mathematics - The frog concerto

A pond contains $24$ waterlilies that are arranged in a rectangular $2\times12$ grid (that is, two rows with twelve waterlilies). One evening $24$ frogs give a croaking concerto for the residents of the pond.

At the beginning of the concert, the $k$th frog is sitting on the $k$th lily pad (where $1\le k\le24$). There is a short break in the middle of the concert, and after this break none of the frogs returns to its old lily pad. It turns out that after the break each of the $24$ lily pads again carries one frog, and that every frog is now sitting on some new pad that is horizontally or vertically adjacent to its old pad from before the break.

Question: How many different seating arrangements are there after the break that match the above description.

(The answer to this question will be a square number. A good solution will clearly explain the reason why a square number shows up here.)

Answer

The answer is

$54289$, or $233^2$

This is because

If we label the frogs 01-12 on the top and 24-13 on the bottom

01 02 03 04 05 06 07 08 09 10 11 12
24 23 22 21 20 19 18 17 16 15 14 13

We see that every odd frog must end on an even position and every even frog must end on an odd position. So the positioning of the even and odd frogs are independent of each other. They are also symmetrical to each other, so the total number of even positions equals the total number of odd positions. Therefore, we can consider only the odd frogs and square the answer to get the total number of seating arrangements.

Now, to calculate the position for 24 frogs, I broke it up into 2 parts:

Part 1: Frog 01 moves to position 24. In this case, the number of positions is the same as calculating the positions for the remaining 11 odd frogs in a 2 x 11 rectangle.

Part 2: Frog 01 moves to position 02. This forces frog 23 to position 24. In this case, the number of positions is the same as calculating the positions for the remaining 10 odd frogs in a 2 x 10 rectangle.

Therefore

$F(x) = F(x-1) + F(x-2)$, so we have a Fibonacci sequence.

$F(1) = 1$
$F(2) = 2$
$F(12) = 233$, so there are 233 positions for 12 odd frogs. Square this to give the total positions for 24 frogs, and we arrive at our answer, $54289$.