Parallelogram 12 Level 1 23 Nov 2023The sound of sequences

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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 Recamán Sequence

A swirly sequence which takes unpredictable turns, where will it land next?

Watch this video from Alex Bellos to explore this unusual sequence and then answer the questions below.

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

1 mark

1.1 What is the rule for deciding the direction in the Recamán Sequence?

  • alternating back then forwards
  • always forward, unless you can’t, in which case go back
  • always back, unless you can’t, in which case go forwards
  • odd lengths go back, even lengths go forward
  • (Not answered)

The Recamán Sequence is described as "always back, unless you can't, then forward"

1 mark

1.2 Will all of the numbers be covered?

  • Yes, definitely
  • Absolutely not
  • We don’t have a proof yet
  • (Not answered)

This is an open question in maths. Perhaps someone who is doing this Parallelogram could figure it out in the future!

2 marks

1.3 How is the musical version of the Recamán Sequence described?

  • always increasing
  • always decreasing
  • contrary motion (going both up and down at the same time)
  • (Not answered)

Like the fugues of Bach and many other fantastic musicians, the Recamán sequence displays the musical concept of contrary motion.

2 marks

1.4 What are the first five terms of the Recamán Sequence?

  • 0, 1, 3, 6, 2
  • 0, 1, 2, 3, 6
  • 1, 2, 3, 4, 5
  • 1, 2, 4, 7, 3
  • (Not answered)

2. An interesting sequence from a past UKMT challenge

1 mark

2.1"Mad Teacher Disease" is spreading.

Each week it spreads from teacher to teacher, shown as the dots in the pentagonal diagrams.

How many teachers will be affected by week 6?

  • 35
  • 48
  • 51
  • 54
  • 67
  • (Not answered)
Show Hint (–1 mark)
–1 mark

First look at the number of new teachers infected each week, is there another pattern there?

undefined marks

2.2 How many teachers will be infected by week 8?

Correct Solution: 92

By day 8 there will be 35 + 16 + 19 + 22 = 92 cases.

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.