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This is really very well-written and worth reading!

My Comic Relief

So…it’s been a week huh?  I was planning on writing a fun post about how much I’m loving The Totally Awesome Hulk today but I can’t get my head around it.  Reading the news for the last five days has taken me on an emotional journey.  I’ve been spending much time in thought and conversation with loved ones about where our country is heading.  I’ve been struggling to understand, let alone find my place in the events that are unfolding.  As I do this, I keep thinking of something Bruce Springsteen said during a concert in Western Australia last Sunday.   

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On its own, I found Rogue One to be very enjoyable and superbly made. There’s every good praise to be made for its story line, effects and artistic direction.

The basic problem about it – and I think that this is what might generate some reservations among fans – is that the story has very little to do with the essential, central Star Wars story line, which we have come to know as being the downfall and redemption of Anakin Skywalker. Of course, the film absolves itself of this fault by including the phrase A Star Wars Story in its title. This subtitle doesn’t appear on screen, but is part of the film’s official title. So, I think one needs to take the subtitle literally and make an effort to get over the fact that this is not a Skywalker story.

Beware of spoilers from this point onwards, if you have not yet seen Rogue One.

Despite what I said above, Rogue One does center again on a family broken apart by the evil Empire’s plans, much like what happens in the seven episodes of Star Wars. This time round, however, the reason for the breakup is less spiritual, less esoteric than the corresponding events in the Skywalker timeline.

Thumbs Up and Novelties

Well, at last, we see the true scale of an Imperial Cruiser as it hovers above the city of Jedha. I had been wanting to see something like that. (The same goes for TIE fighters and The Force Awakens was the first Star Wars film where we see TIE fighters next to their human pilots.)

The spectacular aerial battle in the rain on Eadu rivals the battle over the lake in The Force Awakens. However, the latter is still my favourite.

The other spectacular battle over Scarif harks back to the one over Endor in Return of the Jedi and I was very happy to hear the various squadron leaders reporting in just before it ensued, this particular detail being reminiscent of the future attack on the Death Star in A New Hope (only a film away!). Oh, and I did enjoy that scene with the Hammerhead Corvette causing those two Star Destroyers to crash!

The ground battle on Scarif was, of course, very impressive and kudos for the beautiful scene of Chirrut’s slow advance forward towards the communications master switch before being shot. It was saddening to watch the heroes falling one by one.

The final scene, when Tarkin orders the destruction of the base, shows the full ruthlessness of the Empire and its indifference even to the lives of its own officers. This is perfectly embodied by Tarkin himself.

And the film leads very nicely to the point where the events of A New Hope will begin, with Tantive IV leaving the scene of the battle and carrying the plans of the Death Star.

How ironic that Galen Erso used the same pet name for the Death Star project and his little girl, knowing that Jyn would thus be able to identify the stored plans when she came across that very name. The irony is carried even further when you realise that the Death Star’s primary function is to destroy planets and reduce them to… stardust.

There is also another interesting irony: The Jedi use the Kaiburr crystals for their lightsabers while the Empire uses them to destroy planets.

Interestingly enough, Rogue One is the first Star Wars film where the names of planets are given on screen. I noticed, however, that no such names were given for Jyn Erso’s home planet and the one where Orson Krennic visited Darth Vader.

Nods and Cameos 

I first saw the name Kaiburr ages ago in Alan Dean Foster’s novel Splinter of the Mind’s Eye.

Chirrut and Baze are described as being “Guardians of the Whills”. I believe the name “Whills” comes from the very first novelisation of Star Wars by George Lucas himself, back in 1977. The prologue to that briefly describes the downfall of the Old Republic and the seizing of power by the “ambitious Senator Palpatine”. Underneath the prologue is the phrase:

From the First Saga
Journal of the Whills

I noticed Dr. Evazan running into Jyn and Cassian in the streets of Jedha. And, then, there was the proverbial chess game being played outside the cell where Cassian, Chirrut, Baze and Bodhi were being held, as well as a Twi’lek dancing.

I smiled when I heard K-2SO nearly phrasing “I’ve got a bad feeling about this”!

Darth Vader

Oh dear! Darth Vader was probably shown at his meanest ever when he butchered all those poor Rebels in one of the last scenes.

Prior to seeing the film, I had predicted that we would see Vader force-choking Krennic to death, but then I thought that that would have been due to Krennic’s inability to stop the Rebels from stealing the Death Star plans. Well, I only got it half-right, since the choking was for another reason and not fatal.

Krennic eventually met his bitter end, brought about by the very machine that he was so proud of…

I noticed Vader’s voice to be somewhat aged in relation to the one in Episodes 4-6, but there’s no escaping reality: It has been 11 years since Revenge of the Sith was released.

Grand Moff Tarkin

Well, that was a surprise. They did a really good job creating a digital image of Moff Tarkin to superimpose on the real actor who portrayed him. You could tell it wasn’t really Peter Cushing, but the resemblance was really very good and I must admit that – while watching the film – I kept wondering “Is it CG?”, “Is it CG?”.

