Parallelogram 42 • Level 2 • 19 Jun 2025Pet Puzzle
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. 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)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
2. Pet Puzzle
A cat and a dog weigh 10 Kg. A dog and a rat weigh 7 Kg. And a cat and a rat weigh 5 Kg.
2.1 What is the combined weight of a dog, cat and rat?
(See if you can find a quick way to solve this problem.)
Correct Solution: 11 kg
SLOW APPROACH
(1) C + D = 10
(2) D + R = 7
(3) C + R = 5
Line (1) tells us that C = 10 – D, so we can swap the C in line (3) to give us…
[10 – D] + R = 5.
This means that R = D – 5, and we can swap this into line (2) to give us …
D + [D – 5] = 7
2D = 12
D = 6.
If D = 6, then line (1) tells us that C = 4.
Then line (3) tells us that R = 1.
So, the total weight is D + C + R = 6 + 4 + 1 = 11 Kg.
QUICK APPROACH
If we add (1) + (2) + (3), then we have two of each animal, which means.
2D + 2C + 2R = 10 + 7 + 5 = 22.
So, D + C + R = 11 Kg.
3. Three Simple Ways to Time Travel
(If you have problems watching the video, right click to open it in a new window)3.1 If you want time to slow down, which one of these will help?
- Do nothing
- Go for a walk
- Climb to the top of a mountain
- Watch paint dry
- (Not answered)
3.2 Because of the stronger gravity, which part of your body ages most slowly? Which part of your body generally experiences time passing more slowly?
- The top of your head
- Your nose
- Your heart
- Your belly button
- Your ankles
- (Not answered)
3.3 How big is a femtosecond?
- Thousandth of a second
- Millionth of a second
- Billionth of a second
- Trillionth of a second
- Quadrillionth of a second
- (Not answered)
3.4 How big is a femtosecond?
- One millionth of one millionth, of a second
- One thousandth of one billionth, of a second
- One hundredth of one billionth, of a second
- One trillionth of one billionth, of a second
- One millionth of one billionth, of a second
- (Not answered)
4. Junior Maths Challenge Problem (UKMT)
4.1 The diagram shows a partially completed magic square, in which all rows, all columns and both main diagonals have the same total.
What is the value of
- 10
- 11
- 12
- 13
- 14
- (Not answered)
Label the number in the top right hand corner as “
The total of the diagonal is (
Let w be the number in the top-right square, as shown in the figure.
Because we are dealing with a magic square, the total of the numbers in the right-hand column is the same as the total of the numbers in the diagonal from the bottom-left square to the top-right square.
This gives
Hence, by subtracting w from both sides, we deduce that
It follows that
5. Junior Maths Challenge Problem (UKMT)
5.1 How many integers are greater than
- 320
- 321
- 322
- 323
- 324
- (Not answered)
We have
The integers that are greater than 38 and less than 360 are the integers from 39 to 359, inclusive.
There are
6. Junior Maths Challenge Problem (UKMT)
6.1 Gill scored a goal halfway through the second quarter of a ‘teachers versus pupils’ netball match.
At that point, what fraction of the whole match remained to be played?
1 4 3 8 1 2 5 8 3 4 - (Not answered)
Because
At the time when Gill scored the goal the first quarter and half the second quarter of the whole match had been completed. So the fraction of the game that had been completed was
Therefore the fraction of the game that remained to be played was
6.2 Gill scored a second goal one-third of the way through the third quarter of the netball match. At that point, what fraction of the whole match remained to be played?
7 12 1 3 5 12 2 3 1 4 - (Not answered)
The second goal was scored after:
So,
7. Junior Maths Challenge Problem (UKMT)
7.1 Beatrix places copies of the L-shape shown on a
She is allowed to turn around or turn over an L-shape.
What is the largest number of L-shapes she can place on the board without overlaps?
- 2
- 3
- 4
- 5
- 6
- (Not answered)
The answer is not 4 shapes.
The figure shows one way of placing 5 L-shapes on the board without overlaps.
The 4 × 4 board is made up of 16 cells. The L-shape is made up of 3 cells.
Therefore, however 5 L-shapes are placed on the board without overlapping, there will be just one cell that remains uncovered. So there will not be space for one more L-shape.
Hence, the largest number of L-shapes that Beatrix can place on the board without overlaps is 5.
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.