Parallelogram 22 Level 3 1 Feb 2024Freezing sports and flipping pancakes

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. 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.

This Parallelogram first appeared at the time of the Beijing Winter Olympics, so it has a winter sports theme!

1. Winter Olympics Puzzle

Every four years, the world’s leading winter athletes – the skiers, skaters, sledders and snowboarders – gather to compete for medals in the Winter Olympic Games.

Of course, among the drama, bravery and triumph, there is plenty of room for maths! So, while you’re cheering on your favourites, don’t miss the chance to tackle this Winter Olympics maths puzzle.

Ice hockey is a high-speed, physical game in which skaters try to hit a small rubber puck into the other team’s goal using a curved stick. Matches consist of three 20-minute periods; if a match is tied, the teams play a fourth ‘overtime’ period, followed by a shoot-out if needed.

In the Olympics, the first round of the tournament is a group phase. The teams are split into groups of four, and the teams in each group play each other once. Teams are awarded:

  • 3 points for a win in regular time (W),
  • 2 points for winning in overtime or in a shootout (OTW),
  • 1 point for losing in overtime or in a shootout (OTL), and
  • 0 points for losing in regular time (L).

The total number of goals each team scores (GF) and concedes (GA) are recorded, and the difference between them (goal difference, or GD) is used as a tie-breaker.

In the 2014 Winter Olympics in Sochi, the final table in Group C (Sweden, Switzerland, Czech Republic, Latvia) looked like this:

1 mark

1.1. What was the score in the Switzerland v Sweden game

  • Switzerland 0, Sweden 0
  • Switzerland 1, Sweden 0
  • Switzerland 0, Sweden 1
  • Switzerland 0, Sweden 2
  • Switzerland 0, Sweden 3
  • (Not answered)

Sweden won all their games, because 9 points means 3 wins. Switzerland only had 1 goal scored against them in all three games, so they must have lost 1-0 to Sweden.

1 mark

1.2. What was the score in the Switzerland v Czech Republic game?

  • Switzerland 0, Czech Republic 0
  • Switzerland 1, Czech Republic 0
  • Switzerland 1, Czech Republic 1
  • Switzerland 0, Czech Republic 1
  • Switzerland 2, Czech Republic 0
  • (Not answered)

Switzerland lost to Sweden (who won all their games), and we know Switzerland won its other two games. However, Switzerland only scored two goals, and the only way to win two games with two goals is you win both games 1-0. So the score must have been 1-0 to Switzerland.

5 marks

1.3. The final table in Group B (Canada, Finland, Austria, Norway) looked like this, but I have blanked out some of the numbers.

How did Austria get its 3 points?

  • 1 win and 2 losses
  • 1 OTW, 1 OTL and 1 loss
  • 3 OTLs
  • (Not answered)

You can actually fill in all the gaps, as explained below. However, to answer the question about Austria, you would only need think about the next paragraph.

To get no points, Norway must have lost all three matches, including their match to Austria. So, if Austria beat Norway, then they lost their remaining two matches (to Finland and Canada).

To win eight points, Canada must have two wins (against Austria and Norway) and an OTW. Canada’s OTW was therefore over Finland, making the left-hand part of the table look like this:

  • Canada’s goal difference is 11 - 2 = +9.
  • Finland must have conceded 7 goals to have a GD of +8.
  • Austria must have scored 7 to have a GD of -8.

The last bit is trickier: the total of the goal difference column must be zero, so Norway’s GD is -9, and they conceded 12 goals. (You can also see this by saying every goal is scored by someone and conceded by someone else – so the totals of the GF and GA columns must be the same.)

The final table therefore looks like this:

This puzzle comes from the Komodo website, which offers a programme of maths, largely aimed at younger children. I’ve used it with my son, and I think it offers a great way to practice core arithmetic skills.

The puzzle was created by Colin Beveridge - a maths tutor, former NASA researcher, and writer of maths books including Basic Maths for Dummies.

2. Flipping pancakes

I realise that Shrove Tuesday is still a little while away, but I wrote this Parallelogram on Pancake Day last year, and I was reminded of this video about the maths of pancake flipping. It mentions a friend of mine, David X. Cohen, who is one of the gods of pancake flipping maths. He is probably better known, however, as one of the writers on The Simpsons and one of the driving forces behind Futurama. A lot of mathematicians write for these two shows, and I will return to them in future Parallelograms.

