# A small pond-crossing

While walking through the flower gardens, you encounter a small group of fairies stuck on a lilypad. There are 2 red fairies, 2 yellow fairies, and 2 green fairies, and they have nothing besides a floating leaf to use as a raft. They explain that they're trapped on the lilypad, as the leaf can only hold 2 fairies at a time, and must always have at least 1 fairy to be operated. Furthermore, the green fairies fear being eaten by the yellow fairies if they're outnumbered, and the yellows have a similar fear of the reds

What is the lowest number of moves required to get them from the lilypad onto dry land?

Ignoring the restrictions, the minimum possible number of moves is nine. Five trips with two fairies from lilypad to land, and four trips back with a single fairy to return the leaf, to get all six fairies on dry land.

The minimum that can be achieved with the restrictions is:

Still nine.
1. RY to dry land
2. Y back to the lilypad
3. GG to dry land
4. R back to the lilypad
5. RR to dry land
6. R back to the lilypad
7. YY to dry land
8. R back to the lilypad
9. RR to dry land

The following works and is minimal in terms of number of trips (although there are other possible options):

RG-out/R-back; YY-out/Y-back; RY-out/R-back; RR-out/G-back; GG-out

• After the first move, the yellows would outnumber the green fairy Nov 24, 2021 at 23:04

Firstly note that the leaf should be occupied by 2 fairies (outgoing) or 1 fairy (incoming) every time, so there's never any outnumbering on the leaf. We just need to make sure no outnumbering happens on dry land or on the lilypad.

1. Move one red and one yellow fairy. On the lilypad: 1 red, 1 yellow, 2 green. On dry land: 1 red, 1 yellow. Nobody is outnumbering anyone, except the green fairies whom nobody fears.

2. One yellow fairy goes back, and two green fairies return. On the lilypad: briefly 1 red, 2 yellow, 2 green, then 1 red, 2 yellow. On dry land: briefly 1 red, then 1 red, 2 green.

3. One red fairy goes back, and two red fairies return. On the lilypad: briefly 2 red, 2 yellow, then 2 yellow. On dry land: briefly 2 green, then 2 red, 2 green.

4. One red fairy goes back, and two yellow fairies return. On the lilypad: briefly 1 red, 2 yellow, then 1 red. On dry land: briefly 1 red, 2 green, then 1 red, 2 yellow, 2 green. Reds don't fear being outnumbered.

5. One red fairy goes back, and two red fairies return. The lilypad is all red, no outnumbering. On dry land: briefly 2 yellow, 2 green, then 2 red, 2 yellow, 2 green and they all continue happily on together.

• I should have mentioned this from the start, but the raft can't be moved back without a fairy in it Nov 24, 2021 at 22:52