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Parallelogram 42 • Level 3 • 19 Jun 2025Miracle Chocolate

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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.
  • 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. An end of year message from Simon Singh

As this is the penultimate Parallelogram of the year, Simon has a special end of year message for you... and a quiz question for you to answer:

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

2 marks

1.1 Which mathematical constant was Simon asking about?

  • e
  • π
  • Τ
  • i
  • (Not answered)

June 28th is Tau (Τ) day, because Tau is equal to 2π, which is approximately 6.28.

2. Paradoxes

Last week, I showed you the missing square illusion, in which some shapes are re-arranged... and somehow the area at the end is less the area at the start. It seemed impossible.

Here is the same illusion made real with the help of a bar of chocolate.

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

These two videos (by James Grime) present the mystery in a different way and explain the maths behind the illusion.

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

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

The missing square illusion could be called a “paradox”, which means “a statement that contradicts itself or a situation which seems to defy logic.”

Here is an example of a paradox:

Your mission is not to accept the mission. Do you accept?

If you accept the mission, then you have failed, because the mission is to not accept the mission. But if you do not accept the mission, then you have also failed, because you did not accept the mission.

Try this one:

Answer truthfully (yes or no) to the following question: Will the next word you say be 'no'?

If your next word is “yes” then you have lied, because your next word was not “no”. And if your next word is “no”, then you have lied, because you said that your next word would not be “no”, but it was “no”.

And paradoxes can also come in the form of pictures. Take a look at this odd elephant.

3 marks

2.1 What is odd about the elephant?

  • Its tusks are uneven.
  • Its ears are too small.
  • It does not have a tail.
  • It has too many legs, or maybe not enough.
  • It has a very poor memory.
  • (Not answered)

3. Junior Maths Challenge Problem (UKMT)

2 marks

3.1 The approximate cost of restoring the Flying Scotsman was £4 million. This was about 500 times the cost of building the steam engine in 1923.

Roughly what did the engine cost to build?

  • £800
  • £2000
  • £8000
  • £20,000
  • £80,000
  • (Not answered)

There are 4,000,000 pounds in £4 million. Therefore, the cost of building the engine, in pounds, was roughly

1500×4000000=4000000500=8000.

So, roughly, the engine cost £8000 to build.

4. Junior Maths Challenge Problem (UKMT)

3 marks

4.1 How many pairs of digits (p,q) are there so that the five-digit integer ‘p869q’ is a multiple of 15?

  • 2
  • 3
  • 4
  • 5
  • 6
  • (Not answered)
Show Hint (–1 mark)
–1 mark

If the number is a multiple of 15, then it must also be a multiple of both 3 and 5, so the last digit (q) must be 0 or 5.

Show Hint (–1 mark)
–1 mark

If q=0, then what values for p would make the number divisible by 3?

If q=5, then what values for p would make the number divisible by 3?

How many combinations of p and q do you have?

We have 15=3×5. Also, 3 and 5 have no common factors other than 1. It follows that the multiples of 15 are just the numbers that are multiples of both 3 and of 5.

For ‘p869q’ to be a multiple of 5 we require that its units digit q is either 0 or 5.

For ‘p869q’ to be a multiple of 3 we require that the sum of its digits is a multiple of 3.

We first consider the case where q is 0. In this case the sum of the digits of ‘p869q’ is given by

p+8+6+9+0=p+23.

Because ‘p869q’ is a five-digit integer, p is not 0. So the possible values of p are 1, 2, 3, 4, 5, 6, 7, 8 and 9. It is easily checked that there are three of these digits, namely 1, 4 and 7, for which p+23 is a multiple of 3.

Similarly, when q is 5, the sum of the digits of ‘p869q’ is

p+8+6+9+5=p+28.

In this case there are also three choices of p, namely 2, 5 and 8, for which p+28 is a multiple of 3.

It follows that the number of pairs of the digits (p,q) for which ‘p869q’ is a multiple of 15 is 3 + 3, that is, 6.

[Note: as we have found above, these pairs are (1, 0), (4, 0), (7, 0), (2, 5), (5, 5) and (8, 5).]

5. Junior Maths Challenge Problem (UKMT)

3 marks

5.1 The areas of the two rectangles in the diagram are 25 cm2 and 13 cm2 as indicated.

What is the value of x?

  • 3
  • 4
  • 5
  • 6
  • 7
  • (Not answered)

The rectangle with area 13 cm2 has width 4 cm. Therefore the height of this rectangle is (13 ÷ 4) cm, that is, 134 cm.

It follows that the height of the rectangle with area 25 cm2 is 3+134 cm.

Because this rectangle has area 25 cm2, its width is 25÷3+134 cm.

Now 3+134=124+134=254. It follows that 25÷3+134=25÷254=25×425=4.

Hence the value of x is 4.

6. Junior Maths Challenge Problem (UKMT)

3 marks

6.1 A single polygon is made by joining dots in the 4×4 grid with straight lines, which meet only at dots at their end points. No dot is at more than one corner.

The diagram shows a five-sided polygon formed in this way.

What is the greatest possible number of sides of a polygon formed by joining the dots using these same rules?

  • 12
  • 13
  • 14
  • 15
  • 16
  • (Not answered)

The number of sides of the polygon is equal to the number of corners it has. As no dot is at more than one corner, the maximum number of corners is 16. So the maximum possible number of sides is 16, provided that a 16-sided figure may be drawn.

The figure on the right shows one of several ways in which this can be achieved.

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.