At her blog, Sue VanHattum mentions this puzzle: Use five 5's to make 37.
That got me thinking about making things out of fives (or some other number), so I wrote a Python script (I'm trying to learn how to use Python to accomplish things, and this seemed like a fun exercise) to help me think about that. I'm never used object oriented programming before, and this seemed like a good opportunity to define class (which I called BuildNum) that would let you build a number out of other numbers (via the operations: add, subtract, multiply, divide -- I guess I left out exponentiation for now, which is needed for Sue's original puzzle). The class keeps track of:
- the current value of the number
- a string representing how the number was formed
- the "generation" of the number, defined as the largest of the generations of the component numbers plus 1
- (e.g., "5" is generation 1, "5 + 5" is generation 2, "(5+5)/5" is generation 3)
- a list of the "seeds" (generation 1 numbers) used to make the current number
Here's some sample execution using this class:
Then I decided (I don't really know why!) to look for ways of combining fives to get a string of fives. So I wrote a function that takes a list of BuildNum objects and builds the next generation.
The example in the first line of this post is in generation 5 (where generation 1 consists of just "5"). There aren't any 555's in any generation below 5.
Some more notes:
- If generation 1 has just "5", then generation 4 (for example) has all ways of combining four 5's, but it also has a lot more than that. Generation 2 can have up to two 5's, generation 3 can have up to four 5's, and generation 4 can have up to eight 5's.
- But generation 4 doesn't have everything with up to eight 5's:
- This (with only seven 5's) is in generation 5, but not in generation four:
[((5+5)+5)*((5+5)+5)] + 5
So, what if we we're interested in everything you can make out of five 5's? What's the best approach? I'm thinking maybe you'd construct all of the trees with five leaves to define the parentheses/groupings, and then inserting all different combinations of operations between the branches.
Someone commented on Sue's posting asking about what we can do with students from this kind of stuff. I'm not a teacher, but my first thought is on growth rates: How many ways of forming a number do you have in generation n? (It would also be interesting to try to figure out -- without brute force, which is gets infeasible at some point -- how many unique numbers there are. But that sounds hard.)
Here's the code. I don't know if I'll spend much more time improving this particular code, or writing more related to this problem. Please understand that I'm just beginning to code in Python, so it's probably terrible! But I'll appreciate any advice you have for a beginner.