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## '2009 Challenge' printed from http://nrich.maths.org/

Thank you for your thoughts on this
problem; some interesting points were suggested about the number
2009. We explored some of these using a spreadsheet; you might also
like to try this. There were more suggestions than those listed
below for which the maths didn't quite work out, but thank you for
those too.

To decide how interesting a fact was we
considered its rarity compared to other numbers, how many different
ways of expressing the same sort of fact we could think of and,
well, how much it caught our attention. In reverse order, the
results are as follows:

6. (David from Ysgol Bryn Alyn, Alex from DGS) 2009 and its
reversal 9002 are divisible by 7. This will
happen next in 2016, 2023 and 2030, so this property seems to be
common. Does this lead to a conjecture?

5. (Daniel from Savile Park) 2009 can be written as the difference
of two squares in 3 ways, and the sum of all of the numbers gives a
number in which the digits are sequential

$$2009 = 1005^2-1004^2=147^2-140^2=45^2-4^2$$

$$1005+1004+147+140+45+4=2345 $$

The next years where the number will be the
difference of two squares in exactly three different ways are 2023,
2028, 2032. So, it seems that this is not an unusual property.
However, would other numbers have this sequential digits
property?

4. (Jamie from Ysgol Bryn Alyn) It is 500 years since Henry VIII
was crowned King. Well, this is certainly
unique, and we found this interesting!

3. (Aaron from Ysgol Bryn Alyn, Harry from The Beacon School, Alex
from DGS) The 2009th Prime (17471) is palindromic (reads the same
forwards as backwards). The next three
years giving rise to this behaviour are 2060, 2083 and 2117. Out of
the first 50000 primes, only 113 are palindromic, which makes this
result quite interesting.

2. (Daniel from Savile Park) $2009=7^4-7^3-7^2.$ This is a very neat little formula, and perhaps
the easiest one to remember

1. (Harry from The Beacon School)

$$

2009\times 2008\times 2007\times\cdots \times 4\times 3\times
2\times 1\mbox{ ends with } 500 \mbox{ zeroes}

$$

Each year since 2005 has had a similar
property, but after 2009 we need to wait until 2410 to get 600
zeroes and the year 4000 to get the 999 zeroes. We decided that
this was notable, and therefore that this was the most interesting
fact about the number 2009. Well done Harry.