In the meantime, watch the video and answer the three questions that follow. In fact, perhaps look at the questions first, so you can listen out for the answers when you watch the video.

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

1 mark

2.1. Using the most efficient method for the worst arrangement, how many flips do you need to reorder three pancakes?

  • 0
  • 1
  • 2
  • 3
  • 4
  • (Not answered)
2 marks

2.2. Using the most efficient method for the worst arrangement, how many flips do you need to reorder nineteen pancakes?

  • 20
  • 21
  • 22
  • 23
  • 24
  • (Not answered)
1 mark

2.3. Bonus question: Which of these would be least interested in pancake flipping maths? The answer is not in the video, but see if you can use your intuition to work it out.

  • Astronomers
  • Data scientists
  • Evolutionary biologists
  • (Not answered)

Flipping pancakes is a bit like re-ordering data, so data scientists are very interested in the maths of pancake flipping. In fact, the only research paper ever written by billionaire computer entrepreneur Bill Gates was on this topic.

Flipping pancakes is a bit like flipping genes, and some species are identical to each other apart from a few flipped genes, so evolutionary biologists are interested in pancake flipping, too.

Astronomers might be interested in pancake flipping, but probably less so than evolutionary biologists and data scientists.

3. Junior Maths Challenge Problem (UKMT)

3 marks

3.1. What is the smallest number of additional squares which must be shaded so that this figure has at least one line of symmetry and rotational symmetry of order 2 ?

  • 3
  • 5
  • 7
  • 9
  • More than 9
  • (Not answered)

In the diagram on the right we have shaded the three additional squares that must be shaded if the square is to have rotational symmetry of order 2. With these squares shaded the square has both the diagonals as lines of symmetry.

So the smallest number of additional squares that need to be shaded is 3.

4. Basketball and a supernova

This is a cool video from Physics Girl, who presents a terrific YouTube channel, full of fascinating videos. Find out how it is possible for a ball to bounce higher than it has been dropped. Physics Girl suggests that you try this for yourself – my advice is to start with two balls, because the demonstration is easier to do and is still pretty impressive.

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

5. Junior Maths Challenge Problem (UKMT)

3 marks

5.1. Pupils have decided to hold a mock election at their school. A candidate receiving more votes than any other wins. The four candidates receive 83 votes between them. What is the smallest number of votes the winner could receive?

  • 21
  • 22
  • 23
  • 41
  • 42
  • (Not answered)

Since ¼(83) = 20.75, the winner must get at least 21 votes. If one candidate gets 21 votes there are 62 votes to be shared between the other three candidates, and so, as 3 × 20 < 62 one of these candidates must get at least 21 votes. So a candidate with 21 votes cannot be the winner.

However if one candidate gets 22 votes and the others 21, 20 and 20 votes respectively, all 83 votes have been used, and the winner receives only 22 votes.

6. Neil deGrasse Tyson – the day his universe changed

Neil deGrasse Tyson is one of the world’s most famous astrophysicists, and in America he is a huge celebrity. He is the American version of Brian Cox. In this video, he explains how his life changed when he met Carl Sagan as a teenager.

At that time, Carl Sagan was probably the most famous astrophysicist in the world, and the video shows the difference that an act of kindness can make.

If you want to find out more about deGrasse Tyson and Sagan, then there are a couple of short videos after you hit the submit button. The Carl Sagan video is one of the most memorable scientific statements of the last century. Absolutely worth a few minutes of your time.

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

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

  • This short video from the Secret Life of Scientists and Engineers (If you have problems watching the video, right click to open it in a new window) reveals a bit more about the astrophysicist Neil deGrasse Tyson

  • This clip contains one of the most moving and extraordinary monologues by a scientist (If you have problems watching the video, right click to open it in a new window). It has gone down in the folklore of science, and is all about a photograph called the Pale Blue Dot.

    Here is how Wikipedia describes the clip:

    “Pale Blue Dot is a photograph of planet Earth taken on February 14, 1990, by the Voyager 1 space probe from a record distance of about 6 billion kilometres… Earth's apparent size is less than a pixel; the planet appears as a tiny dot against the vastness of space, among bands of sunlight reflected by the camera. Voyager 1, which had completed its primary mission and was leaving the Solar System, was commanded by NASA to turn its camera around and take one last photograph of Earth across a great expanse of space, at the request of astronomer and author Carl Sagan.”