– 10*9*8+7+6-5+4*321 Happy New Year!

– It is a bit late for that!

– May be, but isn’t it beautiful?!

– That 10*9*8+7+6-5+4*321 = 2012?

– Yeah, I bet you it does not happen every year!

– How long will it be until another year arrives that can be similarly expressed?

– You mean with interspersing +, -, *, /, or nothing between the numbers in order from 10 to 1?

– Yes.

– May be in one thousand years.

– That is a long time. Amazing!

Problem source: The Museum of Mathematics and the Wolfram Blog.

2012 = [ (1+0+9)+(8+7)*(6+5)*4 ] * 3 + 2*1

2013 = [ (1+0+9)+(8+7)*(6+5)*4 ] * 3 + 2 + 1

2014 = {(1+0+9)*[(8+7+6)*5-4] – 3} *2*1

2015 = (1+0+9+8)*7*(6+5+4+3-2) – 1

2016 = (1+0+9+8)*7*(6+5+4+3-2)*1

2017 = (1+0+9+8)*7*(6+5+4+3-2) + 1

Found using a program I’d already written, to solve problems just like this one. The first operation it tries is addition, hence the preference for expressions that start with a chain of additions of single-digit numbers.

Using just 987654321:

2012 = 9*8*7*(6-5)*4 -3-2+1

Some delights for future years:

2013 = (65-4)*(32+1)

2015 = 8*(7+6+5)*(4+3)*2 – 1

2016 = (7+6-5)*4*3*21

2017 = 8*(7+6+5)*(4+3)*2 + 1

And one may use Mathematika:

http://blog.wolfram.com/2012/02/02/happy-109876-54321/