Numbers Are Crazy

Probably enough time has passed since my silly math problem. If you have spent some time working on it, you will see that you can basically create every number around 21 easily, but getting to 21 is a bit harder.

To start, we might as well try to make 21 by constructing 7x3, thats basically all 21 is, getting there by any sort of addition is going to be pretty tricky. We already have a 7, so how can we make a 3 from {1,5,6}? Well, basically we can't. We do have a 6 though, so if one considers 21 to be 7x6/2, we just need to use the 5 and the 1 to make a 2 (also clearly impossible). The real trick though is to use grade school division trick that a/(b/c)=ac/b, so 21 is also 6/(2/7). We now need to use {1,5,7} to make 2/7, and that one is pretty easy to do. Of course, the solution is:

6 ÷ (1 - (5 ÷ 7))

The neat blind spot is that people tend to dismiss division right out when they are given this puzzle. There is no division that you can do with these numbers that will not exit the integers, and people tend to assume that once you leave the integers you won't get back.

The double division is also the solution for the "{3,3,8,8} makes 24" puzzle, so I won't bother saying the solution specifically, its pretty simple now.