Parallelogram 9 Year 9 14 Nov 2019Clock Calculation

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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. Clock Calculation

Last week I showed you one of Maarten Baas’s remarkable clocks. Here is another one of his very clever clocks.

1 mark

1.1. Which number is created from the preceding number by adding just one segment?

Correct Solution: 6

1 mark

1.2. Which number is created from the preceding number by removing just one segment?

Correct Solution: 9

2 marks

1.3. Creating the digit 2 from the digit 1 requires one segment to be removed and 4 segments to be added, so 5 segments change in total. Creating another digit from the previous digit also requires a total of 5 segment changes. Which digit is this? In other words, which digit (apart from 2) is created using 5 segment changes from the previous digit?

Correct Solution: 7

Creating 7 from 6 requires 4 segment to be removed and 1 segments to be added.

So in total 5 segments change.

2 marks

1.4. Baas’s digital clock writes the time in 24-hour format. How many segments need to be changed when the clock changes from 23:59 to 00:00?

Correct Solution: 11

The diagram below shows that we need to adjust 3 segments in the first three digits, and then 2 segments in the final digit.

2. Chessy (but not chess) puzzle

2 marks

2.1. In the game described in the video below, who wins the game? The player who goes first (Player 1), or the player who goes second (Player 2)? Assume that both players make their moves the best possible way.

  • Player 1
  • Player 2

Player 1 can be guaranteed to win by putting the first pawn in the centre of the board. Player 2 can put a pawn anywhere, and the first player can then place a pawn in the diametrically opposite position. The second player can place another pawn, and the first player can then place a pawn in the diametrically opposite position. This can carry on until the board is almost filled. Then, if the second player can place a pawn, then the first player can definitely place a pawn in the diametrically opposite position. If the second player CANNOT place a pawn, then the first player wins.

Below you can see how player 1 (yellow squares) could make her first 4 moves, and player 2 (red squares) could make his first 3 moves. After the first move, player 1 just places pawns opposite her opponent’s pawns.

This video was from patrickJMT’s YouTube channel, which has tons of maths videos.

3. Intermediate Maths Challenge Problem (UKMT)

3 marks

3.1 In the diagram, XY is a straight line.

What is the value of x?

  • 170
  • 160
  • 150
  • 140
  • 130

Let the angles be as marked in the diagram.

The exterior angle of a triangle is the sum of the two opposite interior angles. Therefore we have that

(1) p + 80 = 150;
(2) q + 40 = p; and
(3) r + 20 = q

From (1), p = 70 . Hence, from (2), q = 30 . Therefore, from (3), r = 10 .

Since the angles on a line add up to 180°, it follows that r+x=180, so x = 170.

4. Another 7-segment display question

3 marks

4.1. There are many ways to light up different segments of a 7-segment display, and many of them do not represent numbers or letters. Here are 16 of the arrangements where 3 or fewer segments are lit, and 16 of the arrangements where at least 3 segments are lit.

How many different ways are there to light up a 7-segment display?

Correct Solution: 128

The easiest way to tackle this problem is to realise there are 7 segments and each one can be in one of 2 states, so the answer is 27 = 128.

It might help to look at a simpler version of the problem. For example, imagine if we had only a 3-segment display. The first segment could be off/on, and for each of those two settings the second segment could be off/on, which gives 2 × 2 = 4 variants. For these 4 variants, the final segment could be off/on, which gives 8 variants, or 23.

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.