---
# System prepended metadata

title: Northern Tunnels Pioneer Slowly

---

# Northern Tunnels Pioneer Slowly

## 2024-09-12 Coin Flips to Outnumber

Flip a coin. Tally heads and tails. If heads exceeds the tail by more than n, the game ends. Average coin flips in a game?

$f(-1) = 0$
$f(x) = 1 + f(x-1)/2 + f(x+1)/2$
$f(0) = 1 + 0 + f(1)/2 = 1+f(1)/2$

$f(1) = 1 + f(0)/2 + f(2)/2 = 1 + (1/2 + f(1)/4) + f(2)/2$
$f(1) = 3/2 + f(1)/4 + f(2)/2 \to 3/4\cdot f(1) = 3/2 + f(2)/2$
$f(1) = 2 + 2/3\cdot f(2)$

## Relay Logic

### NOT Gate

```
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```

### Half Adder

```
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```

## Rope Burning

Two ropes, each which burn for 2 hours when lit at one end.
Different parts of the ropes burn at different rates

1 hr: light both ends
1.5 hr: light both ends of one rope and one of the other, when the first rope is burnt, light the other end of the other

1hr: light the middle of a rope, when it reaching one end, light the other end
30min:
- light two random spots on the rope
- if the middle of the rope burns up, light both ends. 
- if one end burns, light the other end
- if the middle burns after lighting the other end, light a new spot

### Circles

connect the end of the ropes together to form a circle.
1 hr: light one spot
30 min: light two spots, each time the burn fronts meet, light a new spot
20 min: light three spots, each time the fronts meet, light a new spot
15 min: light four spots...
12 min: light five spots...
10 min: light six spots...

two ropes:
40 min = 30 min + 10 min = 20 min + 20
1/n hr + 1/m hr, where m and n are positive integer

all numbers of the form (n+m)/nm
reminiscent of - $(x+n)(x+m) = x^2 + (n+m)x + nm$

## The Tax Collector Problem

you start with the numbers 1 to n, and pick numbers from that list such that:

- each pick is not a factor of a number picked previously
- each pick has at least one factor different from the factors of the numbers picked previously

the goal is to have the highest sum of numbers picked.

| n | sequence | sum/total
| - | -- | --
| 2 | 2 | 2/3
| 3 | 3 | 3/6
| 4 | 3 4 | 7/10
| 5 | 5 4 | 9/15
| 6 | 5 4 6 | 16/21
| 7 | 7 4 6 | 18/28
| 8 | 7 (8 6) | 21/36
| 9 | 7 (8 9) | 24/45
| 10 | 7 (8 9 10) | 34/55
| 11 | 11 (8 9 10) | 38/66
| 12 | 11 (8 9 {10) 12} | 50/78
| 13 | 13 (8 9 {10) 12} | 52/91
| 14 | 13 (8 9 {10 14) 12} | 66/105
| 15 | 13 9 15 10 {8 (14} 12) | 81/120
| 16 | 13 9 15 10 {16 (14} 12) | 89/136
| 17 | 17 9 15 10 {16 (14} 12) | 93/153
| 18 | 17 4 10 15 16 14 12 18 | 106/171
| 19 | 19 4 10 15 16 14 12 18 | 108/190
| 20 | 19 4 20 15 16 14 12 18 | 118/210
| 21 | 19 4 14 21 15 20 16 12 18 | 139/231
| 22 | 19 4 14 21 15 20 16 12 18 22 | 161/253
| 23 | 23 4 14 21 15 20 16 12 18 22 | 165/276
| 24 | 23 4 14 21 15 20 16 24 18 22 | 177/300
| 25 | 23 4 14 21 25 20 16 24 18 22 | 178/325

hueristics:

- pick highest prime first

## equal sum of squares

are there any sums of squares equal to each other, a²+b²=c²+d², such that a and b are odd, and c and d are even?

consider, a² mod 4 and b² mod 4, as and a and b are equivalant to 1 or 3, their squares are equivalant to 1.
so a²+b² is a multiple of 4 plus 2. but c²+d² is a multiple of 4 plus 0. so they cannot be equal.

for 4, we have the same thing mod 9.
0 -> 0, 1 -> 1, 2 -> 4, 3 -> 0, 4 -> 7, 5 -> 7, 6 -> 0, 7 -> 4, 8 -> 1


