Parallelogram 29 • Level 2 • 21 Mar 2024Bedtime stories
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.
- When you finish, remember to hit the SUBMIT button.
- Finish by Sunday night if your whole class is doing parallelograms.
- 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 Beth, Carolyn and George love reading their favourite bedtime stories together.
They take it in turns to read a page, always in the order Beth, then Carolyn, then George.
All twenty pages of the story are read on each occasion.
One evening, Beth is staying at Grandma's house but Carolyn and George still read the same bedtime story and take it in turns to read a page with Carolyn reading the first page.
In total, how many pages that Carolyn and George read are the same as the pages that they would read if Beth was there?
- 1
- 2
- 4
- 6
- 7
- (Not answered)
One way to do this is to write down a list of 20 pages and then label how they would be read usually and then on the night that Beth is away.
| Usual | B | C | G | B | C | G | B | C | G | B |
| Page | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| C | G | C | G | C | G | C | G | C | G |
| Usual | C | G | B | C | G | B | C | G | B | C |
| Page | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| C | G | C | G | C | G | C | G | C | G |
By comparison, you can see that pages 5, 6, 11, 12, 17 & 18 are read by the same person, regardless of Beth’s absence. Next is a more mathematical approach to the problem.
When all three children are present, Carolyn reads pages 2, 5, 8, ... , that is those pages whose number leaves remainder 2 when divided by 3. George reads pages 3, 6, 9, ... that is, those whose number is a multiple of 3. When Beth stays at Grandma's, Carolyn reads pages 1, 3, 5, ... that is the odd numbered pages, and George reads the even numbered pages. So the pages that Carolyn reads both normally and also when Beth is away are those with numbers which have remainder 2 when divided by 3, and are odd. That is, the three pages 5, 11 and 17.
The pages that George reads both normally and also when Beth is away are those with numbers that are multiples of 3 and are even, that is, the three pages 6, 12 and 18. So there are altogether 6 pages which are read by the same person normally and when Beth is away, namely pages 5, 6, 11, 12, 17 and 18.
2.
2.1 Beatrix looks at the word JUNIOR in a mirror. How many of the reflected letters never look the same as the original, no matter how Beatrix holds the mirror?
- 1
- 2
- 3
- 4
- 5
- (Not answered)
The letters J, N and R do not have an axis of symmetry. So these letters cannot look the same when reflected in a mirror, however the mirror is held. The letters U, I and O all have at least one axis of symmetry. So each may look the same when reflected in a mirror.
3.
3.1 A solid wooden cube is painted blue on the outside.
The cube is then cut into eight smaller cubes of equal size.
What fraction of the total surface area of these new cubes is blue?
1 8 1 3 3 8 1 2 3 4 - (Not answered)
As the original cube was divided into eight cubes of equal size, these smaller cubes have side equal to half the side of the original cube.
So each of the new cubes originally occupied one corner of the large cube and hence has three faces painted blue and three faces unpainted.
So the fraction of the total surface area of the new cubes which is blue equals one half.
4.
4.1 How many different digits appear when
- 2
- 3
- 4
- 5
- 6
- (Not answered)
5.
5.1 In the division calculation 952,473 ÷ 18, which two adjacent digits should be swapped in order to increase the result by 100?
- 9 and 5
- 5 and 2
- 2 and 4
- 4 and 7
- 7 and 3
- (Not answered)
If after division by 18 the number has to be increased by 100, then before the division it needs to be increased by 18 × 100 = 1800. So we need to find two adjacent digits in the number 952,473 which when swapped increase it by 1800.
If 952,473 is increased by 1800 it becomes 952,473 + 1 800 = 954,273.
We obtain 954,273 from 952,473 by swapping the adjacent digits 2 and 4.
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.