Parallelogram 23 Year 10 19 Mar 2020Chess In Church

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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.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by Sunday night if your whole class is doing parallelograms.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. Chess in church

2 marks

1.1 Which of these statements is not relevant to the picture below?

  • The queen moves in all directions.
  • The bishop only moves diagonally.
  • The knight moves in an L-shape.

Chess involves queens, bishops and knights, but this picture only shows a queen and a bishop on the checkerboard floor of a church.

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 The coach of the Irish hockey team has a maximum speed of 60 miles per hour. If it travels this speed for two hours, roughly how many kilometres does it travel?

  • 120
  • 160
  • 200
  • 240
  • 280
Show Hint (–1 mark)
–1 mark

This is an unusual question, because it expects you apply some previous knowledge, i.e., how many kilometres equal one mile. Roughly, there are more than 1.5 km per mile, but less than 2 km per mile. The exact value is 1.61 km per mile, or 21/13 km per mile. 21 and 13 are easy to remember because they are in the Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, 21, ...

One mile is approximately equal to 1.6 kilometres so 120 miles are approximately equal to 200 kilometres.

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 In the diagram, MON=130°.

The reflection of OP in OM is OQ and the reflection of OP in ON is OR.

What is the size of the obtuse angle QOR?

  • 100°
  • 120°
  • 140°
  • 150°
  • 160°
Show Hint (–1 mark)
–1 mark

If we say MOP=α, then PON=130°α.

Show Hint (–1 mark)
–1 mark

Then, QOR=α+α+130°α+130°α.

As OQ is the reflection of OP in OM, QOM=POM; similarly, RON=PON.

Hence reflex QOR=2×MON=260°.

Therefore QOR=360°260°=100°.

Alternatively, if we say MOP=α, then PON=130°α.

Then, QOR=α+α+130°α+130°α.

But, if the reflex angle is 260°, the obtuse angle must be 100°.

4. I’m not a robot

This is an interesting video by Tom Scott all about how computers can work out who is human and who is a robot (or internet bot).

(If you have any problem seeing the video then just right-click on the video and open the video in a new window)

2 marks

4.1 Which mathematician’s name is abbreviated as part of the term CAPTCHA? Just type his surname.

Correct Solution: TURING

2 marks

4.2 Which company is making CAPTCHAs and has servers that are probably being used to break CAPTCHAs?

  • NSA
  • CIA
  • KGB
  • MFI
  • Google

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 The diagram shows a 1 × x rectangular plank which fits neatly inside a 10 × 10 square frame. What is the value of x?

  • 10+22
  • 1021
  • 1022
  • 10+2
  • 12
Show Hint (–1 mark)
–1 mark

The trick with this question is to measure the red line, part of the diagonal of the square, which is the same length as the side of the rectangle.

Show Hint (–1 mark)
–1 mark

Here is a close-up of the corner of the square.

The figure shows the top left-hand corner of the complete diagram and we see by symmetry that the perpendicular from the corner to the short side of the rectangle has length 12.

Thus the diagonal of the square may be divided into three sections of length 12, x and 12 respectively.

Thus the length of this diagonal =102+102=200=102, so x=1021.

There will be more next week, so check your email or return to the website on Thursday at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.