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LogicGrades 1–6

Ordering

Who is taller, who is older, who came first? In these tasks a few comparisons have to be joined into one order: by height, by age, by a place in a queue or in a race. For many children this is the first chain of reasoning, in which every clue is a link and none may be left out.

The main move
Draw a line and put everyone on it
What it teaches
To join clues into one picture, and to read a comparison both ways
   tallest ........ shortest
    Ann > Ben > Kate > Dan

The rules of a line

Turn every clue the same way

Build the line clue by clue, after turning each one into “taller than”.

The clueTurned into “taller than”The line so far
“Ann is taller than Ben”Ann is taller than BenAnn > Ben
“Kate is shorter than Ben”Ben is taller than KateAnn > Ben > Kate
“Dan is shorter than Kate”Kate is taller than DanAnn > Ben > Kate > Dan

How to solve: draw the line

Draw a line from the tallest to the shortest, or from the first to the last, and put the clues on it one by one, the certain ones first. Whoever cannot be placed yet waits for the next clue. The examples below go from the simplest to an olympiad's.

  1. Turn every clue the same way
  2. Put the clues on the line, the certain ones first
  3. Read the answer, and check every clue against it

Example 1 · Grades 1–2

Three boats

Three boats raced: a red, a blue and a green one. The red boat finished after the blue one. The green boat finished before the blue one. Which boat won?

  1. Turn the first clue round: the red boat finished after the blue one, so the blue one finished before the red one.
  2. On the line, the first on the left: blue, then red. The green boat finished before the blue one, so it goes in front of both: green, blue, red.
  3. The first on the line is the green boat. Check: red after blue, yes; green before blue, yes.

Answer The green boat won.

Example 2 · Grades 3–4

The canteen queue

Five children stand in a queue at the canteen: Lena, Max, Nina, Oleg and Pavel. Nina stands third. Oleg stands right behind Nina. Max stands somewhere in front of Lena, and Lena is not last. Who stands last?

  1. Put down what is certain: Nina is third, and Oleg, right behind her, is fourth. Places 1, 2 and 5 are left for Max, Lena and Pavel.
  2. Lena is not last, so she is first or second. Max stands somewhere in front of her, so she cannot be first: Lena is second, and Max first.
  3. The only place left, the fifth, is Pavel's. Check every clue: they all hold.

Answer Pavel stands last.

Example 3 · Grades 5–6

Six runners

Six runners finished a race with no ties: Ada, Bob, Cid, Dot, Eli and Fay. Bob finished right after Ada. Cid finished after Dot but before Ada. Fay finished right before Eli, and Eli was not last. Dot was not first. Who won?

  1. Find the blocks: “Bob right after Ada” makes Ada-Bob, and “Fay right before Eli” makes Fay-Eli. Dot, then Cid, then the block Ada-Bob go in that order.
  2. Dot was not first, so someone finished before her. Cid, Ada and Bob all finished after Dot: only Fay and Eli can be in front of her, and they come together.
  3. So Fay-Eli opens the race: Fay, Eli, Dot, Cid, Ada, Bob. Eli is not last, as the clue says. Every clue holds.

Answer Fay won.

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. Line up the family

    Line everyone up at home by height, then by age, and say each clue aloud: “Dad is taller than Mum, so Dad stands first.”

  2. The line before the answer

    Ask for the line on paper before the answer. A drawing shows at once whether a clue is missing or turned the wrong way.

  3. Say it the other way round

    Play a quick game: you say “the cat is smaller than the dog”, and the child says it the other way round: “the dog is bigger than the cat”.

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