Parallelogram 43 • Level 1 • 26 Jun 2025Maths mentalism
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.
- 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. Maths mentalism
It is impossible to read someone’s mind... unless you have a helping hand from maths!
In this video you will see a number trick performed on you. Warning: your mind may be a little blown!
(If you have problems watching the video, right click to open it in a new window)1.1 In this trick, what happens after you choose a number?
- You immediately choose a second number
- You cross out the numbers in the same column as your chosen number
- You cross out the numbers in the same row as your chosen number
- You cross out the numbers in the same column or row as your chosen number
- (Not answered)
1.2 Which of the following options sets of numbers is it possible to choose in this trick?
- 1, 7, 11, 16
- 3, 9, 10, 12
- 2, 5, 12, 15
- 13, 14, 15, 16
- (Not answered)
- 7 and 11 are in the same column, so the first option is impossible
- 9 and 12 are in the same row, so the second option is impossible
- 13, 14, 15, 16 are all in the same row so the final option is impossible
1.3 How many numbers must you end up choosing?
- 3
- 4
- 16
- It depends on which numbers you choose
- (Not answered)
Try choosing your numbers a few different ways.
You will see that you always end up with 4 numbers: one for each row and one for each column.
1.4 The trick works because the total is always 34, regardless of which numbers you choose.
You can find this total by summing the main diagonal (1, 6, 11 and 16).
If we extended the trick to a 5x5 grid, allowing someone to choose five numbers, what would the total be this time?
- 39
- 49
- 65
- 68
- (Not answered)
Here is the 5×5 grid
Here is the 5×5 grid
The main diagonal of the 5x5 grid contains the numbers 1, 7, 13, 19, 25, which sum to 65.
2. Square false teeth and babies
2.1 Because she has square false teeth, my wife makes square bites (with sides of 3 cm) when she eats toast.
What area is left after she takes four bites out of a square slice of toast that measures 12 cm × 12 cm?
- 36 cm 2
- 108 cm 2
- 132 cm 2
- 140 cm 2
- 144 cm 2
- (Not answered)
The area of each square bite is 3 × 3 = 9 cm 2.
The area of a piece of toast is 12 × 12 = 144 cm 2.
So the area remaining after 4 bites is 144 − 4 × 9 = 108 cm 2.
2.2 My baby sister is 1,000 days old today.
How many years old will she be on her next birthday?
- 3
- 4
- 5
- 10
- 12
- (Not answered)
You do not need to do any complicated calculations - just consider how many days, roughly, are in one year.
1 year is roughly 350 days, 2 years roughly 700 days, 3 years roughly 1050, so my sister is now between 2 and 3 years.
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...
- 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.
- 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.
- This was our last Parallelogram of the year, but be sure to come back in September for more puzzles and problems.
Cheerio, Simon.