Parallelogram 31 Year 9 11 Jun 2020Tangrams

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.
  • Tackle each Parallelogram in one go. Don’t get distracted.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • Your score & answer sheet will appear immediately after you hit SUBMIT.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Intermediate Maths Challenge Problem (UKMT)

3 marks

1.1 A ship’s bell is struck every half hour, starting with one bell at 0030, two bells (meaning the bell is struck twice) at 0100, three bells at 0130, and so on until the cycle is complete with eight bells at 0400. The cycle then starts again with one bell at 0430, two bells at 0500 and so on. What is the total number of times the bell is struck between 0015 on one day and 0015 on the following day?

  • 24
  • 48
  • 108
  • 144
  • 216

In a single cycle of 4 hours the bell is struck 1+2+3+4+5+6+7+8=36 times. In a 24 hour day there are 6 of these cycles. So in the 24 hours between 0015 one day and 0015 the next day the bell is struck 6×36=216 times.

2. Missing letter

Here is a puzzle via mathematics specialist teacher and former headteacher Drew Foster (@drewfoster0 on Twitter). If you can’t solve it, blame him, not me.

2 marks

2.1 Which letter replaces the question mark?

Correct Solution: q

The sum of the three numbers in each square corresponds to the letter of the alphabet in the corner of that square, eg:

  • 6 + 4 + 4 = 14 -> the 14th letter of the alphabet is N
  • 4 + 1 + 7 = 12 -> the 12th letter of the alphabet is L
  • 5 + 6 + 10 = 21 -> the 21st letter of the alphabet is U
  • 14 + 1 + 2 = 17 -> the 17th letter of the alphabet is therefore Q

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 A large cube is made by stacking eight dice. The diagram shows the result, except that one of the dice is missing.

Each die has faces with 1, 2, 3, 4, 5 and 6 pips and the total number of pips on opposite faces is 7.

When two dice are placed face to face, the matching faces must have the same number of pips.

What could the missing die look like?

Show Hint (–1 mark)
–1 mark

Although all standard dice have pips adding up to 7 on opposite faces, dice can vary according to their orientation.

From the diagrams above, we see that the faces with 4, 5 and 6 pips can go round the vertex where these faces meet in clockwise order (left) or in an anticlockwise order (right).

Although all standard dice have pips adding up to 7 on opposite faces, dice can vary according to their orientation. From the diagram showing the completed cube on the left below, we see that, for the missing die, the faces with 1, 3 and 5 pips go round the vertex where these faces meet in clockwise order. [In a differently oriented die, these faces would go round anti-clockwise. If you have a die to hand, check how these faces are oriented on it.]

So we need to decide which of the dice given as options have this orientation of these three faces. In the die of option A, the face with 3 pips is on the bottom, and we can visualize that the 1, 3 and 5 pip faces go round in anti-clockwise order. In the die of option B, the face with 5 pips is the opposite face to that with 2 pips, and we can again visualize that the 1, 3 and 5 pip faces go round anticlockwise. In the die of option C the face with 3 pips is opposite the face with 4 pips, and the face with 5 pips is opposite that with 2 pips. You should be able to visualize that this means that the faces with 1, 3 and 5 go round in clockwise order. So C is the correct option.

Of course, to complete the question, we really need to check that in the dice of options D and E the 1, 3 and 5 pip faces go round in anti-clockwise order. This is left to you to do.

4. Tangram

The journalist Alex Bellos, who has written some terrific books about mathematics (e.g., Alex's Adventures in Numberland) and football (e.g., Football School Season 1: Where Football Explains the World), penned an interesting newspaper column about the tangram. According to Alex, tangram was first ever puzzle craze.

First, you should read the article.

Then, have a play with this great tangram webpage from Mathigon, a website dedicated to mathematics education. See if you can create some of the tangram patterns.

5. Intermediate Maths Challenge Problem (UKMT)

4 marks

5.1 Suppose that 8m=27. What is the value of 4m?

  • 3
  • 4
  • 9
  • 13.5
  • there is no such m
Show Hint (–1 mark)
–1 mark

Notice that 8=23.

Show Hint (–1 mark)
–1 mark

Notice that 27=33.

Show Hint (–1 mark)
–1 mark

So 8m=27 may be written as 23m=33.

Notice that 8=23 and 27=33. So 8m=27 may be written as 23m=33.

So 23m=33, and hence 2m3=33. It follows that 2m=3. So 4m=22m=22m=2m2=32=9.

Before you hit the SUBMIT button, here are some quick reminders:

  • You will receive your score immediately, and collect your reward points.
  • You might earn a new badge... if not, then maybe next week.
  • Make sure you go through the solution sheet – it is massively important.
  • A score of less than 50% is ok – it means you can learn lots from your mistakes.
  • The next Parallelogram is next week, at 3pm on Thursday.
  • Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon.