PG 28 9 May 2019Sphere Packing

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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. Intermediate Maths Challenge Problem (UKMT)

2 marks

1.1 Which is the smallest positive integer for which all these are true?

  • (i) It is odd.

  • (ii) It is not prime.

  • (iii) The next largest odd integer is not prime.

  • 9

  • 15

  • 21

  • 25

  • 33

In the context of the IMC we can find the solution by eliminating all but one of the options. We first note that all the options are odd numbers that are not primes. So conditions (i) and (ii) are true for all of them. Next we see that 9 + 2 , 15 + 2 and 21 + 2 are all primes, so for the first three options condition (iii) is not true. However 25 + 2 = 27 is not prime, so all three conditions are satisfied by 25.

The same is the case for 33, but, as 25 < 33 , 25 is the smaller of the given options for which all three conditions are true.

If we were not given the options, we would have to search the sequence of odd numbers to find the first pair of consecutive odd integers that are not primes. The first few odd numbers are:

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, …

where the non-primes are shown in bold. [Don’t forget that the number 1 is not a prime number.] From this list we see that 25 is the smallest positive integer for which all three conditions are true.

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 Find the next pair of consecutive odd numbers after the pair 33, 35 that are both not primes. Add them up. What is the total?

Correct Solution: 100

The next two odd non-primes are 49 and 51, which add to make 100.

3 marks

2.2 There is only one example of three consecutive odd numbers that ARE prime. Find those three prime numbers, add them up, and enter the total below.

Correct Solution: 15

3, 5 and 7 are all consecutive odd numbers and are all prime, and they add up to 15.

In fact, they are the only three consecutive odd prime numbers, and we can prove this because any odd number sequence of three numbers will be (n), (n + 2) and (n + 4). n is either a multiple of 3 (say, 3m) or equals (3m + 1) or (3m + 2). If n = 3m, then it is divisible by 3 and not prime, so the sequences fails. If n = (3m + 1), then the second term is (3m + 3), which is divisible by 3 and not prime, so the sequences fails. And if n = (3m + 2), then the third term is (3m + 6), which is divisible by 3 and not prime, so the sequences fails.

3. Sphere-packing

Last week, you watched one of the world’s greatest mathematicians, Neil Sloane, talking about circle patterns. He is also interested in spheres, and this video introduces you to mathematical problem of sphere packing.

The video – presented by James Grime – will explain it in more detail, but essentially the questions is “what is the best way to pack spheres so that you leave the minimum amount of space between the spheres?” Alternatively, if you have a large (actually, infinite) box, then what is the best way to pack oranges, so that you get the most oranges in the box?”

2 marks

3.1 According Kepler, what was the maximum efficiency for packing spheres?

  • 50%
  • 66%
  • 74%
  • 90%
  • 100%
2 marks

3.2 What was unusual about the proof that showed that “hexagonal packing” was the densest way to pack spheres?

  • It was written entirely on a computer
  • It was written solely by a computer
  • It was checked by a computer
  • It was so large that it had to be stored on a computer
  • It was the first proof announced on the internet

4. Intermediate Maths Challenge Problem (UKMT)

3 marks

4.1. In the 2013 Tour de France, the total length of the Tour was 3404 km and the winner, Chris Froome, took a total time of 83 hours 56 minutes 40 seconds to cover this distance.

Which of these is closest to his average speed over the whole event?

  • 32 km/h
  • 40 km/h
  • 48 km/h
  • 56 km/h
  • 64 km/h

As 3404 is close to 3400 and 83 hours 56 minutes 40 seconds is almost 84 hours, his average speed is close to 340084 km/h.

Now 340084=85021=401021. So, of the given options, 40 km/h is the closest to his average speed.

5. Table tennis puzzle

2 marks

5.1. 32 students enter a knockout competition, where the loser of each match is immediately eliminated. How many matches will be played in order to find a champion?

Correct Solution: 31

There are two ways (at least) to approach this problem. The long way is to work out that there are 16 matches in the first round, then 8 matches in the second round, then 4 matches, 2 matches and the final match, which is 31 matches in total (16 + 8 + 4 + 2 + 1 = 31).

The quick way is to remember that everyone will lose one match, except the champion. This means we have 31 matches, which is (32 – 1) matches.

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.