Princess Leia

I was even more surprised to see her! I was sure there wasn’t any chance of seeing the Princess, just the departure of Tantive IV from the battle scene. I am glad to have been proved wrong. You could tell that she was CG, but it didn’t matter.

Missing Scenes

I could not help noticing that some scenes, which I had got used to seeing in the trailers, didn’t make it to the final cut. Most notable of these was the one where a TIE fighter confronts Jyn Erso on that platform where she had gone to transmit the plans up to the Rebel fleet. I was waiting to see that TIE fighter, but it never appeared. What happened?

And then, I never remember seeing Mon Mothma talking about the Empire’s upcoming “weapons test”, or Orson Krennic saying “the power we are dealing with here is immeasurable”.

I am not sure, but did they also miss out a scene with Cassian and Jyn running on the ground on Scarif with Jyn holding the disk with the Death Star plans?

Conclusion

Rogue One is a Star Wars story indeed, well worth seeing. Here’s to many more!

Do you remember that famous carol “The Twelve Days of Christmas”? You can find some historical information about it at Wikipedia. There is also a section on its financial implications.

What I find interesting about this carol, from a mathematical point of view, is the structure of the underlying collection of gifts as well as the calculation of their total number.

The carol starts with a seemingly innocent gift of a partridge in a pear tree sent on the first day of Christmas, which is 25 December. Then, however, as the days proceed through the Christmas period, the gifts received on any one day are:

  • a completely new set of gifts equal in number to the number of the day within Christmas (e.g. on the third day I get three French hens for the first time, on the fourth I get four calling birds for the first time and so on)
  • the whole lot of gifts that were sent on the preceding day

This process, where the set of gifts on any one day is linked directly to the gifts of the preceding day, is an example of a concept known as recursion. Recursion is widely used in both Mathematics and Computer Programming. Many interesting books have been written, which address this concept. If you have ever come across those beautiful images known as fractals, then I should tell you that recursion plays a great part in their production.

Let me turn to the second reason for my interest in the carol: How do I calculate the total number of gifts received from my true love?

Well, if you examine its verses, you will see that I get:

  • 1 partridge in a pear tree on each of the 12 days of Christmas, hence 1 × 12 partridges in their pear trees
  • 2 turtle-doves on each of the last 11 days, hence 2 × 11 turtle-doves
  • 3 french hens on each of the last 10 days, hence 3 × 10 french hens
  • 4 calling birds on each of the last 9 days, hence 4 × 9 calling birds
  • 5 golden rings on each of the last 8 days, hence 5 × 8 golden rings
  • 6 geese a-laying on each of the last 7 days, hence 6 × 7 geese
  • 7 swans a-swimming on each of the last 6 days, hence 7 × 6 swans
  • 8 maids a-milking on each of the last 5 days, hence 8 × 5 maids
  • 9 ladies dancing on each of the last 4 days, hence 9 × 4 ladies
  • 10 lords a-leaping on each of the last 3 days, hence 10 × 3 lords
  • 11 pipers piping on each of the last 2 days, hence 11 × 2 pipers
  • 12 drummers drumming on the final day, hence 12 × 1 drummers

Phew! That’s the whole lot… So, the total number of gifts is the sum of all the above, namely:

(1 × 12) + (2 × 11) + (3 × 10) + (4 × 9) + (5 × 8) + (6 × 7) +

(7 × 6) + (8 × 5) + (9 × 4) + (10 × 3) + (11 × 2) + (12 × 1)

Now, the individual products of the first line are all repeated once on the second line, so this sum is actually twice the value of the first line i.e.

2 × ((1 × 12) + (2 × 11) + (3 × 10) + (4 × 9) + (5 × 8) + (6 × 7))

or

2 × (12 + 22 + 30 + 36 + 40 + 42)

or by doing the sum in brackets

2 × 182

and finally 364.

Now, isn’t that strange? Because the total we have arrived at is precisely one less than the number of days in a year (forgetting leap-years). So, by sending me all these gifts and with the proviso that I resort to one gift per day, my true love has taken care of me for all but one of the days making up one whole year beginning on Christmas Day. These gifts will take me up to the day before next Christmas Eve. That Christmas Eve there will be no gift to use as it will be the 365th day. I should then spend that day thanking my true love and anticipating the flurry of gifts, which will follow on the next day, if she still has the means!

I cannot help wondering whether the creators of this carol designed it on purpose so that the total would come out to 364 or whether this fact completely escaped them. In any case, it is an interesting calculation.

Thank you for bearing with me. May you have a Merry Christmas and a Happy New Year!

I have every good reason to recommend watching “The Man Who Knew Infinity”, an account of the life of the Indian mathematician Srinivasa Ramanujan.

Ramanujan was an amazing figure who had practically no formal training in Mathematics, yet made extraordinary contributions to various fields of the subject. The film centres on his years at Trinity College, Cambridge University, where he worked closely with the English mathematicians G. H. Hardy and J. E. Littlewood.

The film can easily be followed by non-mathematicians. I particularly liked the scene where Ramanujan tries to explain to his wife (and to the general audience) that his love for Mathematics comes from the tendency of mathematical patterns to appear in ways that cause surprise. Hardy also contributes to this by explaining to his butler what Ramanujan was trying to do when tackling the problem of partitions.

