PG 21 7 Mar 2019Tricky parking problem

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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.
  • 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. Parking puzzle

2 marks

1.1 What is the number of the space where the car is parked?

Correct Solution: 87

In order to solve this puzzle, you need to imagine looking at the numbers from the driver’s point of view. Starting from the left, the driver would see 91 – 90 – 89 – 88 – XX – 86. So, then it is obvious that the hidden number is 87.

2. Cats and dogs

Lots of schools offer students a weekly or monthly maths puzzle competition. If your school does not have such a competition, then you could suggest it to your maths teacher.

And if you are a student or teacher that already takes part in or runs a regular maths puzzle competition, then let us know if you have any questions that we could include in our Parallelograms by emailing us.

In the meantime, here is a question from a London school, posed to its students earlier this year.

2 marks

2.1 Donald is standing on some scales, holding his cat and dog. The reading on the scales is 102kg. Donald weighs 60kg more than the combined weight of his cat and dog. The cat's weight is equivalent to 60% of the dog's weight. How much, in kg, does the dog weigh? (Perfect answer - please do not round up/down any decimal places)

Correct Solution: 13.125 kg

We know that:

(a) Don + Dog + Cat = 102kg
(b) Don = Dog + Cat + 60kg (or Dog + Cat = Don – 60kg)
(c) Cat = 0.6 x Dog

Plugging (b) into (a) tells us:

Don + (Don – 60 kg) = 102 kg.
So, Don = 81 kg

Knowing this, we can then use (a) and (c) to say:

81 + Dog + (0.6 x Dog)= 102 kg
So, 1.6 Dog = 21 kg

Dog = 13.125 kg
So, Cat = 7.875 kg

.

3. How to trick your maths teacher

This is a wonderful magic trick by Aleko. Take a look at the video below, and play along by thinking of a number or picking a random card. When the trick has finished, and after you have stopped being astounded, think hard about how the trick is done. You might need to watch the video again and note down the instructions given to you. If you can’t figure out the magic maths behind the trick, then click on the hint below for a full explanation of how Aleko did it, and how you can perform the trick yourself. You will not lose any marks by clicking on the hint.

By the way, if you are interested in magic or enjoyed being amazed, make sure you check out the video in the additional stuff section at the end. Seriously... you will not want to miss it.

Show Hint (–0 mark)
–0 mark

Imagine I am the magician, and I ask you to you to pick a card or number. We shall call this x.

First, I ask you to double x to get 2x.

Then I ask you to add 2 to get 2x+2.

Then I ask you to multiply by 5 to get 10x+10.

Now let’s say I have a y-card in my hand, then I work out 10y and I ask you to subtract this number from your total, so you end up with 10x+1010y, which is 10x+y.

So the tens number is always x (the card number that you chose), and the units number is always y (the number that I chose). And that’s the mathematical magic behind the trick.

4. Junior Maths Challenge Problem (UKMT)

3 marks

4.1 The diagram shows a trapezium made from three equilateral triangles.

Three copies of the trapezium are placed together, without gaps or overlaps and so that complete edges coincide, to form a polygon with N sides.

How many different values of N are possible?

  • 4
  • 5
  • 6
  • 7
  • 8

The three trapeziums have 12 edges in total. Whenever two of them are joined together, the total number of edges is reduced by at least 2. So the maximum possible value of N is 122×2=8.

The minimum number of sides for a polygon is 3. So there are just 6 values for N, namely 3, 4, 5, 6, 7 and 8, that could possibly occur. The diagrams below show that they can all be achieved.

5. More about factorials

Last week, while continuing to explore codes, we asked: “How many ways can we re-arrange the English alphabet of 26 letters?” Take a look back at last week’s Parallelogram if you need to remind yourself what we did.

In short, there are 26! ways to re-arrange 26 letters, where 26! means 26 × 25 × 24 ... × 1.

The (!) is called factorial, and it can be applied to any number, eg. 4! = 4 × 3 × 2 × 1 = 24.

1 mark

5.1. This is a question from the previous Parallelogram, but it is a good refresher question. How many ways can you re-arrange the alphabet? In other words, find out the value of 26!. You will need to use a calculator and find its (!) button.

The answer is more than 100 million billion billion - how many million billion billions is it? Please provide a three digit answer.

Correct Solution: 403 million billion billion

If you were trying to break a code, where every letter had been replaced with a different letter, you might try to check every rearrangement of the alphabet. Here is one such re-arrangement.

a b c d e f g h i j k l m
U Z G Q P O I V C D A B J
n o p q r s t u v w x y z
H F Y E M S R X N L T W K
1 mark

5.2. How long would it take to check every possible arrangement, assuming that it takes 5 seconds to check each arrangement? Which of the following expressions would give you the correct answer in years?

  • (26! × 5) × 31,557,600 years
  • 31,557,600 ÷ (26! × 5) years
  • (26! × 5) ÷ 31,557,600 years

There are 26! different arrangements, and each one takes 5 seconds to check, so it would take (26! × 5) seconds to check all of them.

There are (60 × 60 × 24 × 365.25) = 31,557,600 seconds in a year of 365.25 days (remember, every fourth year has an extra day!).

So it would take (26! × 5) ÷ 31,557,600 years.

2 marks

5.3. How many years would it take to check every possible arrangement?

  • about 60 million years
  • about 60 billion years
  • about 60 million billion years
  • about 60 billion billion years
  • about 60 trillion years
2 marks

5.4. The answer from question 5.3 is much bigger than the age of the universe. How many times bigger than the age of the universe? Which of these answers is nearest?

  • about double
  • about 50
  • about 5 thousand
  • about 5 million
  • about 5 billion

6. Rocket software

This is a photo of Margaret Hamilton, who spent the 1960s developing the software for the rocket guidance system that took humans to the moon. She helped popularise the term software engineer and was a pioneer in computing, particularly when it came to writing software that could cope with errors, by either fixing them or avoiding them or preventing them.

While a mother in her twenties, having completed a degree in mathematics, Hamilton was a crucial figure in the Apollo moon project. Below, you can see her standing next to a print out of the computer code that she created. I like the fact that the stack was so tall they had to remove the printout at the top so that you could see Hamilton when she stood behind it.

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 video shows a trick from the amazing – truly amazing - René Lavand, a magician from Argentina who performed magic with only one hand, because he lost a hand aged nine, in a car crash.