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Archive for January, 2010

Σήμερα το μεσημέρι, πήγα σε μία τράπεζα για να καταθέσω κάποια χρήματα ως ενίσχυση του έργου που επιτελούν οι Γιατροί Χωρίς Σύνορα στην πληγείσα Αϊτή.

Ο υπάλληλος με ρώτησε τι θα ήθελα να αναγραφεί στο τμήμα “αιτιολογία” του δελτίου κατάθεσης. Εγώ απάντησα “Άγγελος Σφυρής Αϊτή”. Όταν ο υπάλληλος εκτύπωσε το δελτίο και μου το έδωσε να το υπογράψω, είδα αυτό που με ώθησε να γράψω τη σημερινή καταχώριση. Στην αιτιολογία το δελτίο έγραφε:

ΑΓΓΕΛΟΣ ΣΦΥΡΗΣ ΙΤ

“Μα δεν εννοούσα Information Technology”, είπα γρήγορα στον υπάλληλο. “Εννοούσα το νησί… ξέρετε, ο σεισμός…”. Και σκεφτείτε ότι ο συγκεκριμένος υπάλληλος δεν είχε ιδέα ότι εγώ ασχολούμαι με την Πληροφορική. Στη συνέχεια ακύρωσα το δελτίο και ο υπάλληλος διόρθωσε το σφάλμα.

Μας έχουν κυριεύσει τα ακρωνύμια: SMS, PC, IP, ASAP, ΙΜΗΟ κλπ και στέκονται εμπόδιο στη συνεννόησή μας…

Παρεμπιπτόντως, σε περίπτωση που θελήσετε να ενισχύσετε και εσείς το έργο των Γιατρών Χωρίς Σύνορα στην Αϊτή, μπορείτε να επισκεφτείτε τη σχετική σελίδα του ελληνικού τμήματος. Ας ευχηθούμε κάθε επιτυχία σ’ αυτούς, αλλά και σε όλους τους άλλους παράγοντες που βοηθούν αυτές τις μέρες το δύσμοιρο λαό της Αϊτής. Αυτή ήταν μια πολύ άσχημη τραγωδία που τους έτυχε.

Η Ελλάδα – όπως ίσως θα έχετε αντιληφθεί πρόσφατα και από άλλες διαδικτυακές πηγές (blogs, e-mail) – έχει μία ειδική σχέση με την Αϊτή, παρά την τεράστια γεωγραφική απόσταση που υπάρχει μεταξύ τους: Σύμφωνα με την εγκυκλοπαίδεια Πάπυρος – Λαρούς του 1963, η Αϊτή ήταν η πρώτη χώρα που χαιρέτισε την ελληνική επανάσταση του 1821. Στις 15 Ιανουαρίου 1822, ο Γάλλος φιλέλληνας κυβερνήτης της Αϊτής Μπουαγιέ ευχήθηκε την ταχεία αναγέννηση της Ελλάδας, απαντώντας σε μία επιστολή που είχε στείλει η επιτροπή των Παρισίων (Α. Κοραής, Κ. Πολυχρονιάδης, Α. Βογορίδης, Χ. Κλονάρης) με την οποία ζητούσε βοήθεια για τον Αγώνα.

 

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My True Love’s Bill

It is Twelfth Night, the eve of the Twelfth and final day of Christmas, and all through this festive period I did not come across anyone singing that well-known and amusing Christmas carol, “the Twelve Days of Christmas”. I suppose that, since the carol describes a multitude of gifts to the singer all from their (one) true love, this may be due to the current recession. You can find some historical information about the carol at Wikipedia. There is also a section on its financial implications, which shows that “my true love’s bill” would reach much higher than the new Burj Khalifa!

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. 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 directly linked 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. I hope to be able to talk more about this in future posts.

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.

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Happy New Year to all who are reading this. A friend of mine wished me all the best for the new decade and I have also seen things on the Internet and on TV along the lines of “the best … of the last decade”.

I want to point out a very simple numerical truth: It is not the end of the decade. The end of the decade occurs on 31 December 2010 and the new decade begins on 1 January of next year.

To understand why, you need to remember that the current millenium did not begin on 1 January 2000. In fact, this millenium began on 1 January 2001. You see, by definition, the millenium is a period of one thousand years. So, if you begin from 1 AD and count the first one thousand years, then 1 AD is the first year, 2 AD is the second and so on. You will reach your goal on the last day of 1000 AD. On that very day, a full one thousand years will be “behind you” in your count and on the next day – 1 January 1001 – you will have entered the second millenium. Now, continue counting: 1001 (first year), 1002 (second year), 1003 (third year) and so on, until you reach 1999 (999th year). You should now be able to see that 2000 is the 1000th year in your count and the end of the second millenium…

So, 2001 is the first year of the third millenium, which is the one we are in. Now, count the first ten years of this third millenium:

  1. 2001
  2. 2002
  3. 2003
  4. 2004
  5. 2005
  6. 2006
  7. 2007
  8. 2008
  9. 2009
  10. 2010

So the first decade of this millenium will close on 31/12/2010.

I suppose people base their end-of-decade feelings on the presence of the number 10 in 2010, but as I have shown this is not a correct foundation. I will seize the opportunity and say that if we wish to understand phenomena and processes that are part of our natural world and/or our society, it is important to be clear in our understanding of simple matters such as this one. And it isn’t that much of a bother to remember this subtlety.

So, until 31 December 2010: Happy New Year (only).

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