Parallelogram 15 Level 3 14 Dec 2023Chrismaths - Part 2

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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.

It’s nearly Christmas, so time for another instalment of your Christmaths challenge paper.

Good luck and happy Chrismaths. Don’t eat too many mince pie charts!


PS: I want to say thank you to the UK Mathematics Trust, who own the copyright to these questions.


3 marks

One lap of a standard running track is 400m.

How many laps does each athlete run in a 5000m race?

  • A) 4
  • B) 5
  • C) 8
  • D) 10
  • E) 1212
  • (Not answered)

Since each lap has length 400m and each athlete runs 5000m, the number of laps that each athlete runs is:



3 marks

What is the value of


  • A) 2
  • B) 10
  • C) 20
  • D) 40
  • E) 1,024
  • (Not answered)

Each number in the numerator is a multiple of 2. If we take out this common factor, we obtain:


It follows that:



3 marks

The diagram shows five circles placed at the corners of a pentagon. The numbers 1, 2, 3, 4, 5 are placed in the circles shown, one in each, so that the numbers in adjacent circles always differ by more than 1.

What is the sum of the numbers in the two circles adjacent to the circle which contains the number 5?

  • A) 3
  • B) 4
  • C) 5
  • D) 6
  • E) 7
  • (Not answered)

The number 3 cannot be placed next to either 2 or 4. So it will be between 1 and 5. Therefore 5 is next to 3, and not next to 1. Also, 5 cannot be next to 4. Therefore 5 is next to 2. Therefore the sum of the numbers next to 5 is 3 + 2, that is, 5.


4 marks

In the addition sum shown, each letter represents a different non-zero digit.

What digit does X represent?

  • A) 1
  • B) 3
  • C) 5
  • D) 7
  • E) 9
  • (Not answered)

From the units column we see that either E+E=S, or there is a carry, in which case E+E=10+S. However, from the tens column, we deduce that E+ES. So there is a carry from the units column to the tens column.

Therefore, from the tens column we see that: 1+E+E=10+E.

If we subtract E+1 from both sides of this equation, we deduce that E=9. It now follows from the units column that S=8. Hence the sum is

It follows that X represents 7.


5 marks

The diagram shows the first few squares of a ‘spiral’ sequence of squares. All but the first three squares have been numbered.

After the first six squares, the sequence is continued by placing the next square alongside three existing squares – the largest existing square and two others.

The three smallest squares have sides of length 1.

What is the side length of the 12th square?

  • A) 153
  • B) 123
  • C) 83
  • D) 53
  • E) 13
  • (Not answered)

We see from the diagram in the question that the side length of the 4th square is 3, and the side length of the 5th square is 4.

Also, the side length of the 6th square is the sum of the side lengths of the 4th and 5th squares. So the 6th square has side length 3 + 4 = 7. Similarly the side length of the 7th square is the sum of the side lengths of the 5th and 6th squares. So the 7th square has side length 4 + 7 = 11.

The sequence of side lengths continues in this way. After the first three numbers, each number in this sequence is the sum of the previous two numbers. Therefore the first 12 numbers in the sequence are:

1, 1, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123.

We therefore see that the side length of the 12th square is 123.

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.