Parallelogram 7 Level 1 19 Oct 2023Connect four

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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. How do you win every time?

Deciding which player should go first, and other strategies for winning is part of a mathematical field called “Game Theory”.

Watch this video of the legendary Brady Haran explaining the strategy to win Connect Four every time!

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

2 marks

1.1. What is the starting strategy which will result in a certain win, if the game is then played perfectly?

  • Start first, place counter in the outer column
  • Start first, place counter in the middle column
  • Start first, place counter anywhere
  • Start second, place counter in the outer column
  • Start second, place counter in the middle column
  • (Not answered)
2 marks

1.2. There are seven starting moves for player one, after which player two will then place their first counter.

How many “configurations” could the game be in after each player has made their opening move?

Correct Solution: 49

For each of the seven starting positions player one could chose there are another 7 places player 2 could chose, 7x7 = 49

2 marks

1.3. The standard Connect Four grid has 6 rows and 7 columns.

What is the longest possible diagonal connection which could be made?

Correct Solution: 6 counters

2. Maths is all fun and games!

Most, if not all, games can be viewed in a mathematical way. Dominos, noughts and crosses, and (of course) chess can all be examined mathematically.

Have a go at this domino based puzzle from the 2012 UKMT Junior Challenge.

2 marks

Dominic wants to place the six dominoes above in a hexagonal ring so that, for every pair of adjacent dominoes, the numbers of pips match.

The ring on the right indicates how one adjacent pair match.

In a completed ring, how many of the five dominoes can he definitely not place adjacent to the following domino?

  • 1
  • 2
  • 3
  • 4
  • 5
  • (Not answered)

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.