The Function Machine Game

Math concepts: odd numbers, even numbers, greater-than/less-than, rounding off, addition, subtraction, multiplication, division, fractions, negative numbers, prime numbers, square numbers, problem solving, mental math
Number of players: two or more
Equipment: pencil (or pen) and paper for every player

Set Up

Each player needs pencil and paper of his own. One person plays the role of the function machine, and he must make up a rule for calculating with whatever numbers the other players will give him. In math, a function is any rule that turns one number into another*, such as:

  • Double the number, and then add seven.

The function machine should write his function rule on a piece of paper and slip it in his pocket.

All the other players make themselves a chart to keep track of the game.

*[In reality, the definition of a function is a bit more complicated than that, but it suffices for this game.]

How to Play

Play proceeds to the left around the table.

  • Each player in turn says any number he wishes, and all players write that number in their Input columns. The function machine calculates with that number, mentally or on scratch paper, and says only the answer to his calculation.
  • If that input number is too difficult for the player to work with, he should say, “That’s too hard,” and ask for a different number. Occasionally, players will give an input number that does not make sense with the chosen rule, in which case the machine answers, “That does not compute.”
  • Whatever answer the machine gives, all other players write it in their Output columns.
  • After receiving an answer, the player whose turn it is may make a guess at what calculation the machine is doing. Other players may not guess at the function rule until their own turns.


The first player to guess the rule of the function machine wins that round. If you have time to play a full game, where every player has a chance to be the machine, then the winner of the game is the player who gave the most output numbers before the others guessed his function rule.


My math club students have always enjoyed this game, especially when it is their turn to be the function machine with the secret rule. But some freeze mentally at the task of creating their function rule, while other students make up rules that are so complex they are nearly impossible to guess. You may want to make a list of suggested function rules in advance, writing the rules on index cards and letting each player draw a rule from your deck.

Here are some function ideas to get you started:

  • Add 15 to the number.
  • Subtract 4 from the number.
  • Pick your favorite number. Whatever the input, the output is always your number.
  • Multiply the number by 8.
  • Cut the number in half.
  • Cut the number in half, and then add one.
  • Double the number, and then subtract three.
  • Add the next larger number, so an input of 3 gives an output of 3 + 4 = 7.
  • Multiply by the next larger number, so an input of 5 gives an output of
    5 × 6 = 30.
  • Triple the next larger number.
  • Subtract the number from 25.
  • If the input number is even, say that number. If the input is odd, or if it’s not a whole number, then say the next larger even number.
  • Square the number, and then add one.
  • Say the largest multiple of 3 that is less than the number.
  • Round off to the nearest hundred.
  • Double the odd numbers, but cut the even numbers in half.
  • All odd numbers are 17, but all even numbers are 12.
  • Say the ones digit of the number.
  • Add the digits in the number together.
  • If the number is prime, say the number. If the number is not prime, say 1. Remember that one, zero, and all negative numbers are not prime.
  • Say the tenths digit of the number (the first digit after the decimal place). The student may need a calculator to rapidly convert fractions to decimal numbers.


This is a great game to play in the car on family trips, since there are no little game pieces and the paper does not need to be passes around. Even the driver can join in the fun.

Beware that some function rules can be described in more than one way. For example, the rule:

  • Double the number, and then add two.

…could also be written as:

  • Double the next higher number.

It helps to have a referee who knows algebra to judge the guesses. “Double the number, and then add two” is written as 2x + 2 , while “double the next higher number” is 2 \left(x + 1 \right) , and anyone who knows algebra can easily see these are the same. The player who tries to guess the rule does not have to put it in the exact words the function machine used, as long as the statements are equivalent.

Edited to Add

You can play an interactive online version of the game here:

19 thoughts on “The Function Machine Game

  1. I love it. I also play math games in the car. I refer to it as the “captive audience”.
    I do one that I call “What Two Numbers?”, such as, “What two numbers multiply to 12 but add to 7?
    This is a great game to form the foundation concept of factoring quadratics.

  2. My buddy, Neil, uses a variation of this with students. Once they think they know the rule, they put up their hand. When called on they must use the rule on the number Neil provides – like 437 or -21 or 2i when the examples were all number like 1, 2 or 10. It keeps the game going a bit longer!

  3. I just stumbled across your website, and I have to say making math fun is an interesting idea. As a student I never had any trouble learning math so I never found opportunity to try anything like this. Algebra came to me almost naturally but I must admit I have never found doing it to be fun.

    I am curious as to what age group your students/children are in,? I can see where this might have been interesting had I been involved with some friends in junior high, but certainly by high school age I would have found anyone trying to make math fun to be a drag.

  4. My students (at this time) range from elementary school to high school, but mostly I work with students from mid-elementary through middle school/junior high.

    Doing math with friends definitely helps. In my original math club, many of the students were “volunteered” by their parents, but because they were with a group of friends, they ended up enjoying it anyway.

  5. One of my favorite books for teaching functions is “What’s My Rule? Using Problem Soving Strategies” (Grades 6-12) by David Logothetti and Teddy Logothetti from Activity Resources. It shows numbers going into silly heads and numbers coming out the other side.

    You have to figure out the process that went on inside the head. The functions start out pretty easy such as f=x+1 but by the end of the book they get harder to ones such as f=x (squared) + x -2. I used this with children learning Algebra 1 and 2 who loved to see new ways to play with numbers.

  6. I learned of this idea of a class that I took as a part of a grant program and I have used it ever since. I created my function box out of a box that the reams of paper come in and I made it to look like a person that we call Fun Freddy Function box.

    There are so many things that you can do with a function box and at so many levels. You can have the students try to write the rule based on the inputs and outputs and then ask them what the output would be for a larger input number. You could also have the students graph the ordered pairs and discuss the type of graph that is created. It can be basic or you can get into squares, cubics, etc. We also look at patterns as we are trying to decide what the rule is.

    My students love these activities and always do so much more math than if they just had a worksheet.

  7. Yes, my students always love this game, too. Years ago, when I first started my math clubs, I remember being surprised at how much they liked it. I expected them to prefer the more hands-on games, or more concrete activities, but this abstract puzzle game was the definite favorite. Maybe it’s because they like the feeling of knowing the secret answer — a feeling they don’t get often in math class.

    When it came time to pick activities for the math carnival, half of the group wanted to do this. I let two of them collaborate in making a cardboard box machine, but the others had to pick a second-choice activity.

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