Week 7X + Y

This is a preview of Parallel. You have to login or create an account, to be able to answer questions and submit answers.

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. x + y

“x + y” is a great film about a British teenager, Nathan Ellis, who represents the UK at the International Mathematical Olympiad, and faces challenges that go beyond the universe of numbers. Take a look at this clip, in which Nathan is at a maths training camp and is asked to solve a tricky problem about playing cards. See if you can follow any of Nathan’s answer. His approach is not at all obvious.

The video explains the that rules are to put playing cards in a row, some face up and some face down and “a move consists of turning a face down card face up, and turning over the card immediately to the right.” And let’s say that, as Nathan suggests, face down = 1 and face up = 0. Let’s look at some arrangements involving just 5 cards.

1 mark

1.1. What is minimum number of moves needed to make this series of cards all become face up: (0 0 0 1 1)

Correct Solution: 1

If you turn over the 1 on the left, then you will also turn over the card on the right – one move.

1 mark

1.2. What is minimum number of moves needed to make this series of cards all become face up: (0 0 0 1 0)

Correct Solution: 2

You could try (0 0 0 1 0) → (0 0 0 0 1) → (0 0 0 0 0).

Or (0 0 0 1 0) → (0 0 0 1 1) → (0 0 0 0 0).

2 marks

1.3. What is minimum number of moves needed to make this series of cards all become face up: (1 1 0 1 1)

Correct Solution: 2

(1 1 0 1 1) → (0 0 0 1 1) → (0 0 0 0 0) or

(1 1 0 1 1 ) → (1 1 0 0 0) → (0 0 0 0 0).

2. Solve this sum

This puzzle was tweeted by maths education specialist Drew Foster (@drewfoster0 on Twitter).

2 marks

2.1. What number is represented by the white circle? If you are stuck then hit the hint. You will lose one mark for (2.1), but you will have a better chance of gaining marks on (2.2) and (2.3).

Correct Solution: 5

Show Hint (–1 mark)
–1 mark

From the column on the right, we know that 3 white circles add up to something that ends with a white circle. A bit of trial error reveals that the only solution is (5 + 5 + 5) = 15. So, in fact, that gives you the answer, but you will still need to think about the value of the red and blue circles.

So, just enter 5 in the box above and you have earned 1 mark (i.e., 2 original marks – 1 hint mark) for very little effort. I am feeling very kind this week. Now get to work on questions 2.2 and 2.3

1 mark

2.2. What number is represented by the red circle?

Correct Solution: 8

1 mark

2.3. What number is represented by the blue circle?

Correct Solution: 1

3. Intermediate Maths Challenge Problem

2 marks

3.1 When I glanced at my car milometer it showed 24942, a palindromic number. Two days later, I noticed that it showed the next palindromic number.

How many miles did my car travel in those two days?

  • 100
  • 110
  • 200
  • 220
  • 1010

The only 5-digit palindromic number that begins 249 is 24942, so the next palindromic must be greater than 24999.

The only 5-digit palindromic number beginning 250 is 25052, so this is the next palindromic number after 24942.

We have that 25052 - 24942 = 110, and so the answer is 110.

1 mark

3.2 How many more miles must you drive to reach the next palindromic number?

Correct Solution: 100 miles

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 Parallelogams.

Cheerio, Simon.