Parallelogram 29 Year 10 21 May 2020A Trapped Knight

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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.
  • It’s half-term for most of you, so this Parallelogram is a bit longer and you have a bit more time to complete it. Importantly, double badge points!
  • Finish by midnight on Sunday 31 May if your whole class is doing parallelograms.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by Sunday night if your whole class is doing parallelograms.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. A Trapped Knight

This is a fascinating Numberphile video presented by mathematical genius Neil Sloane. Although it relates to the knight in chess, you don’t need to know anything about chess, except that the knight moves in an L-shape, which means moving 2 squares in a line, then a turn by 90°, and moving one more square, as shown by the dots on this chessboard.

(If you have any problems watching the video then just right click and open it in a new window)

2 marks

1.1 The knight dashes around the infinite board according to the rules outlined by Neil Sloane, and it eventually gets trapped at square 2,084.

On its way to this square, it doesn't move through all the smaller numbered squares. What is the smallest numbered square that is not visited by the knight?

Correct Solution: 961

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 How many of the following statements are true?

  • Any number which is divisible by 6 is even.
  • Any number which is divisible by 9 is odd.
  • The sum of any two odd numbers is even.
  • The sum of any two even numbers is odd.
  • 0
  • 1
  • 2
  • 3
  • 4

If a number is divisible by 6 then it must also be divisible by 2, so it is even. ‘The sum of any two odd numbers is even’ is also true. However, not all multiples of 9 are odd (e.g. 18) and the sum of any two even numbers is even rather than odd, so two of the four statements are true.

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 To make porridge, Goldilocks mixes together 3 bags of oats with 1 bag containing 20% wheat bran and 80% oats.

All the bags have the same volume. What percentage of the volume of Goldilocks' porridge mixture is wheat bran?

  • 5%
  • 6 23%
  • 20%
  • 26 23%
  • 60%

One quarter of the bags contains one fifth of a bag of wheat bran. So the proportion of wheat bran in the porridge is 120, that is 5%.

4. Intermediate Maths Challenge Problem (UKMT)

5 marks

4.1 Trinni is fascinated by triangle numbers (1, 3, 6, 10, 15, 21, etc.) and recently, coming across a clock, she found that she could rearrange the twelve numbers 1, 2, 3, … 12 around the face so that each adjacent pair added up to a triangle number.

She left the 12 in its usual place; what number did she put where the 6 would usually be?

  • 1
  • 4
  • 5
  • 10
  • 11
Show Hint (–2 mark)
–2 mark

It’s easy to work out which numbers are either side of the 12. It then does not matter which side the 3 and 9 go. It is then easy to work out which numbers go the other side of the 3 and 9. As you get to the end, it gets trickier, as there are some different options, but just remember that for every option clockwise and anticlockwise, the number as to be same at the bottom of the clock.

For ease of reference, label the points A, B, C,…, K as shown.

First note that A = 3 and K = 9 or vice versa. With no loss of generality, let A = 3. Then the only possible values for B, C, and D are 7, 8 and 2 respectively. This gives E = 1 or 4 and, as K = 9, J = 1 or 6.

If J = 1, then E = 4 and the only remaining possibilities for I, H and G are 5, 10 and 11 respectively.

This means that F = 6, but 11 + 6 is not a triangle number, so J is not 1 and must, therefore, be 6. This means that I = 4 and hence E = 1.

The remaining values may now be assigned: H = 11, G = 10 and F = 5.

5. Professor Wiseman’s puzzles

Another puzzle from the psychologist Professor Wiseman, whose website is worth a visit when you’ve completed this Parallelogram.

2 marks

5.1 What letter comes next in this sequence?

W, L, C, N, I, T, _

Correct Solution: S

Show Hint (–1 mark)
–1 mark

What Letter Comes Next In This Sequence…

Look at the initial letters of the question…

2 marks

5.2 What letter comes next in this sequence?

O, T, T, F, F, S, _

Correct Solution: S

Show Hint (–1 mark)
–1 mark

One, Two, Three…

Just take the initial letter of the words that represent the numbers One, Two, Three, Four, Five, Six, …. Seven.

2 marks

5.3 What letter comes next in this sequence?

A, S, D, F, G, H, _

Correct Solution: J

Show Hint (–1 mark)
–1 mark

Look at your keyboard

The letters are those that appear in the middle row of a keyboard, starting from the left.

