by Bill Graham

People often state that students aren’t as competent at math as they used to be. SAT scores are down. I’m sure there’s truth to those statements, but today’s students are also learning many concepts in math and science at an earlier age than most of us did.

One example is the concept of a function. I never heard of this until my freshman year in college. Now I’m teaching it to my 7th- and 8th-grade students. If you already understand functions, go right to the puzzle. If you don’t, I’ll try to explain them in a simple way.

Think of a function as a math machine with an input and an output. Suppose the function is A(x) = 3x + 1. That means if you put any number into this function machine, the machine will multiply the number by three and add one. (Don’t worry about the A(x). That’s the name of the function and I’m using it to keep all the serious mathematicians happy.) If you put five into the machine, what will come out? Five times three plus one equals 16 for the output.

Suppose you join two function machines so that the output of the first one is connected to the input of the second one. Let’s make the second function B(x) = x². If you put two into the A machine, out comes seven. Then seven is the input to the B machine which squares it, and out comes 49. OK?

Here are five functions.

I am going to connect them in alphabetical order and use two as the input number. The output of the A function would be five. Then the B function squares five (25) and multiplies it by three (75), which goes into the C function. Its output is 526 which goes into D. That function’s output is 276,676. Function E multiplies that number by two and adds seven resulting in the final output of 553,359. Can you use two as the input number, rearrange the functions using each function only once and get a larger output number than I did? What is the largest possible output number?