Parallelogram 16 Level 4 21 Dec 2023Chrismaths - Part 3

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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

All four digits of two 2-digit numbers are different.

What is the largest possible sum of two such numbers?

  • A) 169
  • B) 174
  • C) 183
  • D) 190
  • E) 197
  • (Not answered)

To get the largest possible sum we need to take 9 and 8 as the tens digits, and 7 and 6 as the units digits. For example, 97 + 86 = 183.


3 marks

Auntie Fi’s dog Itchy has a million fleas. His anti-flea shampoo claims to leave no more than 1% of the original number of fleas after use.

What is the least number of fleas that should be eradicated by the treatment?

  • A) 900,000
  • B) 990,000
  • C) 999,000
  • D) 999,990
  • E) 999,999
  • (Not answered)

Since no more than 1% of the fleas will remain, at least 99% of them will be eradicated.

Now 99% of a million is 99100×1000000=99×10000=990,0.


3 marks

What value of x makes the mean of the first three numbers in this list equal to the mean of the last four?

15 ; 5 ; x ; 7 ; 9 ; 17

  • A) 19
  • B) 21
  • C) 24
  • D) 25
  • E) 27
  • (Not answered)

The mean of the first three numbers in the list is 15+5+x3 and the mean of the last four is x+7+9+174.


An alternative method in the context of the IMC would be just to try the given options in turn. This runs the risk of involving a lot of arithmetic, but here, as the first option is the correct answer, the gamble would pay off.


4 marks

The shaded region shown in the diagram is bounded by four arcs, each of the same radius as that of the surrounding circle.

What fraction of the surrounding circle is shaded?

  • A) 4π1
  • B) 1π4
  • C) 12
  • D) 13
  • E) It depends on the radius of the circle
  • (Not answered)

Suppose that the surrounding circle has radius r. In the diagram we have drawn the square with side length 2r which touches the circle at the points where it meets the arcs.

The square has area 2r2=4r2. The unshaded area inside the square is made up of four quarter circles with radius r, and thus has area πr2. Hence the shaded area is 4r2πr2=4πr2.

The circle has area πr2. So the fraction of the circle that is shaded is:



5 marks

All the positive integers are written in the cells of a square grid.

Starting from 1, the numbers spiral anticlockwise. The first part of the spiral is shown in the diagram.

Which number is immediately below 2012?

  • A) 1837
  • B) 2011
  • C) 2013
  • D) 2195
  • E) 2210
  • (Not answered)

The key to the solution is to note that the squares of the odd numbers occur on the diagonal leading downwards and to the right from the cell which contains the number 1, and the squares of the even numbers occur on the diagonal which leads upwards and to the left of the cell which contains the number 4.

The squares of the even numbers have the form (2n)2, that is, 4n2. We see that the number 4n2 + 1 occurs to the left of the cell containing 4n2. Below 4n2 + 1 there occur the numbers 4n2 + 2, 4n2 + 3, .., 4n2 + 2n + 1, and then in the cells to the right of the cell containing 4n2 + 2n + 1, there occur the numbers 4n2 + 2n + 2, 4n2 + 2n + 3,..., 4n2 + 4n + 1 = (2n + 1)2.

Now 442 = 1936 and 452 = 2025. Thus 2011 is in the same row as 2025 and to the left of it, in the sequence 1981, … ,2012, … ,2025, and below these occur the numbers 2163, … , 2209 = 472, with 2208 below 2025 as shown below.

It follows that 2195 is the number below 2012.

In the diagram below the square numbers are shown in bold.

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.