Parallelogram 8 Level 1 26 Oct 2023Pick a card, any card

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 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. Card tricks, magic or mathematics?

Watch this video where Anastasia Chaves shows us an astounding card trick you can try at home

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

It seems magical at first but really it is mathematical (which is a magic in its own way).

2 marks

1.1. When dividing by 3 what are the possible remainders?

  • 1, 2 or 3
  • 0 or 1
  • 0, 1, or 2
  • Any remainder is possible
  • (Not answered)
2 marks

1.2 This trick used 21 cards and split them into groups of three.

Why did this need to be done three times?

  • 3 is a magic number
  • The cards needed to be in piles of 7 and 213 is 7
  • To get a number less than 1 you need to divide 21 by 3 three times
  • It helps with the showmanship of the trick
  • (Not answered)
Show Hint (–1 mark)
–1 mark

To be absolutely certain where the card is, you need to narrow its possible positions down to 1 exact place within the stack.

Round 1: the cards are divided in 3 and the chosen card could be in any of seven positions, 213 = 7.

Round 2: those seven positions are divided up, there are two or three of the original 7 cards in each column 73 ~ 2.333 so 2 or 3.

Round 3: the 2 or 3 cards are divided into 3 again, 2/3 or 3/3 will mean there is 1 or less positions in each column for the chosen card.

So when the column with the card is revealed you know with certainty which card it is.

2 marks

1.3 The way Anastasia did this trick left nothing to chance.

If she had tried to randomly guess which card was the right one, what is the probability she would have picked the right card?

  • 0.5
  • 1 in 3
  • 121
  • 0%
  • (Not answered)

2. Suits you!

Three monkeys met for tea in their favourite cafe, taking off their hats as they arrived. When they left, they each put on one of the hats at random.

2 marks

2.1 What is the probability that they all left wearing the wrong hat?

  • 13
  • 12
  • 14
  • 16
  • (Not answered)
Show Hint (–1 mark)
–1 mark

You could make a grid of possible options.

There are three monkeys, so they can pick their hats up in one of six different orders:

Of those six orders, only two of them involve all three monkeys picking up the wrong hat - or 13 of the options.

2 marks

2.2 A pattern repeats every six symbols as shown below:

What are the 100th and 101st symbols, in that order, in the pattern?

  • Diamond, heart.
  • Heart, diamond.
  • Heart, club.
  • Spade, heart.
  • Club, heart.
  • (Not answered)

The pattern repeats every 6 symbols.

Since 96 is a multiple of 6, the symbols positions 97 to 102 are the same as those in positions 1 to 6.

Therefore the 100th and 101st symbols are the same as the 5th and 6th symbol.

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 and Ayliean.