Parallelogram 43 • Level 3 • 26 Jun 2025Nice dice
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. A strange dice trick
Rolling a die is a great way to generate random numbers.
So how could Ben Sparks possibly know the total you get from rolling and stacking three dice?
Watch how this trick is done and maybe fool some of your friends with it later!
(If you have problems watching the video, right click to open it in a new window)1.1 What do the opposite sides of a dice always total?
- 6
- 7
- 8
- 9
- (Not answered)
1.2 How do you find the total for the touching faces on the stack?
- 21 - the top number
- The top number × 3
- It's always 19
- Mindreading abilities
- (Not answered)
1.3 I have some even more unusual dice:
- The red die has the numbers 1, 1, 6, 6, 8, and 8
- The green die has the numbers 2, 2, 4, 4, 9, and 9
- The blue die has the numbers 3, 3, 5, 5, 7, and 7
If I pick the red die, which die should you pick to have the best chance of rolling a higher number than me?
- Green
- Blue
- The red die is the best
- (Not answered)
These curious dice are called “non-transitive dice”.
Green beats red, red beats blue and blue beats green - so no die is the strongest.
Think about the game rock - paper - scissors, it also shares this property.
2. What are the chances?
2.1 A box contains two white socks, three blue socks and four grey socks.
Three of the socks have holes in them, but you don’t know what colour these socks are.
The room is pitch black and you remove the socks one by one.
How many socks must you take to be certain you have a pair of socks of the same colour without holes?
- 5
- 6
- 7
- 8
- (Not answered)
Imagine you have the worst luck possible:
Sock one has a hole...
sock two, another hole...
Now continue the worst case scenario until you MUST have a pair of matching socks with no holes.
Imagine you pick the three socks with holes in them first, so they can be cast aside.
The next three socks you pick are all different colours, at least they don’t have holes.
Then finally on the 7th sock - it cannot have a hole in it, and it must match one of the other three socks you have!
You did it!
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 and Ayliean.