Parallelogram 20 • Level 3 • 18 Jan 2024A flag of flags

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. Five bets you will always win against your teacher

Here are five puzzles, riddles, challenges or brain-busters designed to baffle the world's greatest brains, including your teachers.

As each problem is posed, pause the video and see if you can find a solution.

(If you have problems watching the video, right click to open it in a new window)

2. Junior Maths Challenge Problem (UKMT)

Some of you may have already taken part in the Junior Maths Challenge, and others might be taking part this year for the first time. Either way, each Parallelogram usually contains one or two Junior Maths Challenge questions to help you prepare for the challenge day. These questions are excellent because they encourage you to think in a more creative way, and with practice you might achieve a bronze, silver or gold certificate.

(If you are not taking part in the Junior Maths Challenge, then it does not matter – these questions are still great for developing your mathematical skills.)

2 marks

2.1. A square is divided into three congruent rectangles. The middle rectangle is removed and replaced on the side of the original square to form an octagon as shown.

What is the ratio of the length of the perimeter of the square to the length of the perimeter of the octagon?

3:5
2:3
5:8
1:2
1:1
(Not answered)

Suppose that the square has side length 3d. Then the square has perimeter of length 4 × 3d = 12d.

Each rectangle has size 3d × d. So we see that the octagon has a perimeter length:

4d + 3d + 4d + d + 3d + d + 3d + d = 20d

Hence the ratio of the lengths is 12d:20d = 3:5

3. Junior Maths Challenge Problem (UKMT)

2 marks

3.1. What is the smallest possible difference between two different 9-digit integers, each of which includes all the digits 1 to 9?

9
18
27
36
45
(Not answered)

Let x and y be two different numbers each made up of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9, with x>y . To make x−y as small as possible, it would be best if they differed just in the units digit, but, clearly, this is not possible. So we aim to find x and y which are the same except for their tens and units digits. These could be 21 and 12, for example with x = 987654321 and y = 987654312. This gives x−y = 9.

Can we do any better? We now use a fact we've seen before, that the remainder when a positive integer is divided by 9 is equal to the remainder when the sum of its digits is divided by 9, to show we cannot get a smaller difference.

As x and y are made up of the same digits, which add up to 45, they have the remainder 0 when we divide them by 9. So x−y also has remainder 0 when we divide by 9. So, as x≠y, 9 is the least possible value for x−y.

4. Junior Maths Challenge Problem (UKMT)

What is the largest difference between two 9-digit integers whose digits use each of the digits 1 to 9 once and once only? Once you've worked it out, fill in the larger number, the smaller number, and the difference between them, below:

1 mark

4.1 Larger number:

Correct Solution: 987654321

1 mark

4.2 Smaller number:

Correct Solution: 123456789

1 mark

4.3 Difference between them:

Correct Solution: 864197532

Now we have the opposite problem, so we want the biggest difference in the “hundreds of millions column”, so let’s have 9--------- and 1--------. Then we want the next biggest difference in the tens of millions column, so let’s have 98------- and 12-------. And so on, until we have 987,654,321 and 123,456,789, so the difference is 864,197,532.

5. Flag of flags

It turns out that the country with the flaggiest flag is Norway. By ‘flaggiest’, I mean that it contains the flags of six other countries. For example, here is the flag of Thailand.

Next, here is the flag of Norway with a black rectangle showing how it contains the flag of Thailand.

Spot the five other flags. If you are not a vexillologist (an expert in flags), then you might want to check this page to see what the different flags look like.

1 mark

5.1 Which of these flags is contained in the Norway flag?

Iceland
Indonesia
Ireland
Italy
Ivory Coast
(Not answered)

1 mark

5.2 Which of these flags is contained in the Norway flag?

Nepal
Poland
South Africa
Romania
Russian Federation
(Not answered)

1 mark

5.3 Which of these flags is contained in the Norway flag?

Denmark
Dominican Republic
Finland
Gabon
Germany
(Not answered)

1 mark

5.4 Which of these flags is contained in the Norway flag?

Sweden
Austria
Faroe Islands
France
Yemen
(Not answered)

1 mark

5.5 Which of these flags is contained in the Norway flag?

USA
UK
India
Pakistan
Netherlands
(Not answered)

6. Abbott & Costello - Loan Me 50 cents

Last week’s Parallelogram had some disturbing arithmetic courtesy of Lou Costello, one of America’s most famous comedians. Here he is again, this time being conned by his partner Bud Abbott. Abbott and Costello are one of America’s most famous comedy double-acts.

Listen carefully and see if you can work out why the maths is fundamentally wrong, and how Abbot is conning Costello. It might help to know that "dime" is the nickname for a 10 cent coin. Also, answer the question below.

(BTW, visit the Find Out More section below for a link to the most famous and funniest of all the sketches by Abbott and Costello.)

(If you have problems watching the video, right click to open it in a new window)

2 marks

6.1 Abbott says that 40 + 10 = 50, but Costello says that can’t be right because:

15 + 35 = 50
20 + 30 = 50
25 + 25 = 50
27 + 23 = 50
5 + 0 = 50
(Not answered)

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.

Additional Stuff

From now on, Parallelograms will often contain this Additional Stuff section, which carries no mark, but which you might find interesting. Why not take a look? However, it is optional, so you can also just skip to the SUBMIT button and click.

Here is the most famous sketch by Abbott and Costello – “Who’s on first?” – watch and enjoy:

This is a video presented by James Grime for the Numberphile channel. It is all about the world’s most mathematical flag:

You can find lots of other Numberphile videos by just googling Numberphile. "Numberphile" means “someone who loves numbers”, so if you love maths then it is the channel for you.