## Secret Message

tzmwup
vjet
tjotj
djosdj
vjseda

pb    td kg
 m     n  ŋ
   fv sz    h
       r
       l

pbtdkg
mn
fvcszj
wh
qx
lr
aeiouy

ABCDEFGHIJKLMNOPQRSTUVWXYZ
ETSKIVPWOFGRNMUBXLZDYCHQAJ

like this
rogi dwoz

hello my name is the pink elephant
wirru na meni oz dwi bomg iribwemd

maybe change it to bandersnatch
neati swempi od du temkilzmedsw

Two people share a substitution cipher.

each part of the message is written:
x = ABCDEFGHIJKLMNOPQRSTUVWXYZ
f(alpha) = (secret)

       x = ABCDEFGHIJKLMNOPQRSTUVWXYZ
g(alpha) = ZFNXKBWLTOSEPGIACMRDQHUYJV
g(part-of-message)
          x = ABCDEFGHIJKLMNOPQRSTUVWXYZ
g(h(alpha)) = FLDIGQEHPAJRMYCKTXBSOUVNWZ
  h(alpha)  =       L  P     E         A
h'(g'(alp)) = JSOCGAEHDKPBMXUIFLTQVWYRNZ
h(part-of-message)

g'(ZAAEK) = APPLE

apple trees going bloom
f(g(alpha))
g(apple) = ZAAEK
g(h(alpha))
h(trees)

…

## Why does water stay together?

Why not?

$2H_2O (-237.24 kJ) ⇒ 2H_2 + O_2 + -237.24 kJ$

because we need to get energy out!

$2H_2 + O_2 ⇒ 2H_2O (-237.24 kJ) + 237.24 kJ$


## Chemical Equations

$2H_2O (−237.24 kJ) + 2Na (0) ⇒ 2NaOH (-379.7 kJ) + H_2 (0) + 284.92 kJ$
2mol water + 2mol sodium ⇒ 2mol sodium hydroxide + 1mol hydrogen + 1031.5kJ

## Coin Flips to unbalance

Flip a coin. Tally heads and tails. If one exceeds the other by more than `n`, that one wins. Average coin flips in a game?

| n |average length |
| - | - |
| 1 | 1 |
| 2 | $2/2 + 4/4 + 6/8 + 8/16 + 10/32 + … =2\sum_{n=1}^∞{n\over2^n}=4$
| 3 | $3\cdot1/4 + 5\cdot3/16 + 7\cdot9/64 + 9\cdot27/256 +… =\sum_{n=1}^∞{{2n+1}{3^{n-1}\over4^n}}=9$
| 4 | $4\cdot1/8 + 6\cdot1/8 + 8\cdot7/64 + 10\cdot3/32 + 12\cdot41/512 + …$

### 4 

after 4 turns:
- 4: 2/16 1/8 HHHH|TTTT
- 2: 8/16 1/2 HHHT|TTTH
- 0: 6/16 3/8 HHTT

after 6 turns:
- 4: $1/4\cdot1/2 = 1/8$
- 2: $1/2\cdot1/2 + 1/2\cdot3/8 = 7/16$
- 0: $1/4\cdot1/2 + 1/2\cdot3/8 = 5/16$

after 8 turns:
- 4: $1/4\cdot7/16 = 7/64$
- 2: $1/2\cdot7/16 + 1/2\cdot5/16 = 3/8$
- 0: $1/4\cdot7/16 + 1/2\cdot5/16 = 17/64$

after 10 turns:
- 4: $1/4\cdot3/8 = 3/32$
- 2: $1/2\cdot3/8 + 1/2\cdot17/64 = 41/128$
- 0: $1/4\cdot3/8 + 1/2\cdot17/64 = 29/128$

after 12 turns:
- 4: $1/4\cdot41/128 = 41/512$
 
## Recursive Functions with no Base Case

$f(x) = x\cdot f(1/x)^2$
$f(2) = 2\cdot f(1/2)^2 = 2\cdot1/4\cdot f(2)^4 = 1/2 f(2)^4$
$f(1/2) = 1/2\cdot f(2)^2$
$z = 1/2 z^4 \implies 2 = z^3, z=0$

$f(x) = x\cdot f(1/x)^2 = x\cdot 1/x^2\cdot f(x)^4$
$f(1/x) = 1/x\cdot f(x)^2$
$z = 1/x \cdot z^4$

### Infinite

$f(x) = x/2 + f(x/2) = x$

### Cyclic

cycle of 1: $x$
cycle of 2: $a/x$, $a-x$, $a-{b\over c-x}$
cycle of 3: $a\over b-x$, $a-b/x$
 
## Game?

1. both players start with:

   - a: 3 ants
   - g: 3 grasshoppers
   - B: 2 beetles
   - s: 2 spiders
   - b: 1 bee
   - l: 1 ladybug
   - m: 1 mosquito
 
   in their color in their hand.
   