6. Intermediate Maths Challenge Problem (UKMT)

3 marks

6.1 A shop advertises ‘Buy one, get one at half price’. For this offer, the average cost per item is the same as:

  • Two for the price of one
  • Three for the price of one
  • Three for the price of two
  • Four for the price of three
  • Five for the price of four

The offer gives the purchaser two items for the price of one and a half items.

So the average cost per item is the same as four items for the price of three.

7. Only Connect

I “borrowed” this question from the fantastic BBC quiz show ONLY CONNECT. Catch old episodes and watch clips at the ONLY CONNECT website.

4 marks

7.1 Which option comes next in this sequence?

  • A
  • B
  • C
  • D
  • E
Show Hint (–1 mark)
–1 mark

Each box shows something related to a 7-segment digital display, such as a watch LCD display.

Show Hint (–1 mark)
–1 mark

Each box relates to a number, but which number? Is there a pattern? What is the most obvious pattern? Where would you start this pattern? There are two places to start.

Show Hint (–1 mark)
–1 mark

Look at which segments are missing. You will find that the missing segments make numbers, so what missing segments should be in the fourth box?

The answer is A. If you imagine the missing segments, then you get the pattern 0, 1, 2, ?. So, the next number is 3, but we want the segments left when you remove number three.

8. Intermediate Maths Challenge Problem (UKMT)

4 marks

8.1 Which of the following shaded regions has an area different from the other shaded regions?

Show Hint (–1 mark)
–1 mark

It might be easier to look at the area outside the triangle in order to work out the area of the triangle, or to see which option has a different triangle area.

The area of triangle A=12×2×3=3;
The area of parallelogram B=1×3=3;
The area of triangle C=12×3×2=3;
The area of rectangle E=1×3=3.
However, the area of triangle D=3×33+3+12=212.

9. Social Distancing in San Francisco

I am writing this Parallelogram during the pandemic lockdown (2020), when I received the picture below from Len Fisher, who explained that it was being put up as a reminder to people in San Francisco to keep 6 feet apart.

2 marks

9.1 However, there is a problem with this sign. If the people are 6 feet away from the person in the opposite corner, then they are closer than that to the person in the adjacent (or neighbouring) corner. What is that shorter distance? Give your answer to two significant figures.

Correct Solution: 4.2

According to Pythagoras, a2+b2=c2.

We know that a=b, and that c=6, so:

2a2=36,

So, a=18=4.2.

10. Intermediate Maths Challenge Problem (UKMT)

4 marks

10.1 Platinum is a very rare metal, even rarer than gold. Its density is 21.45 g/cm3.

Assuming that the world production has been about 110 tonnes for each of the past 50 years, and negligible before that, which of the following has a comparable volume to that of the total amount of platinum ever produced?

  • a shoe box
  • a cupboard
  • a house
  • Buckingham Palace
  • the Grand Canyon

The volume of 1kg of platinum is (1000/21.45)cm3, that is approximately 50cm3.

So 1 tonne of platinum has a volume of approximately 50,000cm3, which is 120m3.

The volume of platinum produced per year is therefore about 5m3 and the total volume of platinum ever produced is approximately 250m3.

This is the volume of a cuboid measuring 10m × 5m × 5m, which is comparable to a house.

11. Classroom maths

3 marks

11.1 If you are in a classroom that is 3 m tall, and 5 m long and 5 m wide, then what is the mass of air in the classroom?

The density of air is 1.3 Kg per cubic metre.

  • roughly the same as the mass of a mouse.
  • roughly the same as the mass of a bag of sugar.
  • roughly the same as the mass of a teacher.
  • roughly the same as the mass of an African elephant.

Mass of air = volume × density = 3 × 5 × 5 × 1.3 = 97.5 Kg, roughly the same as an adult human.

12. The Googol machine

This machine physically visualizes a googol (10100). Watch and be awe struck. Each gear has a reduction of 10:1, so when the first wheel makes 10 rotations, the second wheel makes only 1 rotation. Then when the first wheel makes 100 rotations, the second wheel makes only 10 rotations, and the third wheel rotates once. If the first, or when, the first wheel makes a googol rotations, the final wheel will have made one rotation.

(If you have any problems watching the video then just right click and open it in a new window)

The next Parallelogram will be out on Thursday 4 June at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.