Unique Sums

You may have heard someone say that we live in the "Information Age." Certainly one of the most significant technological developments of the late 20th century is our ability to transmit and store information. The speed is astonishing. Even so, we are constantly looking for ways to package information more efficiently, and thus be able to communicate even faster.

Let's look at a very simple example: ordering a pizza over the phone. Options include:
broccoli
mushrooms
pesto
fresh tomato
artichokes
onions
eggplant
tortellini
spicy chicken
Ordering a pizza could be quite a mouthful. You might want it with pesto, mushrooms, fresh tomato, artichokes, spicy chicken, and broccoli. Then your friend wants some other combination. It's a lot to convey and to record, which also means more chance of a mistake in communication. It would be handy if there were a way to identify your particular combination by a single letter or number. Here is one possible system. Each option is assigned a numerical value:
broccoli1
mushrooms3
pesto7
fresh tomato12
artichokes25
onion49
eggplant99
tortellini200
spicy chicken404


Then to place your order, just add up the values of everything you want and then enter that sum on your phone's keypad. If you tell me you want a 64-pizza, I know for sure that it has mushrooms, fresh tomato, and onion.

1)        What does a 45-pizza have on it?

2)        What does a 311-pizza have on it? Can you be sure that everyone in the class will get the same answer as you, or is there more than one correct combination?

3)        It is impossible to have a 46-pizza. Find another example of a pizza that cannot exist.

4)        Suppose the pizza place decided to add an option for red peppers. What number should be assigned to that option? Can you use any number you want, or are you limited in some way?

5)        The numbers I chose for my system do seem a little awkward. Why not just use the numbers one through nine for the nine different options, and then do the adding with those?

6)        Even though it wouldn't work to use the numbers one through nine, I bet you could come up with a system that still works but uses lower numbers than I did. See if you can figure out the lowest possible numbers that can be used to make a unique sum system.

7)        Why not just use the letters of the alphabet or a name or something, and identify each combination individually? You could have combination A, combination B, combination C, etc., until each possible combination had a unique name. Wouldn't that work?

8)        Modify the system you designed in #6 to allow for double portions of any option. Perhaps I want a pizza with double pesto, double onions, and regular artichokes and fresh tomato. Use the lowest possible set of numbers that will accomodate such possibilities.

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