Solution to frog leap game

The seven rocks are laid out in a horizontal line and numbered left to right. The six frogs are evenly divided into a green trio and a brown trio. The green frogs sit on Rocks 1, 2, and 3, facing right. The brown frogs sit on Rocks 5, 6, and 7, facing left.

Rock 4 is vacant. The challenge is to transpose the trios, jumping the green frogs to Rocks 5, 6, and 7 and the brown frogs to Rocks 1, 2, and 3.

Their movement is restricted. A frog can only jump forward, either hopping to a vacant rock one place ahead or leaping over its neighbor frog to a vacant rock two places ahead. Seven rocks and six frogs.

Let me report that I would be a dismal second grader in the People's Republic of China and probably in most other countries as well. Prodding six frogs to opposing destinations along a one-lane road of seven rocks took me far longer than two minutes. To rediscover it, I had to start over. Could I have done better? Remembrance of Things Past. Having no idea what pattern underlay the solution, I began with my only option—spontaneous action.

I jumped my frogs by impulse, observed the result, and tried to get a feel for the situation. For meaningful problems, I keep a running log of what I'm doing and thinking—a substitute for my unreliable memory. That was a mistake: Lacking a catalog of failed strategies as a reference, I wasted time retracing previously unsuccessful paths. Eventually, certain jump sequences, repeated ad nauseam, stuck in my brain.

Recalling them, I eliminated the wrong ones and conjoined the right ones. Before my memory faded again, I recorded the entire sequence of fifteen correct jumps—not in a list, but in a diagram. The correct jump sequence. A diagram of the jumps has a coherence that a list of the jumps lacks. Viewable at a glance, it presents the complete data set all-at-once , integrating the many jumps into a single visual gestalt. With the total configuration in plain sight, there is nothing to recall; the demand on short-term memory is negligible.

A list, on the other hand, must be processed incrementally via many disconnected eye shifts. So, a list is a memory-intensive presentation. I only knew the jump sequence itself and its mirror image, which also solves the puzzle. I wondered if the sequence followed some pattern. Should a pattern exist, anyone who knew it—say, a Chinese second-grader—could implement it and quickly solve the puzzle. Knowing this pattern, one would have a deeper understanding of the solution.

And what if that pattern derived from some general organizing principle that underlies the solutions to many puzzles; puzzles different in their particulars but alike in their form? Knowing the principle, one would have a very deep understanding of the solution; one could easily solve sundry puzzles and seem a miracle worker to naive onlookers.

I examined the solution for patterns, looking for one that governed the jump sequence. There were several of interest.

The aesthetic appeal of certain patterns is entrancing; attention gravitates to them. But sometimes they are secondary—mere byproducts of a primary, generative pattern. Attractive though trivial, they can blind us to truly significant patterns.

These patterns were found to be secondary—but only after their beguiling beauty enticed me down some blind alleys. What were their attractions and their failings? Blind Alley 1: The ordinal pattern in which six uniquely numbered frogs jump. The successful solution requires the frogs to jump in the order shown below. Each frog bears the number of the rock on which it first sits. Put the three green frogs on the rocks on the left and the three brown frogs on the rocks on the right.

There must be an empty rock in the middle Like this:. Green frogs can only hop to the right. Brown frogs can only hop to the left. Frogs can only jump onto an empty rock. A frog can jump over one frog to an empty rock—but only over one. Frogs cannot jump back, only forward. If you get stuck and your frogs can't jump anywhere, move them back to where they started and try again. Guest Posted January 30, Posted January 30, edited. Edited January 30, by Special Edd.

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