Adds initial explanation

Man I forgot all about this, luckily I found my notebook with my initial
proof!

Author: romellem

2019/16/input.js

@@ -1,4 +1,10 @@
-const input = `59766977873078199970107568349014384917072096886862753001181795467415574411535593439580118271423936468093569795214812464528265609129756216554981001419093454383882560114421882354033176205096303121974045739484366182044891267778931831792562035297585485658843180220796069147506364472390622739583789825303426921751073753670825259141712329027078263584903642919122991531729298497467435911779410970734568708255590755424253797639255236759229935298472380039602200033415155467240682533288468148414065641667678718893872482168857631352275667414965503393341925955626006552556064728352731985387163635634298416016700583512112158756656289482437803808487304460165855189`
+const path = require('path');
+const fs = require('fs');
+
+const input = fs
+	.readFileSync(path.join(__dirname, 'input.txt'), 'utf8')
+	.toString()
+	.trim()
 	.split('')
 	.map(v => +v);
 

2019/16/part-two.js

@@ -9,3 +9,81 @@ for (let j = 0; j < 100; j++) {
 }
 
 console.log('8 digits:', slice_to_iterate.slice(0, 8).join(''));
+
+
+/*
+
+Here's the trick:
+
+Let's say your 12 digits number is `123456789123`.
+Look at the table after you get past the "halfway point" down the list:
+
+|  # |   1  |   2  |   3  |   4  |   5  |   6  |   7  |   8  |   9  |   1  |   2  |   3  |
+|----|------|------|------|------|------|------|------|------|------|------|------|------|
+|  1 |   1  |      |  -3  |      |   5  |      |  -7  |      |   9  |      |  -2  |      |
+|  2 |      |   2  |   3  |      |      |  -6  |  -7  |      |      |   1  |   2  |      |
+|  3 |      |      |   3  |   4  |   5  |      |      |      |  -9  |  -1  |  -2  |      |
+|  4 |      |      |      |   4  |   5  |   6  |   7  |      |      |      |      |  -3  |
+|  5 |      |      |      |      |   5  |   6  |   7  |   8  |   9  |      |      |      |
+|  6 |      |      |      |      |      |   6  |   7  |   8  |   9  |   1  |   2  |      |
+|  7 |      |      |      |      |      |      |  *7* |  *8* |  *9* |  *1* |  *2* |  *3* |
+|  8 |      |      |      |      |      |      |      |  *8* |  *9* |  *1* |  *2* |  *3* |
+|  9 |      |      |      |      |      |      |      |      |  *9* |  *1* |  *2* |  *3* |
+| 10 |      |      |      |      |      |      |      |      |      |  *1* |  *2* |  *3* |
+| 11 |      |      |      |      |      |      |      |      |      |      |  *2* |  *3* |
+| 12 |      |      |      |      |      |      |      |      |      |      |      |  *3* |
+
+Starting from row 7 on down, all the numbers are added together! No 0s, no -1s, just the numbers
+added.
+
+Since our "offset" is more than halfway through our full list, we can just immediately jump to the offset
+and compute those numbers on down.
+
+Now adding up all those numbers in the triangle is still a _lot._ But there is another trick!
+
+Look at the very last number, 3. let's call it A.
+
+Go up one row, we have 2 + (3). But wait, 3 is A. So we have 2 + A. Let's call that B.
+
+Go up another row, we have 1 + 2 + (3) -> 1 + (2 + A) -> 1 + B. Call that C.
+
+You see where this is going. Go up another row, 9 + 1 + 2 + (3),
+9 + 1 + (2 + A) -> 9 + (1 + B) -> 9 + C. Call that D.
+
+And so on.
+
+The other trick is, we only need to store the last digit each time we start from the bottom
+and loop up.
+
+Consider
+
+n + ...           = x_n      n + x_n-1 = x_n
+    d + c + b + a = x_4      d + x_3   = x_4
+        c + b + a = x_3  =>  c + x_2   = x_3
+            b + a = x_2      b + x_1   = x_2
+                a = x_1      a         = x_1
+
+Let x_n = D_i..D_1D_0 , where D_n is a single digit.
+(x_n)mod 10 = D_0
+Let `l + x_n` be the next line up we want to add. So l is a single digit number.
+We only care about the last digit of all that addition, so I want to show that
+(l + x_n)mod 10 = (l + D_0)mod 10.
+From the "Addition Property" of modular arithmetic, we know that
+(a + b)mod_n = [(a)mod_n + (b)mod_n]mod_n
+
+So, expanding (l + x_n)mod 10 is
+
+(l + x_n)mod_10 = ((l)mod_10 + (x_n)mod_10)mod_10.
+
+Well (l)mod_10 is just l, since l is less than 10.
+
+                = ((l)       + (x_n)mod_10)mod_10.
+
+Add (x_n)mod_10 = D_0
+
+                = (l         + D0         )mod_10.
+
+QED.
+
+
+*/