### Origin

In the axiomatization of arithmetic, a mathematician is *not* going to say that the integers are 1, 2, 3, and so on. Instead, the mathematician will say:
- 1 is and integer
- adding 1 to an integer creates an integer

That creates all the integers starting from 1 by recursion. That might seem needlessly abstract to you, but that's how mathematicians think. (Mathematicians like recursion and don't like long lists.)
These problems are based on that idea. But the problems go beyond that and stand on their own. They teach recursion and number facility, and of course thinking skills, and maybe still lean a little towards the concepts of axiomatization.

### The Frodo Problem

The basic problem is this:

0 is a frodo.

adding 1 to a frodo makes another frodo.

How many frodos can be made?
I think this is the best way of asking the question. Of course, the question really is, *what* frodos can be made, not how many.

Perhaps ironically, I now like to start with problems that are just as simple but have a little more structure to the answer.

4 is a frodo

adding 3 to a frodo makes a new frodo

How many frodos can be made?

2 is a glodo

adding 2 to a glodo makes a new glodo

How many glodos can you make?

1 is a brodo

adding 2 to a brodo makes a brodo.

Don't ask me why, but on their first try at a problem, everyone so far has just produced one frodo. So I ask them to make another. If they make just one more, I ask them to make another. If they make just one more, I ask them to make all that they can.

### Two Constructors

There are several different small variations to these problems. One is to have two different numbers you can add. For these problems, a great follow-up question is "What is the largest integer you *can't* make with these particular rules. I suspect this is a good "Day 2" problem, which is to say, just do the above on the first day. Or try these, but not the subtraction problems too.

0 is a lodo

Adding 2 to a lodo makes a lodo.

Adding 3 to a lodo makes a lodo.

0 is a jonathodod

Adding 3 to a jonathodo makes a jonathodo.

Adding 4 to a jonathodo makes a jonathodo.

0 is a modo

Adding 6 to a modo makes a modo.

Adding 7 to a modo makes a modo.

### Negatives

Being able to subtract changes things.

10 is a crodo

adding 1 to a crodo makes another crodo.

subtracting 1 from a crodo makes another crodo.

7 is a maxilodo

Adding 5 to a maxilodo makes a maxilodo.

Subtracting 2 to a maxilodo makes a maxilodo.

Make as many different maxilodos as you can.

1 is a zodo

Adding 5 to a zodo makes a zodo

Subtracting 4 from a zodo makes a zodo

0 is a wodo

Adding 183 to a wodo makes a wodo

Subtracting 184 from a wodo makes a wodo
### A Slightly Different Construction Rule

1 is a dodo

adding two dodos makes another dodo

(You can add the same dodo to itself to make a new dodo.)

0 is a stodo.

adding two stodos makes another stodo
### Fractions

0 is a plodo.

adding 1/2 to a plodo makes another plodo.

0 is a spodo.

adding 1/2 to a spodo makes another spodo.

adding 1/3 to a spodo makes another spodo
### Multiple Choice

These problems are more closely tied to the axiomatization of numbers. The first problem is a construction that doesn't work. After doing it, hopefully students will appreciate the construction that does work (which comes second).

0 is a ingimodo.

adding 2 to a ingimodo makes another ingimodo.

adding 3 to a ingimodo makes another ingimodo.

Imagine all the ingimodos and answer these questions.

If you add two ingimodos, do you get a ingimodo?

a. never

b. sometimes (specify)

c. always

If you subtract 1 from a ingimodo, do you get a ingimodo?

a. never

b. sometimes (specify)

c. always

If you multiply two ingimodos, do you get a ingimodo?

a. never

b. sometimes (specify)

c. always

If you divide one ingimodo by another, do you get a ingimodo?

a. never

b. sometimes (specify)

c. always
This is the conventional axiomatization:

0 is a frodo.

adding 1 to a frodo makes another frodo.

Imagine all the frodos and answer these questions.

If you add two frodos, do you get a frodo?

a. never

b. sometimes (specify)

c. always

If you subtract 1 from a frodo, do you get a frodo?

a. never

b. sometimes (specify)

c. always

If you multiply two frodos, do you get a frodo?

a. never

b. sometimes (specify)

c. always

If you divide one frodo by another, do you get a frodo?

a. never

b. sometimes (specify)

c. always

### Going Backwards

Allow your students to make up rules for constructing frodos. Then you, or another student, constructs the set of all frodos. The potential problem is if they get into division, you don't want to be supplying the answers.

Choose some rules that make as few frodos as possible.

Choose some rules that make as many frodos as possible.
### The Ultimate Frodo Problem

1 is a frodo

adding 1 to a frodo makes another frodo

Dividing one frodo by second frodo makes another frodo

How many frodos can you make?
Technically/philosophically, this is even more difficult than it looks. The first rule creates the counting numbers, but not negative numbers or zero. I left out zero because it doesn't work for the second rule. The second rule creates the positive rational numbers. It does not create *all* of the positive numbers, because it does not create the real numbers. I have no idea how students could express their answer. It is more like if they can think the answer, then you can give them the word for the answer.

### Teachers' Guide

I think some students will have no difficulty with the frodo problems as they are stated. Many will need a start, and some will have difficulty. There are some subtle unstated rules, so you probably need to explain things to the students who can't get started.
One explanation is this. In each problem, you are first given "starters". These are numbers that you are told are frodos. Then you are given "production rules". You use these rules to make new frodos. At first, you use the production rules on the starters. When you have made new frodos, you can then use the production rules on them.

The easiest problems seem to be ones in which there is one starter and one adding number. There are, of course, endless variations of these to give to a beginner.

10 is a frodo.

subtracting 1 to a frodo makes another frodo.

1 is a brodo

adding 2 to a brodo makes a brodo.

Some of the problems allow you to add two frodos together, instead of adding a constant to a frodo. In some sense, this is a trivial change, and two of my students hardly noticed it. The third was completely befuddled by it.

### Odd-Even Variations

This is just an extension of the frodo problems, plus they could help develop the conceptual understanding of odd and even numbers. For example, there is a simple proof that the sum of two odd numbers is an even number. To understand this proof, you need to know that the addition of two integers always yields an integer. The frodo problems are designed to teach that concept.
1 and 2 are podos

adding an odd podo to an even podo makes a new podo

2 and 3 are jodos

adding an even jodo to an even jodo makes a new jodo

1 and 3 are nodos

adding an odd nodo to an odd nodo (or, as always, adding an odd nodo to itself) makes a new nodo

### More Problems