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Artful measuring

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Difficulty: Tricky

Holly loves drawing with coloured pencils and her favourite colours have been sharpened a lot. She realises that her most favourite colour, rose, is exactly 3 centimetres long. She measures her other pencils and discovers that the teal pencil is exactly 4 centimetres, and the almond pencil is exactly 6 centimetres (cm).

Holly has left the house with her pencils, and is just about to start drawing when she notices that she forgot her ruler. Then she realises she can use the pencils to measure instead! For example, Holly can find 7 cm by placing the rose pencil (3 cm) and the teal pencil (4 cm) end to end.

What lengths from 1-10 cm can Holly make with her pencils? Is there any length that is impossible?

Bonus question: Can you find a creative way to measure the impossible length?

Need a hint?

We recommend using graph paper. Label 1 through 10 on the page, giving yourself some space to draw for each number. You can represent a pencil by drawing a rectangle around its length (for example, the 3 cm rose pencil fills 3 squares).

Start with the lengths that are already given: 3, 4 and 6. Next, fill in the lengths that can be satisfied with addition. This is shown by putting pencils end-to-end (for example, 3 + 4 = 7).

Here’s the big hint: you can use subtraction to find other lengths. For example, 3 and 4 stacked on top of each other with their back ends lined up measures out 1 cm (the distance between their points).

Here’s the hint for the bonus question: according to the images provided, what is the width of the pencils?

8 is the “impossible” length. Holly can use addition and subtraction of the pencils’ lengths to measure 1 cm through 10 cm except for 8 cm.

We used graphing paper to visualise our solutions, assuming that each square of the graphing paper is 1 cm by 1 cm. We found some lengths using addition, as shown by putting pencils end-to-end. We found other lengths using subtraction, as shown by stacking pencils on top of each other, lining up their back ends and finding the distance between their pointed ends. At least one length (5 cm) needed a combination of addition and subtraction!

Here are the descriptions of the solutions:
1 = 4 – 3: place the 4 cm teal pencil on its side then stack the 3 cm rose pencil on top with their back ends lined up. 1 cm is the distance between their pointed ends.
2 = 6 – 4: place the 6 cm almond pencil on its side then stack the 4 cm teal pencil on top with their back ends lined up. 2 cm is the distance between their pointed ends.
3 = 3: place the 3 cm rose pencil.
4 = 4: place the 4 cm teal pencil.
5 = (6 + 3) – 4: place the 6 cm almond pencil and the 3 cm rose pencil end-to-end to make 9 cm. Then stack the 4 cm teal pencil on top of the almond pencil with their back ends lined up. 5 cm is the distance between the teal pencil’s point and the rose pencil’s point.
6 = 6: place the 6 cm almond pencil.
7 = 3 + 4: place the 3 cm rose pencil and the 4 cm teal pencil end-to-end to make 7 cm.
8 = ?: This is the only length we can’t make with addition and subtraction of the pencil’s lengths. The closest we can get is (6 + 4) – 3, which is another way to make 7.
9 = 3 + 6: place the 3 cm rose pencil and the 6 cm almond pencil end-to-end to make 9 cm.
10 = 6 + 4: place the 6 cm almond pencil and the 4 cm teal pencil end-to-end to make 10 cm.

Look at the graphing paper: each square is 1 cm by 1 cm and each pencil fills the width of 1 square. This means that the width of each pencil is 1 cm. We got to 7 cm with just the rose and teal pencils (3 + 4) and adding the 1 cm width of the almond pencil gets us to 8 cm!

Did you find an alternate solution? Tell us about it in the comments!

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4 responses

1. Martin

Probably not what is exactly asked but: We don’t really need the 6cm pencil. Since the difference between the 4 and the 3 cm pencils is 1cm, all we need to do is keep adding 1cm lots to make any length.

1. Ariel Marcy

Great lateral thinking! Perhaps the 6cm pencil could be used to mark the 1cm lots as you add them up. Thanks so much for sharing your solution.

2. Christine

5 + 3, actually any group of numbers that will add up to 8, including 1 + 1 + 1 + 1 + 1 + 1 + 1 +1

1. Ariel Marcy

Great reasoning! For more of a challenge, can you get to 8 using each of the 3 pencils only once?

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