Consider the following game, which costs $1 to play:

You have 6 boxes, arranged left to right. Initially, each box has 1 penny in it. There are only two things you’re allowed to do:

  1. You can take away 1 penny from any box, and put 2 more in the box directly to the right. Let’s call this operation 1.

  2. You can take away 1 penny from any box, and swap the two boxes directly to the right. Let’s call this operation 2.

You can perform operations 1 and 2 however many times you want, in any order. When you’re finished, you get to collect all the pennies that are left (i.e. all the pennies in all the boxes).

Here are a couple questions to get you started:

  1. Would you pay $1 to play this game if you were not allowed to use operation 2?
  2. Now assuming you can use operation 2, would you pay $1 to play? If so, what’s the most you can win?

I have a bit more to offer than just a puzzle: a playable version of the game!

A few quick notes about the game above:

  • Clicking the blue square below a box is equivalent to performing operation 1 once on that box.

  • Clicking the green square below a box is equivalent to performing operation 2 once on that box.

  • What about the input box with a ‘1’ in it (below each blue box)? Let’s say you have managed to construct a large number, say 30, in one of the boxes, and you want to use operation 1 to double it and move it to the box to the right. You could click the blue box 30 times… but that’s a bit tedious. For convenience, you can input a number, N, into the input box and then click the blue square, and you will perform operation 1 on that box N times. In addition, if you input the number ‘0’, the game will assume you wish to perform operation 1 on all the pennies in that box.

  • Lastly, ‘Your Path’… what is it? It’s a history of all the moves you’ve made (or actions you’ve taken, or operations you’ve performed, however you like to think about it). That way, if you’re competing with a friend to come up with the highest number, you can prove that you’ve achieved your number by sending your path! And they can copy and paste your path into their text box and, assuming it’s not malformed, it will reconstruct your moves in their game so that they can inspect it (by undoing, presumably).