1. A CHRISTMAS TALE
Original Author: Joaquín Collantes
Field Advisor: Antonio Pérez Sanz
English Translation: Francesco Scudellari
-Who are you? -Lucas asked, moving back scared.
-We are the Spirits of Mathematics-, answered one of the three ghosts who
had suddenly appeared before the young man studying in his room.
- I am the first of the Three Spirits of Mathematics –the spirit added. Then
he pointed his finger at his companions, who said:
- I am the second of the Three Spirits of Mathematics.
- And I am the third of the Three Spirits of Mathematics.
But this explanation didn’t calm down the frightened student. And even if
he didn’t believe in ghosts, he startled at the presence of those three
mysterious spirits that had suddenly appeared, and that were floating one
palm over the floor and radiating a weird phosphorescence that lightened
the room with an intense yellowish light.
Two of them were wearing the same clothes, which proved that they
belonged to the same historical period, more or less the 17th or 18th
century: They were wearing a long jacket and short velvet trousers, a
white shirt with wide cuffs, white stockings and black cuir shoes with a small heel and a golden buckle; the Second
Ghost had a dusty, curly wig falling on his shoulders, while the Third Ghost had a simpler one, combed backwards,
with a ponytail. Another difference between them was that the Third Ghost was wearing dark glasses and in his hand he
had a thin stick that he used to help himself to move around, as blind people do.
The First Ghost's clothes were totally different from those of his companions: he was wearing a wide white cotton tunic
with a blue band around the edge, and simple cuir sandals; he was bald with a curly white beard, and looked definitely
like some Greek philosopher or Roman patrician. The only similarity between them was that the three men were loaded
with books, notebooks, spare sheets and scrolls that were continuously falling on the floor, especially the Third Spirit,
who was carrying a handful of books and folders, so many that they reached the ceiling.
Lucas, the frightened maths student, was still scared for the unexpected apparition… until he recalled that on his
desktop he had a book titled “A Christmas Carol”, the classic by Dickens that he was reading at the moment. He
realised that it was happening something very similar to the story told in the book. And so he asked:
- Are you the three Christmas Spirits that appeared to Evenezer Scrooge?
- That appeared to who ?
- To Scrooge, the main character of “A Christmas Carol”, the Charles Dickens’s tale that I’m reading. Scrooge,
the most avaricious, greedy and eselfish man in the world.
- You like this Scrooge, don’t you? – said the Third Ghost ironically.
- Anyway, no, we’re not these three Christmas spirits you’re talking about – said the Third Ghost.
But, you know, it’s Christmas time… and in case you don’t know, the three Christmas spirits appear to Scrooge to teach
him a lesson about his inhuman greed. And these three spirits are: the Ghost of Christmas Past, the Ghost of Christmas
Present and the Ghost of Christmas Future. That’s the reason why, when I saw you three here, I thought that you could
be the Three Spirits looking for Scrooge… and that you got the wrong address!

2. First of all, we don’t know this Charles Dickens’s tale; second, we are not the spirits of Christmas and third, … well….
third…. ehm…. I don’t have a third thing to say! – the First Spirit said, a little ashamed.
-Well, we could say that we are not the spirits you are talking about: we are, as we told you before, the Spirits of
Mathematics, or at least three of them: there are as many spirits of mathematics as great mathematicians who lived on
Earth. – Added the Third Spirit to help his companion.
-That’s certain: to be a Spirit of Mathematics you have to be a dead mathematician. – added the Third Spirit, as an
unnecessary explication.
-So, you didn’t get the wrong address, and you didn’t come to show me the past, present and future Christmas. So why
did you come here? - Lucas asked, and added: - By the way, maybe I’m dreaming?
-No, man, you’re not dreaming – answered the First Spirit – What’s happening to you is real. And we appeared
precisely to you, because you (even if you still don’t know it) will be a great mathematician. And as we know it all, we
appear to young future mathematicians to encourage them.
-Will I get the Fields Medal? – Lucas asked.
-Hey, don’t ask too much. For the time being, just know that you will be a great mathematician, that’s not bad, isn’t it?
By the way, what’s that grid on your desktop?
-A very simple problem I was preparing for a friend of mine. You know, we invent problems and propose them to each
other…
-To disturb each other.... - said the First Spirit with a smile.
-No, we do it because we like mathematics.
-Well, let me see it – said the First Spirit, taking the sheet on the desktop, while some scrolls were falling from his
hands… and read:
“Complete the square”
1 3 ?
13 1 5
8 21 2
-Well, you’re right, this problem is not that simple, even though I’m an expert in geometry…. you know, I’m not that
familiar with Arabic numerals, because I learned them when I was already a ghost: in my hometown, Syracuse, we
didn’t even have Roman numeration!
Luckily, being a ghost, I could follow the evolution of mathematics, from my century until now: because of this I know
the Arabic numerical system, that (I have to admit that!) is very good, was brought to Europe by the great Fibonacci in
the 13th century, even though it didn’t become popular until the 15th century.
-Well, yes, that was more than 300 years after he brought it – Lucas said, to make the old man in the tunic aware that he
knew what he was talking about.
- But, in the end, we can say that this problem is rather simple. By the way, sir Newton, maybe we could suggest this
apprentice mathematician a problem that’s a little more difficult than the previous one, just to disturb his friend, what do
you think?
- Very good, we could propose him that problem about....- he answered
- One moment, one moment... are you Isaac Newton? – Lucas asked, unable to control his surprise. And before the
Second Spirit could answer, he turned his head to the Third Spirit of Mathematics and asked: -And so you, surely you
are...

