Frog Jumping & Stack Up

by Spencer Bowen

This week’s activity is a great puzzle to work on with materials lying around the house, or using our online applet.  This problem was first inspired by and their activity Jumping Frogs, with Stack Up developed by the Julia Robinson Mathematics Festival and the Math Teachers’ Circle Network.

Jumping Frogs

The main goal of this puzzle is to get a row of frogs to jump onto each other to form a single stack. 
Let’s use cups to visualize our frogs:  

One frog can jump one space.  A stack of two frogs jumps two spaces, no more or less. A stack of three frogs jumps three spaces, and so on. Stacks must jump a number of spaces equal to the number of cups in the stack.  Jumps must be straight to the left or right–no backtracking when making jumps.

And the last rule is that frogs can only jump onto other frogs.  They cannot land on empty spaces.

You can start the puzzle off with as many frogs as you want.  You can try something like four or five frogs in a row, or you can make the problem as big as you want!  In fact, you can ask students (or yourself) if you could place one hundred frogs in a row and have them jump into a single stack.

Students working on the puzzle at a JRMF hosted by The American Institute of Mathematics.

This is a fantastic problem to get students thinking in general terms. Have them start small and build up.  Ask them to find methods to make harder problems easy.  A big picture question is if they can find a general method that works for any number of frogs.

This is actually just one question you can ask about these frogs.  For a few other ideas, you can watch founder Gord Hamilton talking about the problem on Numberphile:

Stack Up

After playing with Jumping Frogs for a bit, you can take the puzzle into a new direction… literally! Jumping Frogs has all the frogs laid out in a row, but what if we used two dimensions and laid them out in rectangles or even weirder patterns?  We would need to define our movement in 2D.

  • A stack of n frogs must jump n spaces. No more. No less.
  • Stacks of frogs can jump horizontally or vertically by the same rules as before.
  • Stacks of frogs can jump diagonally. To count diagonal jump distance, treat the diagonal as the hypotenuse of a right triangle. Add the triangle’s horizontal and vertical sides to get distance.
  • And just like before, frogs have to take the shortest path. No backtracking is allowed.

You can then take the earlier question into the second dimension as well. Instead of asking if you can solve any row of frogs into a single stack, now see if you can solve for any array (rows and columns). Additionally, you can create a myriad of challenging puzzles using these rules. Here is an online applet that you can use to play that activity.

Or you can continue the activity at home using cups, poker chips, or any other stackable object. Here is a set of puzzles in two dimensions created by members of the Morgan Hill Math Teachers’ Circle.

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