The Lumber Room

[Needs cleanup… just dumping here for now.]

Mark Jason Dominus tweeted and later blogged about this puzzle:

From the four numbers [6, 6, 5, 2], using only the binary operations [+, -, *, /], form the number 17.

When he tweeted the first time, I thought about it a little bit (while walking from my desk to the restroom or something like that), but forgot about it pretty soon and didn’t give it much further thought. When he posted again, I gave it another serious try, failed, and so gave up and wrote a computer program.

This is what I thought this time.

Idea

Any expression is formed as a binary tree. For example, 28 = 6 + (2 * (5 + 6)) is formed as this binary tree (TODO make a proper diagram with DOT or something):

                   +
               6       *
                     2    +
                        5   6

And 8 = (2 + 6) / (6 – 5) is this binary tree:

                  /
              +       -
            2   6   6   5

Alternatively, any expression is built up from the 4 given numbers [a, b, c, d] as follows:
Take any two of the numbers and perform any operation on them, and replace the two numbers with the result. Then repeat, until you have only one number, which is the final result.

Thus the above two expressions 28 = 6 + (2 * (5 + 6)) and 8 = (2 + 6) / (6 – 5) can be formed, respectively, as:

1. Start with [6, 6, 5, 2]. Replace (5, 6) with 5+6=11 to get [6, 11, 2]. Replace (11, 2) with 11*2=22 to get [6, 22]. Replace (6, 22) with 6+22=28, and that’s your result.
2. Start with [6, 6, 5, 2]. Replace (2, 6) with 2+6=8 to get [8, 6, 5]. Replace (6, 5) with 6-5=1 to get [8, 1]. Replace (8, 1) with 8/1=8 and that’s your result.

So my idea was to generate all possible such expressions out of [6, 6, 5, 2], and see if 17 was one of them. (I suspected it may be possible by doing divisions and going via non-integers, but couldn’t see how.)

(In hindsight it seems odd that my first attempt was to answer whether 17 could be generated, rather than how: I guess at this point, despite the author’s assurance that there are no underhanded tricks involved, I still wanted to test whether 17 could be generated in this usual way, if only to ensure that my understanding of the puzzle was correct.)

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