Why is this formula correct?

Question

You start off with blue frogs on the left and green frogs on the right and a gap in the middle. YOU HAVE TO SWAP THE FROGS OVER
Blue Frogs - Only move right
Green Frogs - Only move Left
Frogs can move into a free space or leapfrog a frog of an opposite colour

For the same amount of frogs on each side;
n(n+2) - Why is this formula right for the same number of frogs on each side? (n = Position Number)

For different number of frogs:
n+m+nm - Why is this formula right for any number of frogs?
(n = No. of Blue Frogs and m = No. of Green Frogs)

For more go to http://mathsnet.net/puzzles/leapfrog/index.html

Answer 1

1

Answer 2

Let G be a green frog, B be a Blue frog and X = a free space.
With 3 frogs on each side we have the following:
0.BBBXGGG Either G-->X or B-->X. Choose G-->X, getting:
1. BBBGXGG If G-->X then there is only one move left, so
2. BBXGBGG Now if G-->X then BBGXBGG, which fails, so
3. BXBGBGG All over if B--> X , so
4. BGBXBGG All over if B--> X , so
5. BGBGBXG
6. BGBGBGX Only move
7. BGBGXGB Only move. Now if G-->X, no more moves, so
8. BGXGBGB If G--> X no more moves, so
9. XGBGBGB
10.GXBGBGB
11.GGBXBGB
12.GGBGBXB
13.GGBGXBB
14.GGXGBBB
15.GGGXBBB
So three frogs on each side takes 15 moves
Two frogs on each side can be shown to take 8 moves and 1 frog on each side can be shown to take 3 moves.

The formula clearly works for n =1 {1(1+2)= 3}
So ny inductio you can prove that it works for all n>1.

Answer 3

To be honest, I think you could forget about the formulas, you can only switch the frogs over if you have a small number of frogs i.e. only 1 or 2 on each side.

I get like 3 moves with the larger numbers, and I'm stuck.

They need to reveal a solution for the larger numbers cause I can't figure it out.

As for why they are correct, you just have to keep testing the different numbers and do some trial and error guessing to work out the formula.

I suppose because the formulas just are.

Answer 4

I under no circumstances boiled the water. we've an R/O equipment that has 5 ranges of filtration in the previous we had that I used bottled water at room temp. To warmth it i could placed warm water in a huge cup and microwave it for a pair of million minute then I placed the bottle into the warm water and enable it heat up. I relatively have seen this question from others and it relatively is have been given me questioning what could make you think of you are able to desire to boil the water, i'm purely curious as a results of fact i've got under no circumstances heard that throughout the time of the previous. besides it is how I did with my 2 boys and am now doing with my 10 month previous daughter. wish I helped.

Answer 5

so bob, how's the traffic look?
The light is red and he's turning right.
This is a REALLY small town...

btw, who here has ever seen a blue frog?

Answer 6

yeah its 10 to 7

Answer 7

Shouldn't this be in the cooking section? Yummy!

Answer 8

because of the wonderful wizard of australia, duh!