Here’s a question that’s been in the back of my head for a while:
Starting from 1 and with just the arithmetic operations + – × ÷ available to you , how quickly can you build up to a given target number?
You can do this on pen and paper, or even just in your head, but I thought it’d make a good online game. That’s what this post is about.
You can play it now: Number Builder!
While playing, I came up with loads of questions and conjectures. The big one is: is there a good strategy for finding the shortest route to the target?
One thing I enjoy is how well quick approximations work: you can often proceed much more quickly by finding a number of the right order of magnitude away from the target, rather than by calculating exactly the number that you need to use.
Of course, I’m not the first person to think about this: I found a relevant sequence in the OEIS (does that need a spoiler warning??) and a paper, “the cost of computing integers”, talking about the fewest number of moves to solve this game (but without division).
There’s an option to have a timer; I’m not the kind of person who enjoys timed challenges, but I know there will be people who will relish the challenge of beating their best time. When you’re trying to minimise time taken instead of number of steps, your strategy necessarily moves away from exact arithmetic to quickly finding anything that gets you a bit closer to the target.
Apart from the powers of 2, will it turn out that there are other numbers available in the first few moves that are particularly useful? For example, I usually end up making myself a 10, when I just want to knock out the next digit directly instead of doing a hard subtraction in my head. And I’m still not sure how useful it is to have a 3.
Like the “is this prime?” game, I’ve set it up to record each game, so I’ll be able to do answer some statistical questions once enough games have been played.