Parallelogram 34 Level 5 25 Apr 2024A Trapped Knight

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteau 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. A Trapped Knight

This is a fascinating Numberphile video presented by mathematical genius Neil Sloane. Although it relates to the knight in chess, you don’t need to know anything about chess, except that the knight moves in an L-shape, which means moving 2 squares in a line, then a turn by 90°, and moving one more square, as shown by the dots on this chessboard.

(If you have problems watching the video, right click to open it in a new window)

2 marks

1.1 The knight dashes around the infinite board according to the rules outlined by Neil Sloane, and it eventually gets trapped at square 2,084.

On its way to this square, it doesn't move through all the smaller numbered squares. What is the smallest numbered square that is not visited by the knight?

Correct Solution: 961

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 To make porridge, Goldilocks mixes together 3 bags of oats with 1 bag containing 20% wheat bran and 80% oats.

All the bags have the same volume. What percentage of the volume of Goldilocks' porridge mixture is wheat bran?

  • 5%
  • 6 23%
  • 20%
  • 26 23%
  • 60%
  • (Not answered)

One quarter of the bags contains one fifth of a bag of wheat bran. So the proportion of wheat bran in the porridge is 120, that is 5%.

5. Professor Wiseman’s puzzles

Another puzzle from the psychologist Professor Wiseman, whose website is worth a visit when you’ve completed this Parallelogram.

2 marks

5.1 What letter comes next in this sequence?

W, L, C, N, I, T, _

Correct Solution: S

Show Hint (–1 mark)
–1 mark

What Letter Comes Next In This Sequence…

Look at the initial letters of the question…

2 marks

5.2 What letter comes next in this sequence?

O, T, T, F, F, S, _

Correct Solution: S

Show Hint (–1 mark)
–1 mark

One, Two, Three…

Just take the initial letter of the words that represent the numbers One, Two, Three, Four, Five, Six, …. Seven.

2 marks

5.3 What letter comes next in this sequence?

A, S, D, F, G, H, _

Correct Solution: J

Show Hint (–1 mark)
–1 mark

Look at your keyboard

The letters are those that appear in the middle row of a keyboard, starting from the left.

7. Only Connect

I “borrowed” this question from the fantastic BBC quiz show ONLY CONNECT. Catch old episodes and watch clips at the ONLY CONNECT website.

4 marks

7.1 Which option comes next in this sequence?

  • A
  • B
  • C
  • D
  • E
  • (Not answered)
Show Hint (–1 mark)
–1 mark

Each box shows something related to a 7-segment digital display, such as a watch LCD display.

Show Hint (–1 mark)
–1 mark

Each box relates to a number, but which number? Is there a pattern? What is the most obvious pattern? Where would you start this pattern? There are two places to start.

Show Hint (–1 mark)
–1 mark

Look at which segments are missing. You will find that the missing segments make numbers, so what missing segments should be in the fourth box?

The answer is A. If you imagine the missing segments, then you get the pattern 0, 1, 2, ?. So, the next number is 3, but we want the segments left when you remove the number three.

9. Social Distancing in San Francisco

I am writing this Parallelogram during the pandemic lockdown (2020), when I received the picture below from Len Fisher, who explained that it was being put up as a reminder to people in San Francisco to keep 6 feet apart.

2 marks

9.1 However, there is a problem with this sign. If each person is 6 feet away from the person in the opposite corner, then each person is closer than that to the person in the adjacent (or neighbouring) corner.

What is that shorter distance? Give your answer to two significant figures.

Correct Solution: 4.2 feet

According to Pythagoras, a2+b2=c2.

We know that a=b, and that c=6, so:


So, a=18=4.2.

11. Classroom maths

3 marks

11.1 If you are in a classroom that is 3 m tall, and 5 m long and 5 m wide, then what is the mass of air in the classroom?

The density of air is 1.3 Kg per cubic metre.

  • roughly the same as the mass of a mouse.
  • roughly the same as the mass of a bag of sugar.
  • roughly the same as the mass of a teacher.
  • roughly the same as the mass of an African elephant.
  • (Not answered)

Mass of air = volume × density = 3 × 5 × 5 × 1.3 = 97.5 Kg, roughly the same as an adult human.

13. Parallel emails

If you use Parallel on an email address from your school, some school IT systems prevent you from receiving the emails we send out to remind you when a new Parallelogram is released, or when we have another exciting and nerdy maths thing to tell you about. To avoid this, you could give us a different email address we can contact you on - either a personal email address, or one for you parents or guardian. If you'd like to do that, you can put it in this form, but please ask a parent first if you are not yet 13 years old.

1 mark

13.1 If you do want to give us an alternate email address, be sure to click the link above (if you haven’t already). Either way, here's a free mark, just for being a fan of Parallel.

  • Thank you!
  • (Not answered)

The next Parallelogram will be out next 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.