The film presents us with some lovely poetic images of India and I cannot forget that scene where Ramanujan’s wife looks on, as her husband sails away in the boat that will take him to a ship bound for England.

Jeremy Irons gives a brilliant performance as the cantankerous G. H. Hardy and at the end of the film quotes from Hardy’s famous and haunting book A Mathematician’s Apology. Dev Patel is also very convincing as the youthful and enthusiastic Ramanujan. The culture shock that he experienced at Cambridge is illustrated well and one can only feel sorry for him, as well for the fact that his life ended so soon.

The only departure from historical fact that I managed to pinpoint concerns the exchange between Ramanujan and Hardy concerning the number 1729. The film shows this as taking place when Hardy bids farewell to Ramanujan, as the latter sets off on his return journey to India. In fact, the exchange took place when Ramanujan was in hospital.

In closing, I must again say ‘bravo’ to the filming world for yet another good film about Mathematics and mathematicians. It has already given us ‘Agora’, ‘A Beautiful Mind’ and ‘The Imitation Game’. I should also include ‘The Theory of Everything’, as Stephen Hawking has used Mathematics so much in his explorations as a cosmologist.

posthumous prize = awarded to one after one’s departure from this world
post-hummus prize = awarded after eating that particular delight

Recently, the following question was set in a GCSE maths exam:

Liz buys packets of coloured buttons.

There are 8 red buttons in each packet of red buttons.
There are 6 silver buttons in each packet of silver buttons.
There are 5 gold buttons in each packet of gold buttons.

Liz buys equal numbers of red buttons, silver buttons and gold buttons.

How many packets of each colour of buttons did Liz buy?

A certain student answered by calculating 8 × 6 × 5 = 240, concluding that this was the number of beads of each colour and then deducing that Liz bought 30 red, 40 silver and 48 gold buttons.

The student lost marks, apparently because the instructions given to the examiners stated that the answer, that had to be arrived at for the number of buttons, of each colour was 120.

Yet it is the question that is at fault, because although 120 is the least common multiple of 8, 6 and 5, the wording does not include any information pointing to the fact that it is this, which is being sought. In fact any common multiple will do, hence the easy choice of 240.

This is quite unfair on the student, but the damage most probably cannot be undone.

There are better ways of phrasing the question and I have thought of one, which in my view also serves to instruct and teach, besides stating the question for the purposes of the exam:

Liz buys packets of coloured buttons.

There are 8 red buttons in each packet of red buttons.
There are 6 silver buttons in each packet of silver buttons.
There are 5 gold buttons in each packet of gold buttons.

Liz wishes to buy the same number of buttons of each of these three colours and, in such a way, so as to obtain the least possible number of buttons.

Calculate how many packets of each colour of buttons Liz should buy, the number of buttons of each colour thus bought and the total number of buttons.

I believe this way of phrasing the question has some benefits:

  1. It illustrates a more natural way in which the problem could occur in real life (Why miss an opportunity to bring Maths a little closer to our daily lives for the sake of students?)
  2. The last sentence helps the student make a distinction between the number of packets and the number of buttons. This is important, because one quite easily could mistake packets for buttons.

And here is a solution to the problem:

Whatever the number of packets of red buttons Liz chooses, she will always end up with a multiple of 8. Similarly, she will end up with a multiple of 6 for the silver buttons and a multiple of 5 for the gold buttons.

Therefore, if Liz wants the same number of buttons of each colour, this common number must be a common multiple of 8, 6 and 5. Since she wants this number to be the least possible, it follows that it must the least common multiple (LCM) of these numbers.

To calculate it, let’s decompose these numbers into prime factors:

8 = 2³

6 = 2 × 3

5 is prime

Hence the LCM of these three will be 2³ × 3 × 5 = 24 × 5 = 120.

[Digression: If one is not happy with this calculation, then one might consider that LCM(8, 6, 5) = LCM(LCM(8, 6), 5).

For the LCM of 8 and 6, we list the multiples of these two:

for 8: 8, 16, 24, …

for 6: 6, 12, 18, 24, …

from whence it follows that LCM(8, 6) = 24. The multiples of 24 and 5, now, are:

24: 24, 48, 72, 96, 120, …

5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125…

and the result follows.]

So, Liz must buy 120 buttons of each colour. To achieve this, she will have to buy:

120 ÷ 8 = 15 packets of red buttons

120 ÷ 6 = 20 packets of silver buttons

120 ÷ 5 = 24 packets of gold buttons

The total number of buttons is 3 × 120 = 360.

 

 

Eidomeni

Today, in #Eidomeni, Northern Greece, on the border with FYROM, there are around 14000 refugees; 7000 of them are children and 70% of these children are ill.
 
The camp, which has been set up there can only hold 4000 people, which means that 10000 are most probably living “outdoors”. To keep warm, they burn everything that they can, including plastic, which means that the air is polluted.
 
The local community chief said that the locals would like to see the 10000 surplus refugees leave, because they are currently occupying their farmland, which is thus rendered unusable.
 
The total number of immigrants in Greece today is around 36400. Out of these, 90% are refugees and 10% are not.
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