3. -Leonhard Euler, at your service – he answered, bowing.
-And you, your outfit makes me think that you are Archimedes, aren’t you?
- You guessed correctly... even if people usually represent me very bad. As portraits of Newton and Euler do exist, you
can easily imagine how they looked like, but as of me... – the Spirit of Archimedes answered.
-At first, I thought you were the Three Wise Men in disguise! This situation is becoming surrealistic. Now I am really
sure that this is a dream.
-Why? – Newton’s Spirit asked – You believe that we are the Three Spirits of Mathematics, that we are the Three Wise
Men in disguise .... and you don’t believe that we are Archimedes, Euler and I.
-Because I was influenced by Dickens’s tale.... and because it’s Christmas time! – Lucas answered, and added, calm
enough- Anyway, what was that problem you were talking about?
-Well – said the Spirit of Archimedes, give him the problem, sir Newton. And as you also are an astronomer, give him a
problem about planets.
And the Spirit of Newton wrote on a piece of paper a short text and drew a picture, offering the paper to Lucas, who
read the text aloud:
“Two planets turn around the same star. The external one takes twelve years to complete an orbit, and the internal one
takes ten. In this moment they are aligned with the star. When will they be aligned the next time?
-Isn’t that too difficult for a 21st century young man? Consider that nowadays, with so much television and Playstation,
teenagers’ neurons are a little bit ..... – said the Spirit of Archimedes.
-Difficult? I think it’s rather normal.... I would propose an even harder one. Anyone of my thirteen children could solve
this planetary problem at the first attempt – said Euler’s Spirit, ducking again to pick up, helped by his companions,
some of his books that had fallen on the floor.
-All these books, papers and scrolls falling on the floor are making me nervous. Why are you carrying so many pieces
of paper and books? – Lucas asked.
- That’s because we always take with us the essential part of our works, just in case. And, of course, in my case I chose
the three books of my masterpiece’s first edition, Philosophiae naturalis principia matematica. In the case of
Archimedes, that’s a little bit harder, as in his times they didn’t bind sheets in the form of books he has to travel
carrying all these scrolls: that’s rather uncomfortable. And as of Euler, look at him, he is constantly loaded with his
complete works. And he published more than 500 books and articles, writing an average 800 pages per year: he was the
most prolific mathematician in History – and, helping Euler to pick up some of his books who had fallen on the ground,
he added – And, as he is blind (he lived his last days of his earthly life in this condition), we have to help him and we
always end up tired of all this…Our body can’t stand it… well, really, we haven’t got a body at all, as we’re ghosts!
When all of Euler’s papers had been collected, an uncomfortable silence fell in the room, until Newton’s spirit says: -
Well, we’re leaving.
-Why do you go away so early? – Lucas asked, already used to the spirits’ presence.
-Because we have to appear to three more maths students to tell them that they will become great mathematicians in the
future. The problem is, one of them lives in France and another one lives in Germany, well, that’s rather close… but the
third student lives in Australia. We also need to consider the time shift, and we have to appear in the night because it's
more spectacular!- added the Spirit of Archimedes.
At this point, Euler’s spirit asked: -Do you want a harder problem to propose to your friend?
-Yes, that sounds good.

4. -Very good, so please note the problem’s formulation, that’s very easy to write down, even if the problem is hard.
“Find all the natural numbers made of 4 digits, and equal to the cube of the sum of their digits.”
-Is that all? – Lucas asked, astonished –Does this simple formulation hide such a hard problem?
-Try to solve it –Euler answered, smiling.
And the Spirit of Newton, going ahead and considering the apparition over, held his hand out so Lucas could shake it,
and said:
-Well, Lucas, good luck in your studies and remember: if you succeed that will be because you studied hard!
Lucas shook the hands of Newton and Archimedes, but couldn’t do the same with Euler’s, because Euler’s hands were
carrying his huge amount of works. And in an instant, as quickly as they had appeared, they disappeared leaving behind
them their phosphorescent light, which did not vanish until the morning, and a penetrating, weird smell: something like
a mix of incense, roquefort cheese and vanilla.
On the following day, Lucas’s mother, upon entering his son’s bedroom to wake him up and remember him to go to
school, said:
-For heaven’s sake, Lucas, your bedroom smells awfully! I told you a thousand times to ventilate your room from time
to time!
P.S.: Lucas was awarded the Fields Medal in the International Congress of Mathematics of 2032 …. And even though
the spirits of Archimedes, Newton and Euler already knew it, they clapped their hands enthusiastically.
THE END
QUESTIONS
NOTE: The answers to the questions marked with an asterisk (*) are not found in the above text. You can look for them
on the Internet or in books.
1. In the beginning, who does Lucas think the three spirits are?
2. In which time of the year does this tale take place?
3. What book was Lucas reading and who is its author?
4. One of the Three Spirits of Mathematics in this tale is Archimedes. What was his hometown?
5. Who are the two other spirits of mathematics?
6. The Arabic numerical system is the system that's commonly used nowadays. What's the name of the
mathematician who introduced it in Europe? In which century did he introduce it?
7. How many books and articles did Euler write in his life?
8. (*)What is the Fields Medal?
9. (*)The “Fibonacci's succession” is a particular series of numbers. What are the first 10 numbers of this
succession?
10. Try to complete the square proposed in the text above.