Week 8Checkmate

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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. Chess puzzles

Before half-term, I mentioned that there would be some chess puzzles coming up, and here they are. If you had never played chess, then I encouraged you to visit Chesskid and learn the basics in order to prepare for today. If you forgot to do that, then:

  • Visit https://www.chesskid.com/ and learn some chess basics if you have not done so already. Or ask a friend or someone in your family to teach you or help with the questions.
  • With 15 minutes of learning, you will be able to answer a couple of the questions.
  • This chess section is only a small part of this week’s Parallelogram. There are loads of other points to earn afterwards.
1 mark

1.1. Which of these chess pieces is most valuable?

  • Bishop
  • Knight
  • Pawn
  • Queen
  • Rook/Castle
1 mark

1.2. The aim of chess is to checkmate your opponent, which means trapping the…

  • Bishop
  • King
  • Knight
  • Knight and Bishop
  • Queen
1 mark

1.3. The game has reached the position below. It is white’s move. What black pieces are being attacked by the white queen? In other words, which pieces could the white queen take on the next move?

  • King
  • Rook
  • Queen and 1 pawn
  • Rook and 2 pawns
  • 2 pawns
2 marks

1.4. It is white’s move. Moving which piece will result in checkmate?

  • Bishop
  • King
  • Knight
  • Pawn
  • Queen

The Queen can move to the (d8) square, and then the King is trapped by the Queen.

1 mark

1.5. This is not an easy question, but it will stretch those of you who are more than beginners. It is white's turn to move. Find white’s best move, then work out black’s only response and then identify which piece white would take on the next move.

  • Pawn
  • Queen
  • Rook/Castle

White moves the Queen to H8, which puts the King in check. The black King is then forced to move to d7. This allows the white Queen to take the black Rook/Castle.

2. October Puzzle

This nice puzzle was set by the University of Cambridge’s NRICH project, and it appeared on Radio 4’s Today Programme.

2 marks

2.1. One year, there were exactly four Tuesdays and exactly four Fridays in October. On what day of the week did Halloween, October 31st, fall that year?

  • Monday
  • Tuesday
  • Wednesday
  • Thursday
  • Friday
  • Saturday
  • Sunday
Show Hint (–1 mark)
–1 mark

1 October cannot be a Tuesday, because then 8, 15, 22 and 29 October would also be Tuesdays, and that would be 5 Tuesdays. The question says there are only 4 Tuesdays.

Show Hint (–1 mark)
–1 mark

Similarly, 2/3 October cannot be a Friday, because then 9/10, 16/17, 23/24 and 30/31 October would also be Fridays, and that would be 5 Fridays. The question says there are only 4 Fridays.

1/2/3 October cannot be a Tuesday, because then 8/9/10, 15/16/17, 22/23/24 and 29/30/31 October would also be Tuesdays, and that would be 5 Tuesdays. The question says there are only 4 Tuesdays.

Exactly the same is true for Friday.

Therefore, the first Tuesday and Friday must both fall on either 4, 5, 6 or 7 October. This can only work if Tuesday is on 4 October and Friday is on 7 October. This means that the last Friday of the month will be on 28 October, which means that 31 October will be on a Monday.

3. Intermediate Maths Challenge Problem

4 marks

3.1 A square piece of card has a square of side 2 cm cut out from each of its corners.

The remaining card is then folded along the dotted lines shown to form an open box whose total internal surface area is 180 cm2.

What is the volume of the open box in cm3?

  • 100
  • 128
  • 162
  • 180
  • 200
Show Hint (–1 mark)
–1 mark

Suppose that the dotted square corresponding to the base of the box has side length x cm. Then the base has area x2 cm, and each of the sides of the box has area 2x cm.

Hence the internal surface area of the box, in cm2, is x2 +4×2x , that is, x2 +8x. Therefore x2 +8x=180.

Suppose that the dotted square corresponding to the base of the box has side length x cm. Then the base has area x2 cm, and each of the sides of the box has area 2x cm.

Hence the internal surface area of the box, in cm2, is x2 +4×2x , that is, x2 +8x. Therefore x2 +8x=180. Now
x2 +8x=180 if and only if x2 +8x180=0
if and only if x+18x10=0

with solutions x=18 and x=10. Since x must be positive, we deduce that x=10.

The volume of the open box is the height multiplied by the area of the base, and, in cm3, this is 2x2, that is, 200.

4. Schiphol Airport Clock

This is an amazing clock at Schiphol Airport, near Amsterdam.

3 marks

4.1. How many painters are employed to paint the time?

  • 0 painters
  • 1 painter who works every day for 24 hours.
  • 2 painters who swap every 12 hours.
  • 3 painters who swap every 8 hours.
  • 24 painters who each take responsibility for one hour.
Show Hint (–1 mark)
–1 mark

Can you actually see the painter? Could you really trust the painters to keep perfect time?

This is a work of art by Maarten Baas, who designs interesting clocks. The painter and clock hands are just a projection, so there is no painter inside the clock.

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.