PG 30 23 May 2019Mathematical Plaques

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux 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.
  • It’s half-term for most of you, so this Parallelogram is a bit longer and you have a bit more time to complete it. Importantly, double badge points!
  • Finish by midnight on Sunday 2 June 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 marks

1.1 Looking at twentieth century mathematicians 1900 to 1999, what is the surname of the mathematician who is best known for proving Fermat’s Last Theorem?

Correct Solution: WILES

2 marks

1.2 Looking at twentieth century mathematicians 1900 to 1999, what is the surname of the mathematician who played a critical role in breaking the German Enigma cipher?

Correct Solution: TURING

2 marks

1.3 Looking at twentieth century mathematicians 1900 to 1999, what is the surname of the mathematician who pioneered fractal geometry and the mathematics of chaos?

Correct Solution: MANDELBROT

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

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)

3 marks

4.1 At the age of twenty-six, Gill has passed her driving test and bought a car. Her car uses p litres of petrol per 100 km travelled.

How many litres of petrol would be required for a journey of d km?

  • pd100
  • 100pd
  • 100dp
  • 100pd
  • p100d

The car consumes p litres of petrol when travelling 100 km, and so uses p100 litres for each km.

Therefore in travelling d km the number of litres of petrol used is p100×d=pd100.

Note that there is an easy way to check this answer. We know the car uses p litres when it travels 100 km, so the correct formula must have value p when d=100. So only options A and B could possibly be correct.

However, the formula of option B implies that the larger d is, that is, the further you travel, the less petrol you use. This cannot be right.

This leaves pd100 as the only option which could be correct.

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 plaque and answer the three questions below.

2 marks

5.1 Who spotted the unusual property of 1,729?

  • Euler
  • Gauss
  • Nightingale
  • Ramanujan
  • Venn
2 marks

5.2 1,729 is known as…

  • a car number
  • a lorry number
  • a taxicab number
  • a double-decker bus number
  • a steam train number
2 marks

5.3 1,729 is the smallest number that can be expressed as the sum of two positive _____ in 2 distinct ways. What is the missing word?

  • Squares
  • Cubes
  • Primes
  • Fractions
  • Polygons

6. Intermediate Maths Challenge Problem (UKMT)

2 marks

6.1 Consider looking from the origin (0,0) towards all the points (m, n), where each of m and n is an integer.

Some points are hidden, because they are directly in line with another nearer point. For example, (2,2) is hidden by (1,1).

How many of the points (6,2), (6,3), (6,4) and (6,5) are not hidden points?

  • 0
  • 1
  • 2
  • 3
  • 4

We can see that:

  • the point (6,2) is hidden behind (3,1),
  • the point (6,3) is hidden behind (2,1), and
  • the point (6,4) is hidden behind (3,2).

But we can see that the point (6,5) is not hidden behind any other point.

So just 1 of the four given points is not hidden.

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 can, 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.
2 marks

8.1 Which country finished top?

Correct Solution: Russia

Show Hint (–1 mark)
–1 mark

Look at each team. If the statements say that a team has finished below any other team, then it did not finish top.

I drew up a 13 by 13 grid, so I could input all the information above. From (1), we know Turkey and Mexico finished above Italy and New Zealand, so the Turkey and Mexico columns had ticks where they crossed the Italy and New Zealand rows.

I then tallied up all ticks and crosses and every team had at least one cross, except Russia, so every team finished below another team, except presumably Russia.

2 marks

8.2 Which country finished second?

Correct Solution: FRANCE

Show Hint (–1 mark)
–1 mark

Look at each team. If the statements say that a team has finished below two or more teams, then it did not finish top or second.

Every team finished below at least two teams, apart from Russia (which we know came top) and France, which must have come second.

2 marks

8.3 Which country finished third? This is much tougher than the previous two questions. If you cannot reach a definite answer, then make your best intelligent guess.

Correct Solution: ALGERIA

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 on Thursday 6 June at 3pm.

Cheerio, Simon.