2. the first player plays one of their pieces anywhere
3. the second player plays one of their pieces adjacent to that pieces
4. subsequently, players take turn placing or moving one of their pieces
   - pieces must always be connected all other pieces on the board.
   - pieces may not be placed touching the other player's pieces,
     but may be moved to next to them.
   - pieces may not be moved until the bee of that color is played.
   - each type of piece moves in it's own way:
     - ants can move to an empty space anywhere along the surface that they are on
     - grasshoppers can jump over any number of pieces in a straight line to an empty space, but may not cross empty space
     - beetles can move one space along the surface or climb onto a piece they are next to. this prevents the underneath piece from moving.
     - spiders move on the surface three times in the same direction

```
   d x x x
a b c   x
 x x x x
```
3 ant 3 grasshopper 2 beetle 2 spider 1 bee 1 ladybug 1 mosquito

 
 ## $f(x)=x+f({1\over1-x})$
 
 f(2) = 2+f(-1) = 3/2+f(2)
 f(-1) = -1+f(1/2) = -1/2+f(2)
 f(1/2) = 1/2+f(2)
 
 1. $x + {1\over1-x} + {x-1\over x} = 0$
 2. $x(1-x) + 1 + {(x-1)(1-x)\over x} = 0$
 3. $x(1-x)x + 1x + (x-1)(1-x) = 0$
 4. $(-x^3 + x^2) + x + (-x^2+2x-1) = 0$
 5. $-x^3 + 3x - 1 = 0, x\ne0,1$
 
 [wolfram alpha](https://www.wolframalpha.com/input?i=-x%5E3+%2B+3x+-+1+%3D+0)

$(x^2-3)(-x) = 1$
$(x-\sqrt3)(x+\sqrt3)(-x) = 1$

$x^3 - 3x + 1 = 0$
$x^3 + 3x^2 + 3x +1 = 3x^2 + 6x$
$(x+1)^3 = 3x(x+2)$

| x  | |
| -  | -
| -2 | 1
| -1 | -3
| 0  | -1
| ⅓  | -1/27
| 2/5| 17/125
| ½  | ⅝
| 1  | 1
| 2  | -3
| 3  | -19

## Making Arrangements by Jumping 3 Point on a Grid

P, Q, and R are points on the grid (i.e. Gaussian Integers)

* type 1: everything with integer distances between at least two of the three 
  ```
  P ------- Q ------- R

  P
  |
  | 
  |
  Q ------- R
  
  P
   \
    \  (some of these)
     \
      Q ------- R
  ```

* type 2: two distances with square roots

  ```
  P ----- R
   \     /
    \   /
     \ /
      Q
  ```
  
* type 3: three distances with square roots

  ```
  P-._
   \  ''-R
    \   /
     \ /
      Q
  ```
  

### Some examples

#### Line of Distance 1 → Line of Distance $\sqrt5$

P: 0,0 Q: 0,1 R: 0,2  Q/R→1,2
P: 0,0 Q: 1,2 R: 0,2  R/P→2,0
P: 0,0 Q: 1,2 R: 2,0  R/Q→2,4
P: 0,0 Q: 1,2 R: 2,4

#### Line of Distance 1 → Line of Distance $\sqrt2$?

```
PQR
------
P RQ
------
P  Q
   R
------
P
   R

Q
------
Q
P
   R
------
------
P
   Q
      R
------
P
   Q
R
------
 P

 R
Q
------
P R
 Q
------
P
 Q
  R
```

## Making Arrangements by Jumping 3 Points On Number Line

P Q     b'=-b
P R     b'=-(2c+b)
Q P     b'=-b   c'=2b+c
Q R     b'=2c+b c'=-c
R P     c'=-(2b+c)
R Q     c'=-c

lemma: all jumps are invertable

Q P: b' = -b c' = 2b+c   b''=-b' = b  c'' = 2b' + c' = -2b + 2b + c = c

because we can't go from b=2 c=2 to b=1 c=1, we can't go the other way either

theorm: all jumps preserve greatest common factor of b and c

conjecture: we can make any arrangements that preserve gcf

```
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```

## Double Sums of Two Squares (≥1)

50: $2\cdot5^2$ |7,1|=|5,5|
65: $5\cdot 13$ |
85: $5\cdot 17$
125: $5^3$
625: $5^4$ |24,7|=|20,15|$
3125: $5^5$

## Probability Thing

Four people are participating in four raffles, each for a different equal prizes.
They start out with four tickets that can be entered in any raffle.

### All different.

