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A 14-1 \leftarrow 4 machine operates by converting any four dots in one box into an antidot one place to the left (and converts four antidots in one box to an actual dot one place to the left).
I9S9D - Image142

  1. This machine is a base machine:
    I9S9D - Image143
    Explain why xx equals 4-4.

  2. What is the representation of the number one hundred in this machine? What is the representation of the number negative one hundred in this machine?

  3. Verify that 23122|-3|-1|-2 is a representation of some number in this machine. Which number?
    Write down another representation for this same number.

  4. Write the fraction 13\dfrac{1}{3} as a “decimal” in base 4-4 by performing long division in a 14-1 \leftarrow 4 machine. Is your answer the only way to represent 13\dfrac{1}{3} in this base?


Consider a strange machine (invented by Dr. Dan V.) following the rule 1100021|1|0 \leftarrow 0|0|2. Here any pair of dots in a box are replaced by two consecutive dots just to their left.

Put in one dot and you get the code 11.
Put in two dots and you get the code 110110.
Three dots give the code 111111.
Four dots gives 112=220=1300=12100=120100=1200100=1200100=...112=220=1300=12100=120100=1200100=1200100=... . We have an infinite string.

  1. Show that this machine is a base negative two machine.
  2. Show that one dot next to two dots anywhere in the machine have combined value zero.

Thus, in this machine, we can delete any 121|2s we see in a code. Consequently, there is a well-defined code for four dots, namely, 100100.

  1. What are the 1100021|1|0 \leftarrow 0|0|2 machine codes for the numbers five through twenty?

Extra: Play with a 1100031|1|0 \leftarrow 0|0|3 machine. (What base is it?)


Consider another strange machine 1000111|0|0 \leftrightarrow 0|1|1. Here two dots in consecutive boxes can be replaced with a single dot one place to the left of the pair and, conversely, any single dot can be replaced with a pair of consecutive dots to its right.
I9S9D - Image141

Since this machine can move both to the left and to the right, let’s give it its full range of “decimals” as well.
I9S9D - Image152

  1. Show that, in this machine, the number 11 can be represented as 0.10101010101...0.10101010101... (It can also be represented just as 11!)
  2. Show that the number 22 can be represented as 10.0110.01.
  3. Show that the number 33 can be represented as 100.01100.01.
  4. Explain why each number can be represented in terms of 00s and 11s with no two consecutive 11s.
    (TOUGH: Are such representations unique?)

Let’s now address the question: What base is this machine?

  1. Show that in this machine we need xn+2=xn+1+xnx{n+2}=x{n+1}+x^{n} for all nn.
  2. Dividing throughout by xnx^{n} this tells us that xx must be a number satisfying x2=x+1x^2=x+1. There are two numbers that work. What is the positive number that works?
  3. Represent the numbers 44 through 2020 in this machine with no consecutive 11s. Any patterns?


The Fibonacci numbers are given by: 1,1,2,3,5,8,13,21,34,...1, 1, 2, 3, 5, 8, 13, 21, 34, ....

They have the property that each number is the sum of the previous two terms.

In 1939, Edouard Zeckendorf proved (and then published in 1972) that every positive integer can be written as a sum of Fibonacci numbers with no two consecutive Fibonacci numbers appearing in the sum.

For example


(Note that 1717 also equals 8+5+3+18+5+3+1 but this involves consecutive Fibonacci numbers.)

Moreover, Zeckendorf proved that the representations are unique.

Each positive integer can be written as a sum of non-consecutive Fibonacci numbers in precisely one way.

This result has the “feel” of a base machine at its base.

Cconstruct a base machine related to the Fibonacci numbers in some way and use it to establish Zeckendorf’s result.

Comment: Of course, one can prove Zeckendorf's result without the aid of a base machine. (To prove that a number NN has a Zeckendorf representation adopt a "greedy" approach: subtract the largest Fibonacci number smaller than NN from it, and repeat. To prove uniqueness, set two supposed different representations of the same number equal to each other and cancel matching Fibonacci numbers. Use the relation F(n+2)=F(n+1)+F(n)F\left( n+2\right) =F\left( n+1\right) +F\left(n\right) to keep canceling.)


Invent other crazy machines.

  1. Invent abcdefa|b|c \leftarrow d|e|f machines for some wild numbers a,b,c,d,e,fa, b, c, d, e, f.

  2. Invent a base half machine.

  3. Invent a base negative two-thirds machine.

  4. Invent a machine that has one rule for boxes in even positions and a different rule for boxes in odd positions.

  5. Invent a base ii machine or some other complex number machine.

How does long division work in your crazy machine?

What is the fraction 13\dfrac{1}{3} in your crazy machine?

Do numbers have unique representations in your machines or multiple representations?

Go wild and see what crazy mathematics you can discover!

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