Parallelogram 21 • Level 4 • 25 Jan 2024A 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.

2 marks

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
(Not answered)

981 is wrong, because it leads to 81 (9 x 9).
983 looks okay, because 83 seems prime, and 3 is prime.
985 is not prime, because it ends in 5.
987 is a multiple of 3, because (9 + 8 + 7 = 24), and 24 is a multiple of 3.
989 is not prime, because it is clearly divisible by 3.

So the answer must be 983.

3 marks

1.2. Which of these numbers is NOT a left truncatable prime?

643
647
651
673
683
(Not answered)

651 leads to 51 (3 x 17).

3 marks

1.3. Which of these numbers is NOT a left truncatable prime?

937
947
953
971
997
(Not answered)

Show Hint (–1 mark)

–1 mark

What odd number is not a prime?

This is a tricky one. The answer is 971. Although 971 is prime, and 71 is prime, the number 1 is NOT prime. Some people argue about whether or not 1 is prime, but there are good reasons why mathematicians agree to define 1 as NOT prime.

2 marks

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
(Not answered)

As of January 2019, the largest known prime was [ 282−1 ], a number with 24,862,048 digits.

2 marks

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
(Not answered)

743 is not a right-truncatable prime, because it leads to 74, which is even and therefore not prime.

2 marks

1.6. Which numbers are more common?

right-truncatable primes
left-truncatable primes
left- and right-truncatable primes are roughly equally common
(Not answered)

Left-truncatable primes are more common, and right-truncatable primes less common, because the right-most digit is very important in determining whether or not a number is prime. All primes above 5 end with the digit 1, 3, 7 or 9, so a right-truncatable prime can only contain those digits after the digit on the extreme right.

The largest right-truncatable prime is the 8-digit 73,939,133, compared to the largest left-truncatable prime, which is the 24-digit 357,686,312,646,216,567,629,137.

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.

1 mark

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
(Not answered)

£1 = 20 shillings, so the change would be 19 shillings, if the magazine cost 1 shilling, but it costs an additional 6 pennies.

1 shilling = 12 pennies, so the answer is 18 shillings and 6 pennies.

2 marks

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
(Not answered)

Sector 4 contains the letters ILLI of the word SHILLING, but if you rotate the coin by 180 degrees you could read ILLI as 1771.

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 marks

4.1 Given any positive integer n, Paul adds together the distinct factors of n, other than n itself.

Which of these numbers can never be Paul’s answer?

1
3
5
7
9
(Not answered)

The only factor of 2, other than 2 itself is 1. So the sum of these factors is 1. [Note that whenever p is a prime number, the sum of the factors of p, other than p, is 1.]

The factors of 4, other than 4 itself, are 1 and 2, whose sum is 3.

The factors of 8, other than 8 itself, are 1, 2, 4, whose sum is 7.

The factors of 15, other than 15 itself, are 1, 3, 5, whose sum is 9.

So 1, 3, 7 and 9 could each be Paul’s answer.

In the context of the IMC this is enough for us to be able to select 5 as the correct option. However, to give a mathematically complete answer, we need to give a reason why the sum of the factors of n, other than n itself, cannot equal 5.

Clearly, we need only consider the case where n>1. So, one of the factors of n, other than n itself, is 1. Suppose that the other factors are a,b,… . Then a,b,… are distinct, none of them is 1, and 1 + a + b + ... = 5. So a + b + ... = 4. However, there is no way of expressing 4 as the sum of more than one distinct positive integer none of which is 1. So the only possibility is that 4 is the only factor of n, other than 1 and n. However, this is impossible, since if 4 is a factor of n, then so also is 2. Therefore the answer could never be 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.