Parallelogram 33 Level 3 18 Apr 2024The North Pole Riddle

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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.

1. The North Pole riddle – the genius version

This video explores in great depth a question that we have covered in two recent Parallelograms.

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

2 marks

1.1. You are standing on the surface of the Earth. You walk 1 km south, 1 km west and 1 km north, and you end up exactly where you started. According to Musk’s full answer, which of the following is most accurate?

  • You are near the North Pole.
  • You are near the South Pole.
  • You can be near either the North or South Poles.
  • You are on the Equator.
  • You are in the centre of the Earth.
  • (Not answered)
3 marks

1.2. Following on from the previous question, which of the following is most accurate?

  • There is no possible location.
  • There is one possible location.
  • There are two possible locations.
  • There are 72 possible locations.
  • There is an infinite number of possible locations.
  • (Not answered)

2. Changing the world

Scientists and engineers like Elon Musk are literally changing the world... almost certainly for the better. If you also want to change the world then studying mathematics and then becoming an engineer or scientist is probably a very good path to follow.

Technology companies like Apple and Microsoft have changed the way we communicate with each other and the way we appreciate art and music... indeed they have changed the way we live our lives.

Once upon a time, Steve Jobs wanted to persuade the head of Pepsi to join him at Apple, when it was a relatively small company. The head of Pepsi could see no reason why he should leave a global giant like Pepsi to join the relatively small Apple Corporation, but Jobs won him over by saying: “Do you want to sell sugar water for the rest of your life, or do you want to come with me and change the world?”

Studying mathematics is a stepping stone to changing the world.

3. Junior Maths Challenge Problem (UKMT)

2 marks

3.1. After tennis training, Andy collects twice as many balls as Roger and five more than Maria. They collect 35 balls in total. How many balls does Andy collect?

  • 20
  • 19
  • 18
  • 16
  • 8
  • (Not answered)

Suppose that Andy collects x balls. Since Andy collects twice as many balls as Roger, Roger collects 12x balls. Andy collects five more balls than Maria, so Maria collects x5 balls. So between them, Andy, Roger and Maria collect x+12x+x5=35 balls in total.

Multiplying this equation by 2 gives 2x+x+2x10=70

This is equivalent to 5x=80.

It follows that x=16.

Note: As we are asked for the number of balls that Andy collects, it is natural to begin the problem by letting x be this number. However, we see then see that this leads to an equation which includes a fraction. If you look ahead, you might prefer to let the number of balls that Roger collects be x.

Then Andy collects 2x balls and Maria collects 2x5 balls.

We then obtain the equation: 2x+x+2x5=35 with no fractions in it.

This equation is equivalent to 5x=40, from which we deduce that x=8.

This calculation is easier that the one we gave above, but, if you use this method, you need to remember that the answer we are asked for is 2x and not just x.

4. As I was going to St Ives

The action movie Die Hard With A Vengeance has a plot in which the two heroes McClane and Carver (played by Bruce Willis and Samuel L. Jackson) have to solve a series of riddles in order to save the city from the evil villain Simon Gruber.

3 marks

4.1. Have a go at this 200-year-old riddle featured in the film:

As I was going to St. Ives I met a man with seven wives,
Each wife had seven sacks, each sack had seven cats,
Each cat had seven kits: kits, cats, sacks and wives,
How many were going to St. Ives?

(BTW, make sure you check the answer sheet when it is available, which will include a clip from Die Hard With A Vengeance.)

Correct Solution: 1

Lots of people give the answer 2,801 (because there is 1 man + 7 wives + 49 sacks + 343 cats + 2,401 kittens). However, the answer is 1, because the rhyme starts with “As I was going to St Ives, I met...”, so only the “I” was going to St Ives, because everyone else was going the other way. Here is how the riddle was solved in the movie.

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

In fact, I am not sure Samuel L. Jackson is right when he is trying to work out the number of items going away from St Ives. Luckily, in the Sesame Street clip below there is a much better and clearer explanation:

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

5. Junior Maths Challenge Problem (UKMT)

2 marks

5.1. Two identical rulers are placed together, as shown (not to scale).

Each ruler is exactly 10 cm long and is marked in centimetres from 0 to 10. The 3 cm mark on each ruler is aligned with the 4 cm mark on the other.

The overall length is L cm. What is the value of L?

  • 13
  • 14
  • 15
  • 16
  • 17
  • (Not answered)

The distance between the 4 cm mark and the 10 cm mark at the end of the ruler is (10 - 4) cm = 6 cm.

The overall length is made up of the length on each ruler from the 4 cm mark to the 10 cm mark, plus the length from the 3 cm mark to the 4 cm mark. This latter distance is 1 cm.

So, L=6+6+1=13.

6. More Möbius

Last week’s Parallelogram looked at the strange properties of Möbius loops. If you recall, making a Möbius loop involves taking a strip of paper, forming a loop, then making half a twist in one end, and finally joining the two ends with sticky tape. If you follow that process, you will create an object that only has one side. Every piece of paper, of course, is supposed to have two sides – a top and a bottom, or maybe a back and a front. However, the Möbius loop has only one side! (By the way, a Möbius loop is sometimes called a Möbius strip or a Möbius band.)

This is a terrific video by the terrific Vi Hart, who explores the wonders of Möbius loops by making them from something called “Fruit by the Foot”.

By all means watch it to the end, but it gets a bit (only a bit) complicated after 2 minutes 30 seconds.

(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 clip is from the Numberphile series and is about Klein bottles, featuring the amazing Cliff Stoll. I can sit and watch him talk about mathematics all day. He also has lots of other interests. For example, in 1986 he investigated and captured the computer hacker Markus Hess, who was working for the Russian KGB and trying to breach American military security.

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