their neighbours white, and so on. As the drawing shows, every edge joins a black corner and a white one.
  2. So the colour changes every minute: after 1 minute the spider is on black, after 2 on white, and after any odd number of minutes on black.
  3. 7 is odd, and the start is white. The spider cannot be back.
  ○-------●
 /|      /|
●-------○ |
| ●-----|-○
|/      |/
○-------●

Answer No.

Where children go wrong

MathTrail explains every wrong answer by the trap it fell into. These are the commonest in this topic, with something you can say to your child.

How to help at home

  1. Odd or even?

    Hide a handful of buttons in your fist and let the child guess: odd or even? Then check together by putting the buttons in pairs.

  2. Who makes the 10th move?

    In any game where two players take turns, ask before the end: who will make the 10th move? And the 15th? The answer comes from even and odd, not from playing on.

  3. House numbers

    On a street, look at the house numbers: even on one side, odd on the other. Which side will number 37 be on?

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