You Can Make Any Number Out of Four 4s Because Math Is Amazing

Here’s a fun math puzzle to brighten your day. Say you’ve got four 4s—4 4 4 4—and you’re allowed to place any normal math symbols around them. How many different numbers can you make?

According to the fantastic YouTube channel Numberphile, you can make all of them. Really. You just have to have some fun and get creative.

When you first start out, the problem seems pretty simple. So, for example, 4 – 4 + 4 – 4 = 0. To make 1, you can do 4 / 4 + 4 – 4. In fact, you can make all the numbers up to about 20 using only the basic arithmetic operations of addition, subtraction, multiplication, and division.

But soon that’s not enough. To start reaching bigger numbers, the video explains, you must pull in more sophisticated operations like square roots, exponents, factorials (4!, or 4 x 3 x 2 x 1), and concatenation (basically, turning 4 and 4 into 44).

This way will get you up to about 100, but going really big requires one more tool: the log. Here, things start to get more complicated, but the back half of the video explains how incorporating the log into the four 4s problem allows you to work out ways to make every number up to infinity.

The solution, involving multiple logs and square roots, broke open this puzzle in the 1930s. But it’s still a fun exercise if you ever find yourself in possession of a lot of free time and a big piece of paper.