How to solve Sinking Leaves

The frog sits on its leaf. Find its route: the frog visits every leaf exactly once and ends on the leaf with the flower.

Every jump goes straight up, down, left or right, from leaf to leaf, and can be any length.

A leaf sinks as soon as the frog jumps off it, including the leaf it starts on.

The frog may fly over water and over a sunken leaf, but never over a leaf still afloat. The leaf with the flower floats to the very end, so the frog never flies over it.

Tap a leaf in the frog’s row or column and the frog jumps onto it. Tapping a sunken leaf takes the frog back to that leaf.

With Crosses on, tapping a square puts a cross on it or takes it off. Crosses are your own notes; the game does not check them.

Nothing is given but the layout of the leaves. Every board has exactly one route, and it can be found without guessing. A 7 × 7 can be solved with the moves from the first chapters — counting jumps, watching the leaves still afloat and steering clear of loops — and at least once it makes you think about the sinking. On an 8 × 8 at least one jump is decided only by a look at the whole pond — at the leaves before the flower or at the only link — and on a 9 × 9 at least three.

The rules above say it all; here each one is drawn the way the board shows it: a leaf the frog has already jumped off sinks and leaves a numbered ring behind. A dashed line without a cross is a jump that is allowed, and with a cross a jump that is not: the board will not make it, and says in one sentence which rule it breaks.

On the board the frog is led jump by jump, but when solving, it is easiest to think of the jumps as lines between leaves, in no order — the way they are drawn in pencil on paper. Every leaf has exactly two of them; the frog’s leaf and the flower’s leaf have one each. A jump joins only leaves in the same row or the same column, so it is enough to count how many leaves a leaf can still be joined to. If that is as many as the jumps it needs, all those jumps are settled. And a leaf that already has its jumps gets no more. That is how every board begins.

Now the rule the game takes its name from. A leaf sinks only once the frog jumps off it, and the frog may not fly over a leaf still afloat. So from the leaf the settled jumps have brought the frog to, the next jump always lands on the first leaf afloat in the direction it takes — longer jumps are out. And the leaf with the flower floats to the very end, so the frog never flies over it — every jump over the flower is out. A 7 × 7 needs one of these moves at least once.

The route is one chain of jumps from the frog to the flower. A jump that would close a few leaves into a loop is out: the route never comes back to a leaf, and there would be no way out of the loop to the other leaves. Also out is a jump that would join the piece of the route starting at the frog to the piece ending at the flower before every leaf is on them: the route would be finished with some leaf left unvisited.

When counting and the moves from the previous chapters get you no further, it pays to look from the other end. Settled jumps leading to the flower are the end of the route: the frog visits their leaves last of all, so until then they stay afloat — like the flower itself. So no other jump may fly over them. Nearly every 8 × 8 and 9 × 9 needs this move. More generally: the frog may fly over a leaf only if it visited that leaf before both ends of the jump, and when the settled jumps say otherwise, the jump is out.

Sometimes you have to look at the whole pond at once. If the leaves fall into two parts that only one possible jump still joins, the route must take that jump — otherwise the frog could not get from one part to the other. On a 9 × 9 this comes in handy about one day in four, on an 8 × 8 rarely.