(10 + 98 – 7 × 6 – 5) × (4 × 3 + 21)

 
Yesterday’s post was inspired by Peter Norvig’s new year’s greeting, which was:

Happy 10/9!*8!*7!-6!*5!/4!+3!*2!+1!
 

I thought his greeting was wonderful, but it bothered me that it relied so heavily on factorials, which many people don’t understand. I wondered whether it would be possible to use only add, subtract and multiply, plus concatenation (eg: 1 and 2 can become 12), so that more people would be able to read the message. For variety, I decided to put the digits in forward order.

My strategy for finding a solution was to start with 2013 and work backwards, trying to whittle things down to zero. First I tried 2013 – 1234, but that gave 779, and I couldn’t see how to get from 779 down to zero using just the remaining digits 5,6,7,8,9.

So I then backtracked and tried 2013 – 123. That gave me 1890, and I knew I was in luck, since 1890 is a multiple of 90, so it must also be a multiple of 45. That gave (2013 – 123) / 45 = 42, which is just 7 × 6 — two more digits down.

At this point it was done, except for incorporating 8 and 9. Fortunately (8 – 9) is -1, which is easy to slip in, giving the final result: 0 + 123 – 45 × 6 × 7 × (8 – 9) = 2013.

Still, there was something very cool about the fact that Peter Norvig’s sequence was an actual countdown from 10 to 1 — just like the countdown to midnight at Times Square on New Years Eve. So this morning I figured I’d try such a countdown, again using only “easy to read” arithmetic.

Since 2013 = 61 × 33, this time I took a different approach:

61  →  10 + 56 – 5  →  10 + (98 – 42) – 5  →  10 + 98 – 7 × 6 – 5

33  →  12 + 21  →  4 × 3 + 21

 
And there you have it — the title of today’s post.