Remember "factorials" from high school math? (Yeah, I know, that was a long time ago.) Factorials is the chapter where we talked about how "5!" means 5+4+3+2+1, (for example).
I've been playing this sudoku-esque game lately called Kakuro lately, where you have to add numbers (1-9 with no repeats) together to equal a certain sum in a number crossword fashion (you can take a look at www.kakuro.com or on wikipedia). It's made me play with factorials (without even realizing it) because I've come to recognized that if the total is 15 and it has 5 boxes, it has to be 1,2,3,4,5 and a total of 35 in 5 boxes is 9,8,7,6,5. I started to notice some patterns...
1+2 = 3 = 1x3
1+2+3 = 6 = 2x3
1+2+3+4 = 10 = 2x5
1+2+3+4+5 = 15 = 3x5
1+2+3+4+5+6 = 21 = 3x7
1+2+3+4+5+6+7 = 28 = 4x7
1+2+3+4+5+6+7+8 = 36 = 4x9
1+2+3+4+5+6+7+8+9 = 45 = 5x9
1+2+3+4+5+6+7+8+9+10 = 55 = 5x11
In every case, the two numbers that multiply to equal the factorial (sum of the reducing numbers) increases by one every second time for the first number, and two every second time for the second number.
The first multiple is the biggest number added together divided by two (rounded up if the biggest number is odd). The second number is the biggest number added plus zero (if the biggest number is odd) or one (if the biggest number is even).
So any number plus the sum of all the numbers beneath it (n!) equals that number divided by two, times that number (maybe plus one) (n/2 x n+1 for even numbers).
Memories from math tell me that this can also be written as (n^2+n)/2 = n! (it's a little clearer when it's not restricted by typing). And that works even for the odd numbers - the rounding up and the adding one offset each other.
That means that if you want to know the sum of a number plus every number beneath it, you times the number by itself, add it in again, and divide the whole thing by two!
I think that this is the formula they taught us in high school - I haven't actually discovered anything new here - but I am super pumped to have figured it out by myself. =)
I'm not so good at Kakuro though... I only get the really hard ones that my Mom saves me from her city's Saturday comics (it shares a page with the crossword and wordsearch), so I can't build up my skills - I'm thrown into the five star puzzles and consistently not getting very far. Knowing the formula to solve factorials isn't actually helping me. But I'm having fun with this!
