Frogs and Toads Puzzle: Solution

The rules

Tap an animal to move it. It can slide onto the empty pad next to it, or jump over one animal of the other kind onto the empty pad just beyond. A frog never jumps a frog and nobody moves backwards. The puzzle is solved when the toads are all on the left and the frogs all on the right. If nobody can move, you are stuck and must undo.

Every solution is the same length

Here is a surprising fact: if you finish at all, you finish in exactly the same number of moves. Every frog has to get past every toad, and the only way two animals pass each other is a jump, so there are always m × n jumps with m frogs and n toads. Each animal also has to travel a fixed distance, and the slides make up the rest: always m + n of them. Three and three take 9 jumps and 6 slides, 15 moves in all.

So the challenge is not to be quick but to be clean. The game counts every move you make, even the ones you undo, and three stars go to a solve with no wasted moves.

How you get stuck

The trap is two animals of the same kind side by side between the empty pad and the animals they still have to pass. Say the row is frog, frog, empty, toad, toad and you slide a frog forward, then the other frog: now two frogs face two toads with no gap between them, and nobody can move. A toad can jump only one frog, never two.

Three rules keep you out of trouble every time, whichever animal you move first:

1. Jump whenever you can. There is never more than one jump to choose from.
2. If nobody can jump, slide an animal of the same kind as the one that just moved.
3. If none of that kind can slide, slide the other kind. There will be just one slide left.

The pattern

Follow those rules and the two kinds take turns in runs that grow and then shrink. For three and three it goes:

• one frog slides,
• two toads move (a jump and a slide),
• three frogs move,
• three toads move,
• three frogs move,
• two toads move,
• one frog slides.

Halfway through, the row is a perfect zigzag of toad, frog, toad, frog with the empty pad at one end.

Two and Two, move by move

A good warm-up. The pads are numbered 1 to 5 from the left: frogs on 1 and 2, toads on 4 and 5.

1. Frog on pad 2 slides to 3.
2. Toad on pad 4 jumps to 2.
3. Toad on pad 5 slides to 4.
4. Frog on pad 3 jumps to 5.
5. Frog on pad 1 jumps to 3.
6. Toad on pad 2 slides to 1.
7. Toad on pad 4 jumps to 2.
8. Frog on pad 3 slides to 4.

Eight moves: four jumps, one for each frog and toad that pass, and four slides.

Three and Three, move by move

The pads are numbered 1 to 7 from the left. Frogs start on 1, 2 and 3, toads on 5, 6 and 7.

1. Frog on pad 3 slides to 4.
2. Toad on pad 5 jumps to 3.
3. Toad on pad 6 slides to 5.
4. Frog on pad 4 jumps to 6.
5. Frog on pad 2 jumps to 4.
6. Frog on pad 1 slides to 2.
7. Toad on pad 3 jumps to 1.
8. Toad on pad 5 jumps to 3.
9. Toad on pad 7 jumps to 5.
10. Frog on pad 6 slides to 7.
11. Frog on pad 4 jumps to 6.
12. Frog on pad 2 jumps to 4.
13. Toad on pad 3 slides to 2.
14. Toad on pad 5 jumps to 3.
15. Frog on pad 4 slides to 5.

Fifteen moves, three times three plus three plus three.

Other sizes

• One and one: 3 moves. Slide the frog, jump the toad, slide the frog.
• Two and two: 8 moves. One frog, then two toads, two frogs, two toads and one frog.
• Three and two: 11 moves. Uneven groups follow the same three rules, and either kind can go first.
• Four and three, four and four, five and five: 19, 24 and 35 moves. Exactly the same pattern, with longer runs. For five and five the runs go 1, 2, 3, 4, 5, 5, 5, 4, 3, 2, 1.

The puzzle is old: versions with coins and pegs instead of frogs and toads appear in puzzle books from the nineteenth century.