Parallelogram 35 • Level 4 • 2 May 2024Mathematical Plaques

• 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. History of Mathematics

As mentioned last week, the terrific Mathigon website has loads of great material, including a timeline showing the great mathematicians of past centuries. Visit the timeline (click and it will open up in a new tab) and answer the three questions below. Just enter the name given in the plum box.

So, if the answer is John Napier, just enter Napier, because that is the name that appears in the plum-coloured box on the timeline:

2. Intermediate Maths Challenge Problem (UKMT)

2 marks

2.1 You are given that m is an even integer and n is an odd integer. Which of these is an odd integer?

• A: 3m+4n
• B: 5mn
• C: m+3n2
• D: m3n2
• E: 5m+6n
• (Not answered)

We make repeated use of the standard facts that the sum of two even integers is even, the sum of two odd integers is also even, and the sum of an even integer and an odd integer is odd. Also, the product of two integers is odd if, and only if, both integers are odd. Note that the following tables give us a convenient way to summarize these facts.

We can now check the options, one by one.

• A: As m is even, 3m is even. As 4 is even, 4n is even. So 3m+4n is even.
• B: As m is even, mn is even. So 5mn is even.
• C: As 3 and n are both odd, 3n is odd. Therefore, as m is even, m+3n is odd. So m+3n2 is odd.

We could stop here, as we are entitled to assume that there is just one correct answer amongst the given options. For a complete solution, we would need to check that options D and E are also even.

Note that there is a very quick method here which depends on the assumption that whether a particular option is odd or even depends only on whether m and n are odd or even, and not on their actual values. Granted this assumption, we can check the options by substituting any even number for m and any odd number for n. The arithmetic is easiest if we make the choices m=0 and n=1. Then the values of the options are 4, 0, 9, 0 and 6 respectively. It is then easy to see that only 9, corresponding to option C, is odd.

3. Spot the mathematician

All over the UK, there blue plaques on buildings where famous people lived or worked. Sometimes people put up “alternative” or “unofficial” plaques on buildings that have not yet been given blue plaque status.

3 marks

3.1 This photo by David Harrison shows a plaque erected next the Drypool Bridge in Hull.

What is the surname of the mathematician who is being remembered? I have removed his name from the middle section of the plaque.

Correct Solution: Venn

The answer is VENN, because John Venn invented the Venn diagram and he was both (mathematician, philosopher, priest) + (had a strong beard).

4. Intermediate Maths Challenge Problem (UKMT)

5. Another unofficial mathematical plaque

In 2017, I put up an unofficial plaque to recognise a building where two mathematicians had a curious conversation about the number 1,729. Visit this web page about the number 1729 and answer the three questions below.

6. Intermediate Maths Challenge Problem (UKMT)

7. Our first instinct is far too often wrong

I read an interesting article in the Financial Times written by Tim Harford, who is a respected journalist covering economics, all about whether or not to go with your first instinct or whether it is better to change your mind if you have doubts. Tim wrote:

"Most people would advise that the initial answer is usually better than the doubt-plagued second guess. Three-quarters of students think so, according to various surveys over the years. College instructors think so too, by a majority of 55 to 16 per cent. The 2000 edition of Barron’s How to Prepare for the GRE Test is very clear that students should be wary of switching: “Experience indicates that many students who change answers change to the wrong answer.”

This confidence would be reassuring, were it not utterly erroneous. Researchers have been studying this question since the 1920s. They have overwhelmingly concluded both that individual answer changes are more likely to be from wrong to right, and that students who change their answers tend to improve their scores."

This is something to bear in mind next time you take a test. I would also encourage you to read the whole article if you are able to (it is behind a paywall), as it covers some interesting idea and explains how you test a theory about changing your mind.

8. Who came top?

This problem was set by a school as one of its monthly maths challenges.

Thirteen nations competed in a sports tournament. From the following pieces of information, the challenge is to identify the position of all 13 teams in the league table.

I would encourage you draw up the complete league table, but for this Parallelogram you just need to answer three questions below:

1. Turkey and Mexico both finished above Italy and New Zealand.
2. Portugal finished above Venezuela, Mexico, Spain and Romania.
3. Romania finished below Algeria, Greece, Spain and Serbia.
4. Serbia finished above Turkey and Portugal, both of whom finished below Algeria and Russia.
5. Russia finished above France and Algeria.
6. Algeria finished below France but above Serbia and Spain.
7. Italy finished below Greece and Venezuela, but above New Zealand.
8. Venezuela finished above New Zealand but below Greece.
9. Greece finished below Turkey, who came below France.
10. Portugal finished below Greece and France.
11. France finished above Serbia, who came above Mexico.
12. Venezuela finished below Mexico, and New Zealand came above Spain.

9. Parallel emails

If you use Parallel on an email address from your school, some school IT systems prevent you from receiving the emails we send out to remind you when a new Parallelogram is released, or when we have another exciting and nerdy maths thing to tell you about. To avoid this, you could give us a different email address we can contact you on - either a personal email address, or one for you parents or guardian. If you'd like to do that, you can put it in this form, but please ask a parent first if you are not yet 13 years old.

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.
• 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.
• The next Parallelogram will be out next Thursday at 3pm.

Cheerio, Simon.