```
4 0 0 0
0 4 0 0
0 0 4 0
0 0 0 4
```

each person gets 1 prize.

### All evenly.

```
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
```

each person gets 1 prize on average, but with the following probabilities:

0: ${3\over4}^4 = {81\over 256} \approx 1/3$
1: ${1/4*{3\over 4}^3 + 3/4(1/4{3\over 4}^2 + 3/4(1/4*{3\over4} + 3/4(1/4)))} = 1/4*{3\over4}^3 * 4 = {3\over4}^3 = {27\over64} \approx ½$
2: $6*3/4^2*1/4^2 = 6*9/16*1/16 = 54/256 \approx 1/5$
3: $4*3/4*1/4^3 = 4*3/4*1/64 = 12/256 \approx 1/21$
4: $1/4^4 = 1/256$

$(3/4 + 1/4)^4 → (x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

### Each 2 in 2.

```
2 2 0 0
0 0 2 2
0 2 2 0
2 0 0 2
```

0: 1/4
1: 1/2
2: 1/4

### What's the best strategy?

Give other people's tickets, and you go last:

4 5 9 2

#### 0 0 0 4

0/4 0/5 0/9 4/6 = 2/3

#### 1 0 0 3

1/5 0/5 0/9 3/5

0: $4/5*2/5 = 8/25$
1: $1/5*2/5 + 4/5*3/5 = 2/25+12/25 = 14/25$
2: $1/5*3/5 = 3/25$

average: $0*8/25 + 1*14/25 + 2*3/25 = 4/5$

#### 1 1 0 2

1/5 1/6 0/9 2/4

0: $4/5*5/6*2/4 = 20/60 = 1/3$
1: $1/5*5/6*2/4 + 4/5*1/6*2/4 + 4/5*5/6*2/4 = 1/12 + 1/15 + 1/3 = 29/60$
2: $1/6$
3: $1/5*1/6*2/4 = 1/60$

average: $0*1/3 + 1*29/60 + 2*10/60 + 3*1/60 = 51/60$

#### 1 1 1 1

1/5 1/6 1/10 1/3

0: $4/5*5/6*9/10*2/3 = 360/900 = 2/5$
1: $(1*5*9*2 + 4*1*9*2 + 4*5*1*2 + 4*5*9*1)/900 = (90 + 72 + 40 + 180)/900 = 382/900$
2: $(900-360-382-20-1)/900 = 137/900$
3: $(4 + 5 + 9 + 2)/900 = 20/900$
4: $1/5*1/6*1/10*1/3 = 1/900$

average: $(382 + 2*137 + 3*20 + 4*1)/900 = 720/900 = 8/10 = 4/5$

## Line Representations

### Two numbers

(a,b), where a is x-intercept, b is y-intercept, except if a=0, then b is slope

$x = c$ → $(c,∞)$
$y = c$ → $(∞,c)$
$ax + by = 1$ → $(1/a, 1/b)$
$ax + by = 0$ → $(0, -b/a)$

### Two possibly transfinite numbers (inverse intercepts)

$ax + by = c$ → $(a/c,b/c)$

$ax + by = 0$ → $(a∞, b∞)$

### Three choices

## Exploring OEIS

### Sequences that agree for a while

- $a(n) = round(n*\phi^{18})$, where $\phi$ is the golden ratio,  https://oeis.org/A004953
- $a(n) = ceiling(n*\phi^{18})$, where $\phi$ is the golden ratio, https://oeis.org/A004973

## Collatz Conjecture

$f(n) = \left\{\begin{array}{c l}n/2 & \textrm{if even}\\3n+1 & \textrm{if odd}\end{array}\right\}$

 1
 2 → 1
3 → 10 → 5 → 16 → 8 → 4 → 2 → 1
6 → 3
 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10
9 → 28 → 14 → 7
12 → 6
15 → 46 → 23 → 70 → 35 → 106 → 53 → 160 → 80 → 40
18 → 9
 19 → 58 → 29 → 88 → 44 → 22
21 → 64 → 32 → 16
24 → 12
