Stepping Stones

Today we'll look at some puzzles based on Neil Sloane's presentation of the stepping stones puzzle, given in this Numberphile video.


Setup

This activity can be done as a paper and pencil activity with graph paper. Here is a nice whiteboard option. It would be better if there were a natural number manipulative to use. The main problem is that each station will need several \(1\)'s but just one of each of the larger numbers, and most sets of tokens include the same number of \(1\)'s as other values. Here are some options in the form of foam dice:

For the first set, students would want \(6\) blue dice (labels \(7-12\)) and \(8+\) yellow dice (labels \(1-6\), with some redundant \(1\)'s). This would allow them them to do puzzles starting with at least three \(1\)'s and place \(2-12\).

For the second set, students would want something like \(6+\) red dice (\(1,1,1,2,3,4\)), \(5\) blue dice (\(5,6,7,8,9\)), \(5\) yellow dice (\(10,11,12,13,14\)), and \(6\) green dice (\(15,16,17,18,19,20\)). (There are some strange but useful redundancies on the second set, so other groupings could work.) This would allow students to do puzzles starting with at least three \(1\)'s and place up to \(20\). Each of these dice is \(< 0.8"\) in side length, so a printout of a \(10 \times 10\) grid with squares \(\geq 0.8"\) should work nicely.

The premise of the puzzles is that a grid starts with some number of squares labeled \(1\). The solver needs to place a \(2\) so that the numbers in its eight neighboring squares all sum to \(2\). (In this case, exactly two neighboring squares would need to contain a \(1\).( Then the solver must place a \(3\) so the neighboring squares sum to \(3\), then a \(4\) so the neighbors sum to \(4\), and so on. It doesn't matter if the neighbors of a number cease to sum to that number at some point down the line -- as long as the neighbor sum was correct when the number was placed, it is fine. The goal is to place as large a number as possible.


Example

Below, we are able to place \(2\), then \(3\), and then \(4\). However, we have no place for \(5\).


Challenges

The puzzles are below are presented in terms of trying to place the maximum number of values \(2-n\) in order, where \(n\) is the known maximal value for each puzzle. You could also simply cast them as challenges about placing as many numbers as possible, in order, starting from \(2\). In each of the puzzles, there may be multiple maximal solutions, but any maximal solution will fit within the less-shaded portion of the provided \(10 \times 10\) grid given the prescribed starting positions for the \(1\)'s. Since a standard checkerboard is \(8 \times 8\) and all of these regions happen to fit in that space, you can use the chunk inside the purple outline to set the puzzle up on an \(8 \times 8\) board. The first ten or so puzzles are likely a challenging enough arc on their own.

  1. Place \(2-5\):

  2. Place \(2-10\):


  3. Place \(2-10\):


  4. Place \(2-12\):

  5. Place \(2-13\):

  6. Place \(2-14\):

  7. Place \(2-15\):

  8. Place \(2-15\):

  9. Place \(2-15\):

  10. Place \(2-15\):

  11. Place \(2-16\):

  12. Place \(2-17\):

  13. Place \(2-17\):

  14. Place \(2-17\):

  15. Place \(2-17\):

  16. Place \(2-17\):

  17. Place \(2-18\):

  18. Place \(2-18\):

  19. Place \(2-19\):

  20. Place \(2-19\):

If you want to try these digitally, here are a couple options:

Notes

  • If a puzzle starts with two \(1\)'s, the largest number we can hope to place is \(16\), as suggested by \(\#11\) above.

  • If a puzzle starts with three \(1\)'s, the largest number we can hope to place is \(28\). This requires a \(7 \times 11\) board to allow for all maximal solutions. The board below and prescribed \(1\)'s can be used to investigate this:

    • (This could work nicely with the \(10 \times 15\) whiteboard grid linked above.)


Extra Challenges

Students might enjoy trying to find arrangements of \(1\)'s that allow for them to reach ever-increasing values.

  • If they have an infinite board and as many \(1\)'s as they like, can they figure out a way to get to \(n\) for any value of \(n\)?

  • If they restrict themselves to a \(10 \times 10\) board and are allowed as many \(1\)'s as they like to start, what is the largest number they can get to? (I don't know the answer to this question!)


This post was sponsored by the Julia Robinson Mathematics Festival

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