# 2009 Challenge

##### Stage: 3, 4 and 5 Challenge Level:

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.