 25 → 76 → 38 → 19
27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → 187 → 562 → 281 → 844 → 422 → 211 → 634 → 367 → 1102 → 551 → 1654 → 827 → 2482 → 1241 → 3724 → 1862 → 931 → 2794 → 1397 → 4192 → 2096 → 1048 → 524 → 262 → 131 → 394 → 197 → 592 → 296 → 148 → 74 → 37 → 112 → 56 → 28 → 14
30 → 15
33 → 100 → 50 → 25
36 → 18
39 → 118 → 59 → 178 → 89 → 268 → 134 → 67 → 202 → 101 → 304 → 152 → 76 → 38
42 → 21
 43 → 130 → 65 → 196 → 98 → 49 → 148 → 74 → 37
45 → 136 → 68 → 34
48 → 24
51 → 154 → 77 → 232 → 166 → 84 → 42

83 → 250 → 125 → 376 → 188 → 94

## 2011 Functions 3

Let $f(x)$ be a function such that $f(x) + f({1\over 1 − x}) = x$ for all x not equal to 0 or 1. Find the exact value of $f(2)$.

$f(2) + f(-1) = 2 \implies f(2) = 2 - (-1 - (1/2 - f(2))) = 7/2-f(2)$
$f(-1) + f(1/2) = -1 \implies f(-1) = -1 - (1/2 - f(2)) = -3/2+f(2)$
$f(1/2) + f(2) = 1/2 \implies f(1/2) = 1/2 - f(2)$

$f(2) = 7/2 - f(2) = 7/4$

### Cycles of $1\over1-x$:

a | b   | c   | d
--|-----|-----|---|
1 | ∞   | 0   | 1
2 | -1  | 1/2 | 2
3 | -1/2| 2/3 | 3
4 | -1/3| 3/4 | 4
-2| 1/3 | 3/2 | -2
-3| 1/4 | 4/3 | -3
 x | $1\over 1-x$ | ${1 \over 1-{1\over 1-x}} = {1 \over {1-x \over 1-x} -{1\over 1-x}} = {1 \over {-x\over 1-x}} = {x-1\over x}$ | ${1\over 1- {x-1\over x}} = {1\over {x-(x-1)\over x}} = {x\over x-(x-1)} = x$

## 2007 Functions 3

Let $f_n$ be a positive integer, such that $f_n=f_{n-1}(f_{n-1}-2)+2$ where $f_0 = 3$. Find $f_4$.

265?



| $n$ | $f_n$ | |
| --- | ----- |-|
| 0   | 3     |  2+1
| 1   | 5     |  4+1
| 2   | 17    | 16+1
| 3   | 257   |256+1
| 4   | 65537 | $2^{16}+1 = 1024 * 64 + 1 = 64000 + 3*512 = 65024+512 = 65536$

2048, 4096, 8192, 16384, 32768, 65536

$f_n = 2^{2^n}+1$
$g_n = f_n - 1 = 2^{2^n} = g_{n-1}g_{n-1}$ where $g_0 = 2$

-1: $x(x-2) = 1 \implies x^2-2x-1 = 0 \implies x = {2 ± \sqrt{4+4}\over 2} = 1±\sqrt2$ 

## Function Relations

We observed, for variables $x$ and $y$ and constants $a$, $b$, $c$:

$$f(x)+f(y) = f(x+y) \implies f(x) = a\cdot x$$
$$f(x)+f(y) = f(x*y) \implies f(x) = \log_b x$$
$$f(x)+f(y) = 2c \implies f(x) = c$$
$$f(x)+f(y) = f(x+y)+2c \implies f(x) = a\cdot x + c$$

## Sum of 1..n and difference of n² and sum of 1²…(n-1)².

$$f(x)=x^2-f(x-1) \textrm{ and } f(0)=0 \implies f(x)-x = f(x-1)$$

## 2000 polynomials 2

$p(x) = 2x^3-5x^2-9x+18 = 0$

$(2x-a)(x-b)(x-c) = 0$
$2x^3 - (2c+2b+a)x^2 + (2bc+ac+ab)x - abc = 0$
$2c+2b+a = 5\quad 2bc+ac+ab=-9\quad abc=-18$

$p(-2) = -16 - 20 + 18 + 18 = -36 + 36 = 0$
$p(0) = 18$
$p(1) = 6$
