Parallelogram 21 • Level 4 • 23 Jan 2025A Shilling for your Thoughts
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. 357,686,312,646,216,567,629,137
Watch this video by James Grime from the Numberphile YouTube channel, which is all about the number 357,686,312,646,216,567,629,137. Watch carefully and then answer the questions below.
1.1. James showed you a truncatable prime, whereby the left digits can be removed one by one to create a set of smaller, but still prime, numbers. Which of these is a left-truncatable prime?
- 981
- 983
- 985
- 987
- 989
1.2. Which of these numbers is NOT a left truncatable prime?
- 643
- 647
- 651
- 673
- 683
1.3. Which of these numbers is NOT a left truncatable prime?
- 937
- 947
- 953
- 971
- 997
1.4. James showed you the largest left-trunctable prime number, which has 24 digits. How many digits are in the largest known prime number? You might have to google this.
- Roughly 1,000 digits.
- Roughly 10,000 digits.
- Roughly 100,000 digits.
- Roughly 1,000,000 digits.
- Roughly 10,000,000 digits
1.5. So far, we have been discussing LEFT-truncatable primes, but what about RIGHT-truncatable primes, numbers that remain prime as you successively remove the right-most digit?
Which of these numbers is NOT a right-truncatable prime?
- 719
- 733
- 739
- 743
- 797
1.6. Which numbers are more common?
- right-truncatable primes
- left-truncatable primes
- left- and right-truncatable primes are roughly equally common
2. A shilling coin
Before 1971, Britain had pounds, shillings and pennies (rather than pounds and pence). There were 20 shillings in a pound, and 12 pennies in a shilling, which means 240 pennies in a pound.
2.1. If you bought a magazine costing 1 shilling and 6 pennies, how much change would you get from a £1 note?
- 8 shillings and 4 pence
- 8 shillings and 6 pennies
- 18 shillings and 4 pennies
- 18 shillings and 6 pennies
- 98 shillings and 94 pennies
2.2. This image shows the back of a one shilling coin, and you can see that it is dated 1963 across the middle. However, why might someone mistakenly think the coin was minted in 1771. Which sector of the coin might give this false impression?
- 1
- 2
- 3
- 4
- 5
- 6
3. Proving that four random dots create something wonderful
Some of you may have seen this mathematical amusement from a previous year of the Parallel Project, but please look at it again, because this time you will need to think about it a bit more deeply.
Grab a pencil, paper and ruler.
- Make 4 random dots (or ask someone else to make them for you).
- Join up the dots to make a quadrilateral.
- Put a mark at the midpoint of each line in this first quadrilateral.
- Join up the four new points to make a second quadrilateral.
- The second quadrilateral is a parallelogram.
You can test this by playing around with the interactive graphic on this page on the Math Open Reference website. You can drag the four orange dots wherever you want, and it automatically generates the first quadrilateral, the midpoints and the second internal quadrilateral, which is always a parallelogram.
But why is it true? How can you prove that the second quadrilateral is always a parallelogram?
On the same page as the interactive graphic, if you scroll down you will find a short proof. A mathematical proof is an extraordinary thing. A thing of beauty. Some are just a few lines, while others run for a hundred pages or more. Each proof is a step-by-step argument that shows why something is true or false with absolute 100% confidence. Once something is proven, then the proof remains solid for eternity.
For this reason, the mathematician G. H. Hardy once wrote: “Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean.”
This particular proof is very short, but not simple to follow. Have a go at understanding the proof (remember to scroll down to find it), but don’t worry if it does not make complete sense.
4. Intermediate Maths Challenge Problem (UKMT)
This is one of the toughest questions from a UKMT Intermediate Maths Challenge exam, so be ready to stretch your brain.
4.1 Given any positive integer
Which of these numbers can never be Paul’s answer?
- 1
- 3
- 5
- 7
- 9
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.