Parallelogram 46 • Level 2 • 17 Jul 2025The apple pie bake-off
Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/
- 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.
- 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.
1.1 At the Marldon Apple-Pie-Fayre bake-off, prize money is awarded for 1st, 2nd and 3rd places in the ratio 3 : 2 : 1.
Last year Mrs Keat and Mr Jewell shared third prize equally.
What fraction of the total prize money did Mrs Keat receive?
1 4 1 5 1 6 1 10 1 12 - (Not answered)
Third prize is worth one-sixth of the total prize money, so Mrs Keat received half of that amount, that is one-twelfth of the total.
2.
2.1 What is the smallest four-digit positive integer which has four different digits?
- 1032
- 2012
- 1021
- 1234
- 1023
- (Not answered)
Here it is easy just to check the options that are given. A, D and E are the only options in which all four digits are different. Of these, clearly, E is the smallest.
For a complete solution we need to give an argument to show that 1023 really is the smallest four digit positive integer with four different digits. It is easy to do this.
To get the smallest possible number we must use the four smallest digits, 0, 1, 2 and 3. A four digit number cannot begin with a 0. So we must put the next smallest digit, 1, in the thousands place, as a four-digit number beginning with 2 or 3 is larger than one beginning with a 1. For similar reasons the hundreds digit must be the smallest remaining digit, 0. Similarly the tens digit must be 2 and the units digit must be 3.
So the required number is 1023.
3.
3.1 Which of the following has exactly one factor other than 1 and itself?
- 6
- 8
- 13
- 19
- 25
- (Not answered)
The factors of 6 are 1, 2, 3 and 6; the factors of 8 are 1, 2, 4 and 8; the factors of 13 are 1 and 13; the factors of 19 are 1 and 19; and the factors of 25 are 1, 5 and 25. We see from this that, of the numbers we are given as options, only 25 has exactly one factor other than 1 and itself. It is worth noticing that 25 is the only square number.
4.
4.1. Amy, Ben and Chris are standing in a row. If Amy is to the left of Ben and Chris is to the right of Amy, which of these statements must be true?
- Ben is furthest to the left
- Chris is furthest to the right
- Amy is in the middle
- Amy is furthest to the left
- None of statements A, B, C, D is true
- (Not answered)
From the information, we see that Amy is to the left of both Ben and Chris. So the three are in the order Amy, Ben, Chris or the order Amy, Chris, Ben.
So Amy is certainly furthest to the left, and the others are all false either in one case or in both.
5.
5.1 Each side of an isosceles triangle is a whole number of centimetres. Its perimeter has length 20 cm. How many possibilities are there for the lengths of its sides?
- 3
- 4
- 5
- 6
- 7
- (Not answered)
Let the length of the two equal sides of the isosceles triangle be
In a triangle, the length of one side must be less than the sum of the lengths of the other two sides. So
This gives
Therefore, as
So there are four possibilities for the side lengths of the triangle:
- 6, 6, 8;
- 7, 7, 6;
- 8, 8, 4; and
- 9, 9, 2.
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.