$p(1.5) = 6.75 - 11.25 - 13.5 + 18 = -4.5 - 13.5 + 18 = 0$
$p(2) = -4$
$p(3) = 54 - 45 - 27 + 18 = 9 - 9 = 0$
$p(6) = 432 - 180 - 54 + 18 = 252 - 36 = 216 > 0$

$(2x-3)(x-3)(x+2) = 0$

${p(x)\over dx} = 6x^2 - 10x - 9$

$$x = {10 ± \sqrt{100+216}\over12} = {5±\sqrt{25+54=79}\over6} \approx 2.16,-0.66$$

## 2000 polynomials 1

$6x^5-x^4-35x^3+40x^2-17x+33 \mod 3x^3-5x^2+2x-3$

$6x^5-5x^4+2x^3-3x^2 \implies 2x^2$

$4x^4-37x^3+43x^2-17x+33 \mod 3x^3-5x^2+2x-3$

$4x^4-{20\over3}x^3+{8\over3}x^2-4x\implies {4\over3}x$

$-{91\over3}x^3+{121\over3}x^2-13x+33 \mod 3x^3-5x^2+2x-3$

$-{91\over3}x^3+{455\over9}x^2-{182\over9}x+{91\over3} \implies -{91\over9}$

$-{92\over9}x^2+{65\over9}x+{8\over3}$

### Quadratic Formula

$$ ax^2+bx+c = 0$$

$$\implies x^2 + {b\over a}x = -{c\over a} $$

$$\implies x^2 + {b\over a}x + ({b\over 2a})^2 = ({b\over 2a})^2 - {c\over a} $$

$$\implies (x + {b\over 2a})^2 = {b^2 - 4ac\over 4a^2} $$

$$\implies x + {b\over 2a} = ±{\sqrt{b^2 - 4ac}\over 2a} $$

$$\implies x = {-b ± \sqrt{b^2-4ac} \over 2a} $$


$$ x^2 + 5x + 6 = 0 \implies x \in \{-2,-3\} \implies (x+2)(x+3) = 0$$

### Cubic Formula *

$$ ax^3+bx^2+cx+d = 0 $$

$$ \implies x^3+{b\over a}x^2+{c\over a}x = -{d\over a} $$

$$ (x+y)^3 = x^3+3yx^2+3y^2x+y^3 $$

$$ \implies x^3+{b\over a}x^2+{c\over a}x + (3{b\over 3a}^2-{c \over a})x =
(3{b\over 3a}^2-{c \over a})x - {d\over a} $$


---

|  +  |  1  |  0  |  11  |  10  |
|:---:|:---:|:---:|:----:|:----:|
|  1  | 110 |  1  |  0   |  11  |
|  0  |  1  |  0  |  11  |  10  |
| 11  |  0  | 11  |  10  | 1101 |
| 10  | 11  | 10  | 1101 | 1100 |


```rust=
  11
  01

 110
  10

 100
1100

```

----

| +    | 0    | 1      | !   |
|:----:|:----:|:------:|:------:|
| 0    | 0    | 1      | !   |
| 1    | 1   | 1!   | 0   |
| ! | ! | 0  | !1 |

```rust
0!1  0-3+1
!01 -9+0+1
 
!1! -9+3-1
0!0   -3

!0! -9+0-1


0!1
```

```
 1!1
 !01
  101
 0001
 
 1+(!+!) ~ (1+!)+!
 (1+1)+! ~ 1+(1+!)



```

----
```
 01.0
+01.0
----- 
 10.1
 
  010.10
  010.10
  101.00
    1.01
  110.11
  
  11
  11
 111.010101
 
 .0101010101010101
 .0101010101010101
 -----------------
 .10101010101010101
 .00101010101010101
 -----------------
0.11010101010101010
0.01010101010101010
 -------------------
0.10101010101010101
0.10101010101010101

0.(0101) 
0.0101
0.10101
0.00100
-------
0.(11011)

   1.01011
   1.01000
  10.10111
  10.10000
 101.01111
 101.00000
 1010.11111
 1010.00000
10101.11111
10100.00000

   101
   101
  1010.1
  0010.0
  
---
 1
```

